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Possibility Semantics
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Author
Holliday, Wesley Halcrow
Publication Date
2021-10-06
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University of California
Possibility Semantics
Wesley H. Holliday
University of California, Berkeley
wesholliday@berkeley.edu
Previous version in Selected Topics from Contemporary Logics, ed. Melvin
Fitting, Volume 2 of Landscapes in Logic, College Publications, London,
2021, ISBN 97-1-84890-350-0, pp. 363-476. This version corrects § 4.3.
Abstract. In traditional semantics for classical logic and its extensions,
such as modal logic, propositions are interpreted as subsets of a set, as in
discrete duality, or as clopen sets of a Stone space, as in topological duality.
A point in such a set can be viewed as a “possible world,” with the key property of a world being primeness—a world makes a disjunction true only if it
makes one of the disjuncts true—which classically implies totality—for each
proposition, a world either makes the proposition true or makes its negation
true. This chapter surveys a more general approach to logical semantics,
known as possibility semantics, which replaces possible worlds with possibly
partial “possibilities.” In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as
compact regular open sets of an upper Vietoris space, as in the recent theory
of “choice-free Stone duality.” The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling
which disjunct is true. We explain how possibilities may be used in semantics
for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for
traditional semantics, to avoid the nonconstructivity of traditional semantics,
and to provide richer structures for the interpretation of new languages.
Keywords: first-order logic, modal logic, provability logic, intuitionistic logic, inquisitive logic, Boolean algebra, regular open algebra,
canonical extension, MacNeille completion, Heyting algebra, Stone duality, possible world semantics, Kripke frame, axiom of choice, forcing
AMS classification (2010): 03B10, 03B20, 03B45, 03B55, 06D20,
06D22, 03F45, 03G05, 06E25
College Publications, London 2024
Wesley H. Holliday
Contents
1 Introduction
2 Philosophical explanation
2.1 Partiality . . . . . . . .
2.2 Refinement . . . . . . .
2.3 Truth and falsity . . . .
2.4 Regular sets . . . . . . .
2.5 Connectives . . . . . . .
2.6 Summary . . . . . . . .
3
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3 Boolean case
10
3.1 Posets of possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Posets and complete Boolean algebras . . . . . . . . . . . . . . . . 11
3.3 Possibility frames and Boolean algebras . . . . . . . . . . . . . . . 18
3.3.1 Constructing possibilities from nonzero propositions . . . . 19
3.3.2 Constructing possibilities from proper filters of propositions 21
3.4 Spaces of possibilities and Boolean algebras . . . . . . . . . . . . . 24
3.4.1 Topological possibility frames . . . . . . . . . . . . . . . . . 25
3.4.2 UV-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 First-order case
4.1 First-order possibility models . . . . . . . . . . . . . . . . . . . . .
4.2 Soundness of first-order logic with respect to possibility semantics
4.3 Completeness of first-order logic with respect to possibility semantics
4.4 First-order possibility models and forcing . . . . . . . . . . . . . .
29
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36
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42
5 Modal case
5.1 Universal modality and propositional quantification . . .
5.2 Neighborhood frames . . . . . . . . . . . . . . . . . . . .
5.2.1 Basic frames . . . . . . . . . . . . . . . . . . . .
5.2.2 Tree completeness . . . . . . . . . . . . . . . . .
5.2.3 General frames . . . . . . . . . . . . . . . . . . .
5.3 Relational frames . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Accessibility relations . . . . . . . . . . . . . . .
5.3.2 Relational possibility models and Kripke models
5.3.3 From V-BAOs to full and principal frames . . . .
5.3.4 Dual equivalence with complete V-BAOs . . . . .
5.3.5 Quasi-normal possibility frames . . . . . . . . . .
44
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Possibility Semantics
5.4
5.5
5.3.6 Full frames and world incompleteness in provability logic
5.3.7 Principal frames and V-incompleteness in provability logic
5.3.8 General frames . . . . . . . . . . . . . . . . . . . . . . . .
Functional frames . . . . . . . . . . . . . . . . . . . . . . . . . . .
First-order modal logic . . . . . . . . . . . . . . . . . . . . . . . .
6 Further connections and directions
6.1 Interval semantics . . . . . . . . .
6.2 A bimodal perspective . . . . . . .
6.3 Intuitionistic case . . . . . . . . . .
6.4 Inquisitive case . . . . . . . . . . .
6.5 Further language extensions . . . .
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76
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. 94
. 96
. 96
. 101
. 102
7 Conclusion
103
A Appendix
104
1
Introduction
Traditional semantics for classical logic and its extensions, such as modal logic,
can be seen as based on two fundamental relationships:
1. Tarski’s [163] discrete duality between complete and atomic Boolean algebras (CABAs) and sets;
2. Stone’s [160] topological duality between Boolean algebras (BAs) and Stone
spaces.
In the traditional approach, propositions are interpreted as subsets of a set, as in 1,
or as clopen sets of a Stone space, as in 2. Negation, conjunction, and disjunction
of propositions are interpreted as set-theoretic complementation, intersection, and
union, respectively. As a consequence, the points in such sets can be viewed as
“possible worlds,” with the key property of a possible world being:
• primeness: a world makes a disjunction true only if it makes one of the
disjuncts true.
This follows from the interpretation of disjunction as union. Given the interpretation of negation as complementation, classical worlds also have the property of:
• totality: for each proposition, a world either makes the proposition true or
makes its negation true.
Wesley H. Holliday
If we start with an algebra of propositions, then such worlds can be recovered as
atoms—if the algebra of propositions is a CABA—or prime filters—if the algebra
of propositions is an arbitrary BA and we avail ourselves of the Axiom of Choice.1
In this chapter, we survey a more general approach to logical semantics, known
as possibility semantics, which replaces possible worlds with possibly partial “possibilities.” The classical version of possibility semantics is based on the following
two relationships:
1′ . the representation of complete Boolean algebras (CBAs) using posets, exploited in set-theoretic forcing [162];
2′ . the duality between BAs and upper Vietoris spaces (UV-spaces), developed
in the recent theory of “choice-free Stone duality” [27].
In this approach, propositions are interpreted as regular open sets of a poset,
as in 1′ , or as compact regular open sets of a UV-space, as in 2′ (all of these
notions will be defined in what follows). The partial order ⊑ in the poset of
possibilities represents a relationship of refinement between partial possibilities:
roughly speaking, x ⊑ y means that x contains all the information that y does and
possibly more. Instead of interpreting negation and disjunction as set-theoretic
complementation and union, they are interpreted using the refinement relation ⊑:
• a possibility x makes the negation of a proposition true just in case no
refinement x′ ⊑ x makes the proposition true;
• a possibility x makes the disjunction of two propositions true just in case
for every refinement x′ ⊑ x there is a further refinement x′′ ⊑ x′ that makes
one of the disjuncts true.
As a result, a possibility may be neither total nor prime; e.g., a possibility may
settle the proposition “p or not p” as true and yet not settle which disjunct is
true, leaving this to be settled by refinements of the possibility.
By endowing posets of possibilities with additional structure, one can give possibility semantics for first-order logic, as done by Fine [61] and van Benthem [10],
and modal logics, as done by Humberstone [104], who coined the term ‘possibility
semantics’. As we will see, for modal logic the move from worlds to possibilities
allows us to overcome some incompleteness results for traditional semantics.
There is also a natural generalization of possibility semantics for intuitionistic
logic. Possible world semantics for intuitionistic logic [118] is based on:
1
The Prime Filter Theorem needed for Stone duality is weaker than the Axiom of Choice
[85], but still beyond ZF set theory [59] (as well as ZF with Dependent Choice [145]).
Possibility Semantics
3. the representation of certain complete Heyting algebras—namely those that
are generated by completely join-prime elements—using posets, generalizing
Birkhoff’s [28] representation of finite distributive lattices.2
In this traditional approach, propositions are interpreted as downward-closed subsets (downsets) of the poset.3 Intuitionistic implication and negation are interpreted using the relation ⊑ of the poset just as in possibility semantics. As a
consequence, the elements of the poset need not have the property of totality.
However, disjunction is still interpreted as union, so the elements still have the
primeness property of worlds, explaining our use of the term “possible world semantics for intuitionistic logic.” By contrast, possibility semantics for intuitionistic
logic is based on the following (for the definition of a nucleus, see § 6.3):
3′ . the representation of arbitrary complete Heyting algebras using posets
equipped with a nucleus on the algebra of downsets, in the manner of
Dragalin [51].
The nucleus can be realized concretely in various ways (the most concrete way uses
a distinguished subrelation of the partial order), as we will review later. In this
approach, propositions are interpreted as fixpoints of the nucleus. The disjunction
of two propositions is interpreted by applying the nucleus to the union of the
interpretations of the disjuncts. Since the result may strictly extend the union, a
possibility may settle a disjunction as true without settling which disjunct is true,
so the primeness property need not hold. Possibility semantics for classical logic is
the special case of this approach using the nucleus of double negation (see § 6.3).
Possibility semantics for classical logic may also be seen as a special case of
possibility semantics for orthologic, the propositional logic of ortholattices. This
is based on the representation of ortholattices using symmetric and reflexive compatibility relations (or symmetric and irreflexive orthogonality relations) on a set
[79, 80, 99, 132]. Propositions are fixpoints of a closure operator defined using
the compatibility relation. The disjunction of two propositions is interpreted by
applying the closure operator to the union of the interpretations of the disjuncts.
Classical possibility semantics is a special case where additional properties are
assumed to hold of compatibility [99, Example 3.16]. Rather than covering possibility semantics for orthologic here, we refer the reader to [102] for an introduction
with applications to epistemic modals and conditionals.
2
In the case of topological duality, there is Esakia’s [55, 56] duality between Heyting algebras
and Esakia spaces. For a possibility-based topological duality for Heyting algebras, several ideas
have been proposed, but the advantages of different proposals are not yet clear.
3
It is more common in the intuitionistic literature to think in terms of upward-closed subsets
(upsets) (for an exception, see [51]), but in line with the predominant practice in set-theoretic
forcing, we think in terms of downsets in this chapter.
Wesley H. Holliday
There are at least three benefits in the move from worlds to possibilities:
1. In the setting of discrete duality, by lifting the restriction to atomic CBAs,
possibility semantics allows us to characterize logics that cannot be characterized using possible world semantics based on CABAs. Examples are
given in §§ 5.1, 5.2.1, 5.3.6, and 5.5.
2. In the setting of topological duality, while world-based topological dualities
require nonconstructive choice principles, possibility-based topological dualities can be carried out in the setting of quasi-constructive mathematics,
defined as “mathematics that permits conventional rules of reasoning plus
ZF + DC, but no stronger forms of Choice” [151, § 14.76].4
3. Even when atomicity or nonconstructivity are not concerns (e.g., because
finite algebras are sufficient for one’s purposes), the extra refinement structure in possibility semantics makes possible the interpretation of extended
languages that cannot be handled in a purely world-based semantics. An
example is given in § 6.4 with the language of inquisitive logic.
The rest of the chapter is organized as follows. In § 2, our philosophical
preamble, we derive the essential elements of (classical) possibility semantics from
a few basic axioms about truth and falsity. Successive subsections then explain
possibility semantics for different algebras/logics: Boolean algebra/logic (§ 3),
first-order logic (§ 4), modal algebra/logic (§ 5), Heyting algebra/intuitionistic
logic (§ 6.3), and inquisitive logic (§ 6.4). §§ 6.1-6.2 provide perspectives on
possibility semantics from the point of view of interval semantics and traditional
possible world semantics. In addition to surveying the basic theory and existing
results for possibility semantics, we highlight a number of open problems.
How to read this chapter
This chapter is designed for the reader to dip in and out of sections based on the
reader’s interest. The main prerequisites for later sections are §§ 3.1, 3.2, and 3.3.
Readers mainly interested in possibility semantics for propositional modal logic
can skip § 4; readers mainly interesting in possibility semantics for intuitionistic
propositional logic can skip §§ 4-5, etc. Results from previous sections are cited
explicitly rather than assumed. Theorems are numbered by subsection. Thus,
e.g., Theorem 3.1.1 is the first theorem in § 3.1. If Theorem 3.1.1 has multiple
parts, we refer to Theorem 3.1.1.1 for the first part, 3.1.1.2 for the second, etc.
4
In fact, ZF suffices for the dualities, though Dependent Choice may be useful in applications.
Possibility Semantics
2
Philosophical explanation
In this section, we explain the core ideas of classical possibility semantics philosophically. This section is not meant to give set-theoretic mathematical definitions
and results, so we do not use definition or theorem environments. But we do give
“proofs” from axioms using classical logic, which could be fully formalized.
For further philosophical motivation, see [104, 106], [105, § 6.44], and [74, 84,
148]. A comparison of possibility semantics to other semantics based on partial
states, such as situation semantics or truthmaker semantics, is beyond the scope
of this chapter; we recommend the surveys in [114] and [63] to interested readers.
2.1
Partiality
Unlike a possible world, a possibility may be partial, in two related respects:
1. It may fail to settle the truth or falsity of a proposition (non-totality).
2. It may settle the truth or falsity of a proposition without settling exactly
how the proposition is made true or false (non-primeness); e.g., it may settle
that the particle is spin up or spin down but not settle that the particle is
spin up and not settle that the particle is spin down.5
2.2
Refinement
Some possibilities are less partial than others, inducing a relation of refinement:
(A0) A possibility x is a refinement of a possibility y (notation: x ⊑ y) iff:
• x settles as true every proposition that y settles as true, and
• x settles as false every proposition that y settles as false.
The relation ⊑ is therefore reflexive and transitive. If we do not distinguish
possibilities that settle as true/false exactly the same propositions, then we may
also assume antisymmetry for ⊑. Then ⊑ is a partial order.
2.3
Truth and falsity
In possibility semantics, the notions of a possibility x settling a proposition P as
true or as false are related as stated in the following axioms:
(A1) x settles P as true iff no possibility refining x settles P as false;
5
Hale [84] calls these types of partiality global incompleteness and local incompleteness.
Wesley H. Holliday
(A2) x settles P as false iff no possibility refining x settles P as true.
If one thinks of settling true as a kind of necessitation, then one can see (A1) and
(A2) as related to Aristotle’s idea of the duality of necessity and possibility [128].
Using (A1) and (A2), we may argue that the ⊑ relation is separative:
• if x is not a refinement of y (x 6⊑ y), then there is a refinement x′ of x that is
incompatible with y, in the sense that x′ and y have no common refinement.
Proof. If x is not a refinement of y, then by (A0), there is a proposition P that (i)
y settles as true but x does not or (ii) y settles as false but x does not. In case (i),
it follows by (A1) that there is a refinement x′ of x that settles P as false, which
by (A2) implies that no refinement x′′ of x′ settles P as true. It follows that no
refinement x′′ of x′ is a refinement of y, because y settles P as true and hence
any refinement of y settles P as true by (A0). Thus, x′ is incompatible with y.
In case (ii), the argument is analogous, flipping the roles of (A1) and (A2) and of
truth and falsity.
2.4
Regular sets
From (A1) and (A2), we have as a consequence:
(C1) x settles P as true iff for every possibility x′ refining x, there is a possibility
x′′ refining x′ that settles P as true.
Proof. From left to right, suppose x settles P as true and x′ is a possibility
refining x. Then by (A1), x′ does not settle P as false. Thus, by (A2), there is a
possibility x′′ refining x′ that settles P as true. From right to left, we prove the
contrapositive. If x does not settle P as true, then by (A1), there is a possibility
x′ refining x that settles P as false. Hence, by (A2), no possibility x′′ refining x′
settles P as true.
We say that a set X of possibilities is regular if:
• x ∈ X iff for every possibility x′ refining x, there is a possibility x′′ refining
x′ such that x′′ ∈ X.
We say that a set X of possibilities corresponds to a proposition P if X contains
exactly the possibilities that settle P as true. Thus, as an immediate consequence
of (C1), we have:
(C2) If a set X of possibilities corresponds to a proposition, then X is regular.
Possibility Semantics
2.5
Connectives
Three final axioms relate truth and falsity for propositions formed using logical
connectives:
(A3) x settles the negation of P as true iff x settles P as false;
(A4) x settles the conjunction of P and Q as true iff x settles P as true and x
settles Q as true;
(A5) x settles the disjunction of P and Q as false iff x settles P as false and x
settles Q as false.
As a consequence of (A3) and (A0), assuming that every proposition has a
negation, we can characterize refinement purely in terms of truth:
(C3) A possibility x is a refinement of a possibility y iff x settles as true every
proposition that y settles as true.
As a consequence of (A3) and (A2), we can give truth conditions for negation
purely in terms of truth:
(C4) x settles the negation of P as true iff no possibility refining x settles P as
true.
Finally, as a consequence of (A5), (A1), and (A2), we can give truth conditions
for disjunction purely in terms of truth:
(C5) x settles the disjunction of P and Q as true iff for every possibility x′
refining x, there is a possibility x′′ refining x′ such that x′′ settles P as true
or x′′ settles Q as true.
Proof. From left to right, suppose x settles the disjunction as true and x′ is a
possibility refining x. Then by (A1), x′ does not settle the disjunction as false.
Thus, by (A5), either x′ does not settle P as false, in which case by (A2) there is
a possibility x′′ refining x′ that settles P as true, or x′ does not settle Q as false,
in which case by (A2) there is a possibility x′′ refining x′ that settles Q as true.
In either case, there is a possibility x′′ refining x′ that settles P as true or settles
Q as true. From right to left, we prove the contrapositive. Suppose x does not
settle the disjunction as true, so by (A1), there is a possibility x′ refining x that
settles the disjunction as false. It follows by (A5) that x′ settles P as false and
settles Q as false, which by (A2) implies that no possibility x′′ refining x′ settles
P as true or settles Q as true.
Wesley H. Holliday
2.6
Summary
From (A0)-(A5), we have derived the essential elements of classical possibility
semantics, which we will see in this chapter:
• there is a partially ordered set of possibilities;
• propositions correspond to regular sets of possibilities;
• truth conditions for negation and disjunction involve quantification over
refinements as in (C3) and (C4).
3
Boolean case
In both classical possible world semantics and classical possibility semantics,
propositions form a Boolean algebra (BA) under the operations of negation, conjunction, and disjunction. The key difference between these semantics is in how
such BAs of propositions are represented. In this section, we review the ways of
representing BAs used in possibility semantics. After some preliminaries concerning partially ordered sets in § 3.1, we review a well-known way of representing
complete BAs in § 3.2: we represent a complete BA as the BA of regular open sets
of a partially ordered set. The way of representing arbitrary BAs in §§ 3.3-3.4 is
the basis for the recent theory of “choice-free Stone duality” [27]: we represent an
arbitrary BA as the BA of compact regular open sets of an appropriate topological space. It is a short step from these representations to semantics for formal
languages: we simply interpret formulas as propositions in the BAs represented
using our partially ordered sets or topological spaces. Thus, the algebraic and
topological material in this section is at the core of possibility semantics.
3.1
Posets of possibilities
A partially ordered set (or poset) is a pair (S, ⊑) of a nonempty set S and a partial
order ⊑ on S. As in § 2, we think of elements of S as possibilities, and for x, y ∈ S,
we take x ⊑ y to mean that x refines y. Define x ⊏ y to mean x ⊑ y and x 6= y.
A set U ⊆ S is a downward-closed subset or downset of (S, ⊑) if U is closed
under refinement: for all x ∈ U and x′ ⊑ x, we have x′ ∈ U . For x ∈ S, the
principal downset of x is the set of all possibilities refining x:
↓x = {y ∈ S | y ⊑ x}.
We are especially interested in posets in which every possibility can be properly
refined (so there are no worlds in the sense of Definition 3.2.8): for all x ∈ S, there
Possibility Semantics
is a y ∈ S such that y ⊏ x. Two running examples will be the full infinite binary
tree and the collection of open intervals of Q ordered by inclusion.
Example 3.1.1. Consider the full infinite binary tree 2<ω :
ǫ
.
..
.
.
.
..
.
.
..
11
..
..
10
.
.
01
..
..
00
1
..
0
We may regard 2<ω as a poset (S, ⊑) where S is the set of all finite sequences of
0’s and 1’s and for any such sequences σ and τ , we have σ ⊑ τ iff τ is an initial
segment of σ. Thus, in the diagram above (and later diagrams), an arrow from
τ to σ indicates σ ⊑ τ . Viewing this as a poset of possibilities, each possibility
may be taken to settle a finite sequence of “yes or no” questions. If σ ⊑ τ , then σ
answers all the questions that τ does in the same way as τ does but may answer
additional questions on which τ is silent. No possibility settles all questions.
Example 3.1.2. Consider the set S = {(a, b) | a, b ∈ Q, a < b} of all nonempty
open intervals (a, b) = {x ∈ Q | a < x < b} of rational numbers. Let ⊑ be the
inclusion order: (a, b) ⊑ (c, d) if (a, b) ⊆ (c, d). Thus, we obtain infinite sequences
of refinements from infinite chains of shrinking intervals:
(
)
(
)
(
)
)
...
(
Adopting the temporal interpretation of the poset (S, ⊑) in [11, p. 59], we may
think of a possibility as settling that we are now temporally located in some
stretch (or “period” or “region”) of time. There is no possibility of a sharpest
localization, i.e., no temporal “instants.”
3.2
Posets and complete Boolean algebras
In this section, we review how a poset of possibilities gives rise to a (complete)
Boolean algebra of propositions.
Wesley H. Holliday
Given a poset (S, ⊑), say that a subset U ⊆ S is regular open in (S, ⊑) if
U = {x ∈ S | ∀x′ ⊑ x ∃x′′ ⊑ x′ : x′′ ∈ U }.
(1)
This is the same notion of regularity introduced in § 2.4. Equivalently, U ⊆ S is
regular open in (S, ⊑) if U satisfies:
• persistence: if x ∈ U and x′ ⊑ x, then x′ ∈ U ;
• refinability: if x 6∈ U , then ∃x′ ⊑ x ∀x′′ ⊑ x′ x′′ 6∈ U .
Persistence is just the condition of being a downset from § 3.1. If we think of U
as a proposition, then persistence is built into our notion of refinement from § 2.2.
As for refinability, the philosophical explanation comes from § 2.3: if x does not
make U true, then there must be a refinement x′ of x that makes U false.
Example 3.2.1. In the infinite binary tree from Example 3.1.1, every principal
downset is regular open: persistence is immediate, and for refinability, if y 6∈ ↓x,
then there is a child y ′ ⊑ y such that for all y ′′ ⊑ y ′ , y ′′ 6∈ ↓x. The union of ↓x
and ↓x′ is also regular open provided x and x′ are not children of the same node.
By contrast, consider ↓00 ∪ ↓01, represented by the area under the dashed curve:
ǫ
.
..
.
.
..
..
.
.
..
11
..
..
10
.
.
01
..
..
00
1
.
0
The set ↓00 ∪ ↓01 is not regular open: for 0 6∈ ↓00 ∪ ↓01, yet there is no y ⊑ 0 such
that for all z ⊑ y, z 6∈ ↓00 ∪ ↓01, so refinability fails for ↓00 ∪ ↓01. Another set
that is not regular open is U = {σ ∈ S | σ contains at least one 1}, represented
by the filled-in black nodes in the diagram above: for ǫ 6∈ U , yet there is no y ⊑ ǫ
such that for all z ⊑ y, z 6∈ U , so refinability fails for U .
Example 3.2.2. In the poset of intervals from Example 3.1.2, every principal
downset is regular open. Moreover, if (ai , bi ) and (aj , bj ) are such that bi < aj ,
then U = ↓(ai , bi ) ∪ ↓(aj , bj ) is regular open. For suppose (x, y) 6∈ U , so that
(x, y) 6⊆ (ai , bi ) and (x, y) 6⊆ (aj , bj ). We choose a subinterval (x′ , y ′ ) ⊆ (x, y) as
follows. If (x, y) ∩ ((ai , bi ) ∪ (aj , bj )) = ∅, let (x′ , y ′ ) = (x, y). Otherwise, suppose
(x, y) ∩ ((ai , bi ) ∪ (aj , bj )) 6= ∅. If (x, y) ∩ (ai , bi ) 6= ∅ and (x, y) ∩ (aj , bj ) 6= ∅, let
(x′ , y ′ ) = (bi , aj ). In a diagram (for the case where ai < x and y < bj ):
Possibility Semantics
y
x
ai
bi
aj
x′
y′
bj
If (x, y) ∩ (ai , bi ) = ∅, then let (x′ , y ′ ) = (x, aj ) if x < aj and (x′ , y ′ ) = (bj , y)
if aj ≤ x and bj < y; similarly, if (x, y) ∩ (aj , bj ) = ∅, then let (x′ , y ′ ) = (x, ai )
if x < ai and (x′ , y ′ ) = (bi , y) if ai ≤ x and bi < y. Then observe that for all
(x′′ , y ′′ ) ⊆ (x′ , y ′ ), we have (x′′ , y ′′ ) 6∈ U , so refinability holds for U .
By contrast, V = ↓(a, b) ∪ ↓(b, c) is not regular open: for (a, c) 6∈ V , yet for
every subinterval of (a, c), there is a further subinterval that is either a subinterval
of (a, b) or a subinterval of (b, c), so refinability fails for V .
We now recall how the regular open sets of a poset form a Boolean algebra of
propositions. The following result is widely used in forcing in set theory [162] and
can be considered the crucial starting point for possibility semantics.
Theorem 3.2.3.
1. For any poset (S, ⊑), the collection RO(S, ⊑) of regular open sets, ordered
by inclusion, forms a complete Boolean algebra with the following operations:
^
_
¬U
= {x ∈ S | ∀x′ ⊑ x x′ 6∈ U };
{Ui | i ∈ I} =
\
{Ui | i ∈ I};
{Ui | i ∈ I} = {x ∈ S | ∀x′ ⊑ x ∃x′′ ⊑ x′ : x′′ ∈
[
{Ui | i ∈ I}}.
Elements of RO(S, ⊑) are precisely those U ⊆ S such that U = ¬¬U .
2. Given any Boolean algebra (B, ≤), let B+ be the result of deleting the
bottom element of B, and let ≤+ be the restriction of ≤ to B+ . Then
(B, ≤) embeds into RO(B+ , ≤+ ) as a regular subalgebra;6 and if (B, ≤) is
complete, then it is isomorphic to RO(B+ , ≤+ ).
Sketch of 2. We claim that the map ϕ : B → RO(B+ , ≤+ ) given by ϕ(b) =
↓+ b := {b′ ∈ B+ | b′ ≤+ b} is a Boolean embedding such that for any family
{ai | i ∈ I} ⊆ B, if the join of {ai | i ∈ I} exists in (B, ≤), then
↓+
_
{ai | i ∈ I} =
_
{↓+ ai | i ∈ I}.
(2)
6
Recall that a regular subalgebra of a Boolean algebra B is a subalgebra B′ such that if a set
S of elements of B′ has a join a in B′ , then a is also the join of S in B. Contrast Theorem 3.2.3.2
with the fact that not every Boolean σ-algebra can be represented as a regular subalgebra of a
powerset algebra (see, e.g., [76, Ch. 25]).
Wesley H. Holliday
It is easy to see that ↓+ b ∈ RO(B+ , ≤+ ), ϕ is injective, and ↓+ ¬b = ¬↓+ b. As
W
S
for (2), let a = {ai | i ∈ I} and A = {↓+ ai | i ∈ I}. Suppose b ∈ B+ is
not in the left-hand side of (2), so b 6≤ a. Then b′ := b ∧ ¬a 6= 0, and for all
b′′ ≤+ b′ , we have b′′ 6∈ A. Thus, b is not in the right-hand side by the definition
of join in RO(B+ , ≤+ ). Now suppose b ∈ B+ is in the left-hand side, so b ≤ a,
and consider b′ ≤+ b. Suppose for contradiction that there is no b′′ ≤+ b′ with
W
b′′ ∈ A. It follows that b′ ∧ ai = 0 for each i ∈ I, so {b′ ∧ ai | i ∈ I} = 0. Then
by the join-infinite distributive law for Boolean algebras, b′ ∧ a = 0, contradicting
b′ ≤ b ≤ a and the fact that b′ ∈ B+ . Thus, we conclude there is a b′′ ≤+ b′ such
that b′′ ∈ A. Hence b is in the right-hand side.
W
Since for each U ∈ RO(B+ , ≤+ ), we have U = {↓+ a | a ∈ U }, if (B, ≤) is
complete, then (2) implies that ϕ is surjective and hence an isomorphism.
Thus, any complete Boolean algebra can be represented as the regular opens
of a poset. The poset (B+ , ≤+ ) is an example of a separative poset, as in § 2.3.
Definition 3.2.4. A poset (S, ⊑) is separative if every principal downset ↓x is
regular open in (S, ⊑); equivalently, for all x, y ∈ S, if y 6⊑ x, then there is a z ⊑ y
such that ↓z ∩ ↓x = ∅.
The posets in Examples 3.1.1 and 3.1.2 are also separative. By contrast, a linear
order with more than one element is an example of a non-separative poset.
Separative posets have the following good properties for our purposes.
Proposition 3.2.5.
1. For each poset, its regular open algebra is isomorphic to the regular open
algebra of a separative poset.
2. The restriction of a separative poset to a regular open subset is again a
separative poset.
3. If (S, ⊑) is a separative poset, then the map x 7→ ↓x is a dense orderembedding of the poset (S, ⊑) into the poset (RO(S, ⊑), ⊆).7
4. If (S, ⊑) is a separative poset and U, V ∈ RO(S, ⊑) are such that U ∨ V is
infinite, then U is infinite or V is infinite.
By a dense order-embedding of a poset (S, ⊑) into a poset (S ′ , ⊑′ ), we mean an orderembedding e of (S, ⊑) into (S ′ , ⊑′ ) such that e[S] is a dense subset of (S ′ , ⊑′ ), i.e., for every
x′ ∈ S ′ there is a y ′ ⊑′ x′ such that y ′ ∈ e[S].
7
Possibility Semantics
Proof. For part 1, given (S, ⊑), the poset (RO(S, ⊑)+ , ⊆+ ) obtained in Theorem
3.2.3.2 from the BA RO(S, ⊑) is such a separative poset. More directly, one can
take the quotient of (S, ⊑) by the equivalence relation ∼ defined by: x ∼ y iff (i)
∀x′ ⊑ x ∃x′′ ⊑ x′ : x′′ ⊑ y and (ii) ∀y ′ ⊑ y ∃y ′′ ⊑ y ′ : y ′′ ⊑ x.
Parts 2-3 are also easy to check.
For part 4,8 by part 2 we may assume without loss of generality that S = U ∨V .
Let x ∼ y iff ↓x ∩ U = ↓y ∩ U . If U is finite, then ∼ partitions the infinite set
S into finitely many cells, one of which must be infinite. Call it I, and define
f : I → ℘(V ) by f (x) = ↓x ∩ V . We claim that f is injective. For if x, y ∈ I and
y 6⊑ x, then by separativity, there is a z ∈ ↓y such that ↓z ∩ ↓x = ∅. It follows,
since ↓x ∩ U = ↓y ∩ U , that ↓z ∩ U = ∅, so z ∈ ¬U , which with z ∈ U ∨ V implies
z ∈ V . Thus, z ∈ f (y) but z 6∈ f (x), so f is injective. Then since I is infinite, it
follows that ℘(V ) is infinite and hence V is infinite
Part 2 of Theorem 3.2.3 can be restated in the form of Proposition 3.2.6 below.
Recall that the MacNeille completion of a Boolean algebra B is the unique (up to
isomorphism) complete Boolean algebra B ∗ such that there is a dense embedding
of B into B ∗ , i.e., a Boolean embedding of B into B ∗ such that every element of
B ∗ is a join of images of elements of B (see, e.g., [76, Ch. 25]).9
Proposition 3.2.6. For any Boolean algebra B, RO(B+ , ≤+ ) is (up to isomorphism) the MacNeille completion of B.
Proof. For A ⊆ B, let Au be the set of all upper bounds of A and Aℓ the set of
all lower bounds of A. Recall that the MacNeille completion of B (as a poset)
can be constructed as ({A | A ⊆ B, A = Auℓ }, ⊆). Note that this is isomorphic to
({A \ {0} | A ⊆ B, A = Auℓ }, ⊆). Now we claim that for any A ⊆ B,
Auℓ \ {0} = ¬¬A
(3)
where ¬¬A = {b ∈ B+ | ∀b′ ≤+ b ∃b′′ ≤+ b′ : b′′ ∈ A}. Since elements of
RO(B+ , ≤+ ) are precisely those A ⊆ B such that A = ¬¬A, equation (3) implies
that ({A \ {0} | A ⊆ B, A = Auℓ }, ⊆) = (RO(B+ , ≤+ ), ⊆). To prove (3), let b be
a nonzero element of B. If b is not in the left-hand side of (3), then there is some
a ∈ Au such that b 6≤ a, which implies b ∧ ¬a 6= 0. Since a ∈ Au , for every c ∈ A,
c ≤ a and hence c ∧ ¬a = 0. Thus, setting b′ = b ∧ ¬a, we have b′ ≤+ b and for
all b′′ ≤+ b′ , b′′ 6∈ A. Hence b is not in the right-hand side. Conversely, suppose
b is not in the right-hand side of (3), so there is some b′ ≤+ b such that for all
8
The proof is nearly the same as in [27, Lemma 2.3] for the case of V = ¬U .
Compare the realization of B ∗ as in Proposition 3.2.6 with its realization as the regular
open algebra of the Stone space of B (see Remark 3.4.12).
9
Wesley H. Holliday
b′′ ≤+ b′ , b′′ ∈
6 A. It follows that for every c ∈ A, b′ ∧ c = 0, so c ≤ ¬b′ . Thus,
′
u
¬b ∈ A , so b′ 6∈ Auℓ and then b 6∈ Auℓ . Hence b is not in the left-hand side.
The reason for calling sets satisfying (1) “regular open” is the following. In a
topological space, a regular open set is an open set U such that
U = int(cl(U ))
where int and cl are the interior and closure operations, respectively. Any poset
(S, ⊑) may be regarded as a topological space (S, τ ), where the family τ of open
sets is the family Down(S, ⊑) of all downsets of (S, ⊑). Such spaces are T0 , i.e., for
any two distinct points, there is an open set containing one point but not the other,
as well as Alexandroff, i.e., the intersection of any family of opens is open. Indeed,
T0 Alexandroff spaces are in one-to-one correspondence with posets. Recall that
the specialization order of a space (S, τ ) is the binary relation on S defined by:
x 6 y iff for every U ∈ τ , x ∈ U implies y ∈ U . Given a T0 Alexandroff space
(S, τ ) with specialization order 6, whose converse is >, the poset (S, >) is such
that (S, Down(S, >)) = (S, τ ).10 For any poset (S, ⊑), the interior and closure
operations of the space (S, Down(S, ⊑)) are given by
int(U ) = {x ∈ S | ∀x′ ⊑ x x′ ∈ U }
cl(U ) = {x ∈ S | ∃x′ ⊑ x : x′ ∈ U },
so that
int(cl(U )) = {x ∈ S | ∀x′ ⊑ x ∃x′′ ⊑ x′ : x′′ ∈ U }.
Thus, a subset U of a poset is regular open in the topological space (S, Down(S, ⊑))
iff U is regular open as defined above Theorem 3.2.3.
Now Theorem 3.2.3.1 is a special case of the following result of Tarski.
Theorem 3.2.7 ([164, 165]). For any topological space X, the collection RO(X)
of regular open sets of X, ordered by inclusion, forms a complete Boolean algebra
with the following operations:
^
_
¬U
= int(X \ U );
\
{Ui | i ∈ I} = int(
{Ui | i ∈ I});
[
{Ui | i ∈ I} = int(cl(
{Ui | i ∈ I}).
10
This is a case where thinking in terms of the family of all upward-closed sets would be more
convenient, but Theorem 3.2.3.2 is a case where thinking in terms of downsets is more convenient,
as we do not have to flip the relation of the Boolean algebra. There is no winning in choosing
up or down.
Possibility Semantics
Theorem 3.2.7 can in turn be seen to follow from a more general theorem about
nuclei on Heyting algebras, explained in § 6.3 (Theorem 6.3.4).
Tarski [163] also observed that for any set W , its powerset ℘(W ) is a complete
and atomic Boolean algebra (CABA), and for any CABA B whose set of atoms is
At(B), ℘(At(B)) is isomorphic to B.11 Let us relate this representation of CABAs
to the representation of complete Boolean algebras (CBAs) in Theorem 3.2.3. For
this we introduce the following notion, inspired by the philosophers’ conception of
a “possible world” as maximally specific and hence having no proper refinements.
Definition 3.2.8. A world in a poset (S, ⊑) is an element x ∈ S such that for all
x′ ∈ S, if x′ ⊑ x then x′ = x.
Proposition 3.2.9. Let (S, ⊑) be a poset in which for every x ∈ S, there is a
world x′ ⊑ x. Then RO(S, ⊑) is isomorphic to the powerset algebra ℘(W ), where
W is the set of worlds of (S, ⊑).
Proof sketch. The isomorphism sends U ∈ RO(S, ⊑) to U ∩ W .
We conclude this section with a basic fact about complete BAs whose proof
shows (i) how the representation in Theorem 3.2.3 can be used to prove facts
about complete BAs and (ii) how every poset has the same regular open algebra
as the union of a poset of only worlds and a poset with no worlds.
Proposition 3.2.10. Any complete BA B is the product of a complete and
atomic BA and a complete and atomless BA.
Proof. Let (B+ , ≤+ ) be the poset given by Theorem 3.2.3.2. Where x <+ y iff
x ≤+ y and y 6≤+ x, let
B⋆ = {x ∈ B+ | ∀x′ <+ x ∃x′′ <+ x′ },
and let ≤⋆ be the restriction of ≤+ to B⋆ . Thus, we have deleted all possibilities
that are properly refined by worlds. Then it is easy to check the following:
1. The map U 7→ U ∩ B⋆ is an isomorphism from RO(B+ , ≤+ ) to RO(B⋆ , ≤⋆ ).
2. B⋆ is the union of its “atomic” part A = {x ∈ B⋆ | ¬∃x′ <⋆ x} and its
“atomless” part C = {x ∈ B⋆ | ∀x′ ≤⋆ x ∃x′′ <⋆ x′ }.
3. Where ≤A and ≤C are the restrictions of ≤⋆ to A and C, respectively, the
BAs RO(A, ≤A ) and RO(C, ≤C ) are atomic and atomless, respectively.
11
For another route to CABAs based on postulates about consistent sets of propositions,
see [71].
Wesley H. Holliday
4. The map (U, V ) 7→ U ∪ V is an isomorphism from RO(A, ≤A ) × RO(C, ≤C )
to RO(B⋆ , ≤⋆ ).
Putting the above facts together, B is isomorphic to RO(A, ≤A )×RO(C, ≤C ).
3.3
Possibility frames and Boolean algebras
We now move from the representation of complete Boolean algebras to that of
arbitrary Boolean algebras. We consider two closely related representations: the
first, given in this section, is in the spirit of “general frame” theory in modal
logic [30, § 5]: propositions are not arbitrary regular open sets of possibilities, but
rather certain distinguished regular open sets. The second representation, given
in § 3.4, “topologizes” the first representation: propositions are not arbitrary
regular open sets of possibilities, but rather those regular open sets with a special
property—namely compactness—in a certain kind of topological space.
The most straightforward way to move from complete to arbitrary Boolean
algebras is to simply add to a poset (S, ⊑) a distinguished collection of regular
open sets, forming a Boolean subalgebra of RO(S, ⊑).
Definition 3.3.1. A possibility frame is a triple F = (S, ⊑, P ) such that:
1. (S, ⊑) is a poset;
2. ∅ 6= P ⊆ RO(S, ⊑) (elements of P are called admissible sets);
3. for all U, V ∈ P , ¬U ∈ P and U ∩ V ∈ P .
A possibility frame F is full if P = RO(S, ⊑). A world frame is a possibility
frame in which ⊑ is the identity relation.
Obviously full possibility frames are in one-to-one correspondence with posets,
and full world frames are in one-to-one correspondence with sets.
Every possibility frame give rise to an associated BA as follows.
Lemma 3.3.2. Given a possibility frame F = (S, ⊑, P ), the collection P forms
a Boolean subalgebra of RO(S, ⊑), which we denote by F b .
Proof. Follows from Theorem 3.2.3.1.
Example 3.3.3. Recall from Example 3.3.3 the full infinite binary tree regarded
as a poset (S, ⊑). A downset U in (S, ⊑) is finitely generated if there is a finite
set U0 ⊆ S such that
U = ↓U0 := {y ∈ S | ∃x ∈ U0 : y ⊑ x}.
Possibility Semantics
world representation uses. . .
possibility representation uses. . .
for CABAs
atoms (or principal
nonzero elements (or principal
or CBAs:
prime filters) of a CABA
proper filters) of a CBA
for BAs:
prime filters of a BA
proper filters of a BA
Figure 1: world vs. possibility representations
Let
P = {U ∈ RO(S, ⊑) | U is finitely generated}.
Since each node has finite depth and the tree is finitely branching, one can check
that for all U, V ∈ P , we have ¬U ∈ P and U ∩ V ∈ P . Hence F = (S, ⊑, P ) is
a possibility frame. The Boolean algebra F b is clearly atomless: for any U ∈ P ,
x ∈ U , and y ⊏ x, we have ↓y ∈ P and ↓y ( U . Then since there are only
countably many finitely generated downsets, F b is the unique (up to isomorphism)
countable atomless Boolean algebra [76, Ch. 16]. Since F b is a dense subalgebra
of RO(S, ⊑), it follows that RO(S, ⊑) is (up to isomorphism) the MacNeille
completion of the countable atomless Boolean algebra.
Example 3.3.4. Another source of examples of possibility frames comes from
semantics for intuitionistic logic. An intuitionistic general frame [37, § 8.1] is
a triple (S, ⊑, P ) where (S, ⊑) is a poset and P is a set of downsets of (S, ⊑)
containing ∅ and closed under ∩, ∪, and the operation → defined by
U → V = {x ∈ U | ∀x′ ⊑ x (x′ ∈ U ⇒ x′ ∈ V )}.
Then (S, ⊑, P ∩RO(S, ⊑)) is a possibility frame, using the fact that ¬U = U → ∅.
Below we will consider two main approaches to constructing possibility frames
from BAs in § 3.3.1 and § 3.3.2, respectively: constructing possibilities as nonzero
propositions or as proper filters of propositions—compare this with the use of
atoms and prime filters in possible world semantics, as in Figure 1. Each of these
two approaches has two versions, giving us four ways of constructing a possibility
frame from a BA, summarized in Figure 2.
3.3.1
Constructing possibilities from nonzero propositions
The first way of constructing possibilities already appeared in Theorem 3.2.3.2:
simply take possibilities to be nonzero propositions. Philosophically, this fits with
how we often describe possibilities in natural language, e.g., the possibility that
Wesley H. Holliday
possibility frame built from a BA B
associated BA
full: Bu = (B+ , ≤+ , RO(B+ , ≤+ ))
(Bu )b is MacNeille of B
principal: Bp = (B+ , ≤+ , {↓+ a | a ∈ B})
(Bp )b ∼
=B
filter: Bf = (PropFilt(B), ⊇, RO(PropFilt(B), ⊇))
(Bf )b is canonical extension of B
general filter: Bg = (PropFilt(B), ⊇, {b
a | a ∈ B})
(Bg )b ∼
=B
Figure 2: four ways of building a possibility frame from a BA
it is raining in Beijing, using that-clauses that philosophers traditionally take to
refer to propositions [133]. On this view, given a BA B, the set of possibilities is
B+ as in Theorem 3.2.3.2.
Proposition 3.3.5 ([96]). Given a Boolean algebra B, define its full frame
Bu = (B+ , ≤+ , RO(B+ , ≤+ ))
and its principal frame
Bp = (B+ , ≤+ , {↓+ a | a ∈ B})
where (B+ , ≤+ ) is as in Theorem 3.2.3.2 and
↓+ a = {b ∈ B+ | b ≤+ a},
noting that where 0 is the bottom element of B, ↓+ 0 = ∅. Then:
1. Bu and Bp are possibility frames;
2. (Bu )b , i.e., RO(B+ , ≤+ ), is (up to isomorphism) the MacNeille completion
of B (see Proposition 3.2.6);
3. (Bp )b is isomorphic to B;12
4. Bu = Bp if and only if B is complete.
12
Note that {↓+ a | a ∈ B} ordered by inclusion is isomorphic to {↓a | a ∈ B} ordered by inclusion. Dana Scott (personal communication) reports having discussed this “baby” representation
of BAs by lattices of sets with Tarski over 60 years ago.
Possibility Semantics
3.3.2
Constructing possibilities from proper filters of propositions
To prepare for the second way of constructing possibilities, recall that in a lattice
L, a filter is a nonempty set F of elements of L such that a, b ∈ L implies a∧b ∈ L,
and a ∈ F and a ≤ b together imply b ∈ F . F is proper if it does not contain all
elements of L. A proper filter F is prime if a ∨ b ∈ F implies that a ∈ F or b ∈ F .
In a Boolean algebra B, the prime filters are exactly the ultrafilters, those proper
filters F such that for all a ∈ B, a ∈ F or ¬a ∈ F .
The second way of constructing possibilities is inspired by the idea of defining
worlds as ultrafilters, or in logical terms, as maximally consistent sets of sentences;
but since possibilities may be partial, we construct them as proper filters, or in
logical terms, as consistent and deductively closed sets of sentence.
Remark 3.3.6. In logic, the idea of constructing canonical models using consistent
and deductively closed sets of formulas appears in [147, 104, 10, 13, 14]. In lattice
theory, a representation of ortholattices using proper filters appears in [79], and
a representation of arbitrary lattices using filters appears in [138].
While obtaining worlds from arbitrary Boolean algebras requires the nonconstructive Boolean Prime Filter Theorem, which is not provable in ZF set theory
[59] or even ZF plus Dependent Choice (ZF+DC) [145], this is not needed for the
construction of possibilities from arbitrary BAs. All results in this section are
provable in ZF, with the exception of Proposition 3.3.13, which uses ZF+DC.
Proposition 3.3.7 ([96]). Given a Boolean algebra B, define its filter frame
Bf = (PropFilt(B), ⊇, RO(PropFilt(B), ⊇))
and general filter frame
b | a ∈ B})
Bg = (PropFilt(B), ⊇, {a
where PropFilt(B) is the set of proper filters of B, and
Then:
b = {F ∈ PropFilt(B) | a ∈ F }.
a
1. Bf and Bg are possibility frames;
2. (Bf )b , i.e., RO(PropFilt(B), ⊇), is (up to isomorphism) the canonical extension of B;13
13
Here we use the “constructive” definition of canonical extension from [75], which is equivalent
Wesley H. Holliday
3. (Bg )b is isomorphic to B.
Part 3 points to a duality between Boolean algebras and those special possibility frames in the image of the (·)g map. They can be given the following characterization, analogous to the definition of descriptive frames in possible world
semantics [77] but using filters instead of ultrafilers.
Proposition 3.3.8 ([96]). For any possibility frame F = (S, ⊑, P ), the following
are equivalent:
1. F is isomorphic to (F b )g ;
2. F satisfies the following conditions:
(a) the separation condition: for all x, y ∈ S, if y 6⊑ x, then there is a
U ∈ P such that x ∈ U but y 6∈ U ;
(b) the filter realization condition: for every proper filter F in F b , there is
an x ∈ S such that F = {U ∈ P | x ∈ U }.
We call an F satisfying (a) separative. We call an F satisfying (a) and (b) filterdescriptive.
The separation condition corresponds to the characterization of refinement in
(C4) of § 2.5. The filter realization condition captures the idea that for any consistent set F of propositions, there is the possibility of all propositions in F being
true, which is included in the frame. In contrast to filter-descriptive frames, full
frames may easily fail the filter realization condition. For example, when we regard the infinite binary tree in Examples 3.1.1, 3.2.1, and 3.3.3 as a full possibility
frame (S, ⊑, RO(S, ⊑)), the filter F generated by the set F0 = {↓0, ↓00, ↓000, . . . }
of propositions is proper, as any finite subset of F0 has a nonempty intersection;
but there is no possibility in S that belongs to all propositions in F0 .
Let us note two useful consequences of separation and filter realization.
Lemma 3.3.9.
1. If F = (S, ⊑, P ) is separative, then the poset (S, ⊑) is separative.
to the standard definition in ZFC but not in ZF: the constructive canonical extension of a Boolean
algebra B is the unique (up to isomorphism) complete Boolean algebra C for which there is an
embedding e of B into C such that every element of C is a join of meets of e-images of elements
of B (or equivalently in this Boolean case, every element of C is a meet of joins of e-images of
VC
WC
elements of B), and for any sets X, Y of
e[X] ≤C
e[Y ], then there are
V elements
W of′ B, if
′
′
′
finite X ⊆ X and Y ⊆ Y such that X ≤ Y . Compare the realization of the canonical
extension in Theorem 3.3.7.2 as the regular open algebra of the poset of proper filters with the
traditional realization as the powerset of the set of prime filters (see Remark 3.4.12 below).
Possibility Semantics
2. If F = (S, ⊑, P ) satisfies filter realization, then for any {Ui | i ∈ I} ⊆ P such
W
W
that S = {Ui | i ∈ I}, there is a finite I0 ⊆ I such that S = {Ui | i ∈ I0 }.
Proof. For part 1, suppose y 6⊑ x. Then by the separation condition, there is a
U ∈ P such that x ∈ U but y 6∈ U . Then since U ∈ RO(S, ⊑), there is a z ⊑ y
such that for all z ′ ⊑ z, z ′′ 6∈ U , which implies ↓z ∩ ↓x = ∅.
For part 2, we first claim that if P0 ⊆ P has the finite intersection property
T
(fip), then P0 6= ∅. For if P0 has the fip, then the filter F in F b generated by
P0 is proper, in which case filter realization implies there is an x ∈ S such that
T
F = {U ∈ P | x ∈ U }, so x ∈ P0 . Now suppose toward a contradiction that for
W
W
every finite J ⊆ I, we have S 6⊆ {Uj | j ∈ J}, so there is an xJ ∈ S \ {Uj |
W
j ∈ J}. Then as {Uj | j ∈ J} ∈ RO(S, ⊑), there is an x′J ⊑ xJ such that
W
V
T
x′J ∈ ¬ {Uj | j ∈ J} = {¬Uj | j ∈ J} = {¬Uj | j ∈ J}. Thus, {¬Ui | i ∈ I}
T
W
has the fip, so {¬Ui | i ∈ I} =
6 ∅, contradicting S = {Ui | i ∈ I}.
To obtain a categorical duality between filter-descriptive frames and BAs, we
introduce morphisms on the possibility side, based on the standard notion of a
p-morphism between relational structures (see, e.g., [37, p. 30]).
Definition 3.3.10. Given possibility frames F = (S, ⊑, P ) and F ′ = (S ′ , ⊑′ , P ′ ),
a p-morphism from F to F ′ is a map h : S → S ′ satisfying the following conditions
for all x, y ∈ S and y ′ ∈ S ′ :
1. for all U ′ ∈ P ′ , h−1 [U ′ ] ∈ P ;
2. ⊑-forth: if y ⊑ x, then h(y) ⊑ h(x);
3. ⊑-back: if y ′ ⊑′ h(x), then ∃y: y ⊑ x and h(y) = y ′ (see Figure 3).
Remark 3.3.11. If F is a full possibility frame, then together ⊑-forth and ⊑-back
imply condition 1 of Definition 3.3.10 (see Fact 3.5 of [96]).
We can now state a choice-free duality theorem for BAs.
Theorem 3.3.12 ([96]). (ZF) The category FiltPoss of filter-descriptive possibility frames with p-morphisms is dually equivalent to the category BA of Boolean
algebras with Boolean homomorphisms.
As an example of how possibility frames can be used to prove facts about BAs,
consider the following basic fact in Proposition 3.3.13, the proof of which is almost
copied verbatim from our proof using upper Vietoris spaces in [27] (note that the
filter realization property in Proposition 3.3.8 is not needed in the proof). Recall
that the Axiom of Dependent Choice (DC) states that if a binary relation R on a
set X is serial, i.e., for every x ∈ X there is a y ∈ X with xRy, then there is an
infinite sequence x0 , x1 , . . . of elements of X such that xi Ri+1 for each i ∈ N.
Wesley H. Holliday
x
h(x)
x
h(x)
∃y
y′
⇒
y′
Figure 3: The ⊑-back condition of p-morphisms, where solid arrows represent the
relation ⊑ and dotted arrows represent the function h.
Proposition 3.3.13. (ZF+DC) Every infinite BA has infinite chains and infinite
anti-chains.
Proof. As the dual of any infinite BA is an infinite separative possibility frame, it
suffices to show that for any infinite separative F = (S, ⊑, P ), there is an infinite
chain U0 ) U1 ) . . . of sets from P , as well as an infinite family of pairwise disjoint
sets from P . For this it suffices to show that for every infinite U ∈ P (note that S
is such a U ), there is an infinite U ′ ∈ P with U ) U ′ and U ∩ ¬U ′ 6= ∅. For then
by DC, there is an infinite chain U0 ) U1 ) . . . of sets from P with Ui ∩¬Ui+1 6= ∅
for each i ∈ N, in which case {U0 ∩ ¬U1 , U1 ∩ ¬U2 , . . . } is our antichain.
Assume U ∈ P is infinite. Since ⊑ antisymmetric, take x, y ∈ U such that
y 6⊑ x. Then since F is separative, there is a V ∈ P such that x ∈ V and y 6∈ V ,
which with y ∈ U and U, V ∈ P implies that there is a z ⊑ y such that z ∈ U ∩¬V .
Since U, V ∈ P , we have U ∩ V, U ∩ ¬V ∈ P ; and since z ∈ U ∩ ¬V and x ∈ U ∩ V ,
we have z ∈ U ∩ ¬(U ∩ V ) 6= ∅ and x ∈ U ∩ ¬(U ∩ ¬V ) 6= ∅. Thus, if U ∩ V
is infinite, then we can set U ′ := U ∩ V , and otherwise we claim that U ∩ ¬V is
infinite, in which case we can set U ′ := U ∩¬V . Let ⊑′ be the restriction of ⊑ to U .
Since (S, ⊑) is a separative poset by Lemma 3.3.9.1, (U, ⊑′ ) is a separative poset
by Proposition 3.2.5.2 and the fact that U ∈ RO(S, ⊑). Given V ∈ RO(S, ⊑),
we have U ∩ V, U ∩ ¬V ∈ RO(U, ⊑′ ) and U ∩ ¬V = ¬′ (U ∩ V ), where ¬′ is the
negation operation in RO(U, ⊑′ ). Then since U is infinite, by Proposition 3.2.5.4
either U ∩ V or ¬′ (U ∩ V ) is infinite, as desired.
3.4
Spaces of possibilities and Boolean algebras
Dualities between categories of algebras and categories of topological structures
play a central role in the study of modal and nonclassical logics (see, e.g., [21]).
Possibility Semantics
Along these lines, in this section we “topologize” the possibility frames of § 3.3
by using the family P of admissible sets to generate a topology on S. This is
analogous to the way that general frames in modal logic give rise to topological spaces (see [149]). We first briefly consider topologizing arbitrary possibility
frames in § 3.4.1 and then filter-descriptive possibility frames in particular in
§ 3.4.2. § 3.4.1 is based on [26], which introduces relational topological possibility
frames for modal logic.
3.4.1
Topological possibility frames
To describe the spaces that arise from topologizing a possibility frame, we need
the following variation on the standard topological notion of a clopen set.
Definition 3.4.1. Let (S, ⊑, τ ) be a triple such that (S, ⊑) is a poset and (S, τ )
is a topological space. For U ⊆ S:
1. U ⊆ S is open in (S, ⊑, τ ) if U is open in (S, τ );
2. U is neg-closed in (S, ⊑, τ ) if ¬U = {x ∈ S | ∀x′ ⊑ x x′ 6∈ U } is open
in (S, τ );
3. U is neg-clopen in (S, ⊑, τ ) if U is both open and neg-closed in (S, ⊑, τ );
4. NegClop(S, ⊑, τ ) is the family of neg-clopen sets;
5. NegClopRO(S, ⊑, τ ) = NegClop(S, ⊑, τ ) ∩ RO(S, ⊑).
Definition 3.4.2. A topological possibility frame is a triple T = (S, ⊑, τ ) such
that (S, ⊑) is a poset, (S, τ ) is a topological space, and NegClopRO(S, ⊑, τ ) is
closed under intersection and forms a basis for τ .
Possibility frames may be turned into topological possibility frames, and vice
versa, as follows.
Proposition 3.4.3.
1. For any possibility frame F = (S, ⊑, P ), the triple T (F) = (S, ⊑, τ ) where
τ is the topology generated by taking the elements of P as basic opens is a
topological possibility frame.
2. For any topological possibility frame T = (S, ⊑, τ ), the triple
F (T ) = (S, ⊑, P ) where P = NegClopRO(S, ⊑, τ ) is a possibility frame.
Wesley H. Holliday
3. If F = (S, ⊑, P ) is a separative possibility frame, then ⊑ is the converse of
the specialization order of (S, τ ) in T (F).14
4. If F is a possibility frame satisfying filter realization, then F (T (F)) = F.
5. For any topological possibility frame T , T (F (T )) = T .
Proof. Parts 1, 2, 3, and 5 are easy to check from the definitions.
For part 4, we must show U ∈ P iff U ∈ NegClopRO(S, ⊑, τ ), where τ is the
topology generated by P . The left-to-right direction is straightforward. From
right to left, suppose U ∈ NegClopRO(S, ⊑, τ ). Then U is open in (S, τ ), so
S
S
U = {A ∈ P | A ⊆ U }, and ¬U is open in (S, τ ), so ¬U = {B ∈ P |
W
B ⊆ ¬U }. Since U, ¬U ∈ RO(S, ⊑), it follows that U = {A ∈ P | A ⊆ U }
W
and V = {B ∈ P | B ⊆ ¬U }, and then since U ∨ ¬U = S, we have S =
W
W
{A ∈ P | A ⊆ U } ∨ {B ∈ P | B ⊆ ¬U }. It follows by Lemma 3.3.9.2
that S = A1 ∨ · · · ∨ An ∨ B1 ∨ · · · ∨ Bm where Ai ⊆ U and Bi ⊆ ¬U . Hence
A1 ∪ · · · ∪ An ⊆ U , which with U ∈ RO(S, ⊑) implies A1 ∨ · · · ∨ An ⊆ U , and
U ∩ (B1 ∪ · · · ∪ Bm ) = ∅, which implies U ∧ (B1 ∨ · · · ∨ Bm ) = ∅. Then:
U
= U ∧ S = U ∧ (A1 ∨ · · · ∨ An ∨ B1 ∨ · · · ∨ Bm )
= (U ∧ (A1 ∨ · · · ∨ An )) ∨ (U ∧ (B1 ∨ · · · ∨ Bm ))
= U ∧ (A1 ∨ · · · ∨ An ).
Since ∧ is intersection, U = U ∧ (A1 ∨ · · · ∨ An ) implies U ⊆ A1 ∨ · · · ∨ An , and
we already derived A1 ∨ · · · ∨ An ⊆ U above. Thus, U = A1 ∨ · · · ∨ An ∈ P .
In light of part 3, there is little harm in dropping ⊑ from the signature of our
structures and simply working with spaces, as we will do in the next section.
3.4.2
UV-spaces
In this section, we characterize the topological spaces corresponding (via Proposition 3.4.3.1) to the filter-descriptive possibility frames of Proposition 3.3.8.
Recall that Stone [161] proved that every distributive lattice L can be represented as the lattice of compact open sets of a spectral space, namely the spectral
space S(L) of prime filters of L topologized with basic open sets of the form
β(a) = {F ∈ PrimeFilt(L) | a ∈ F }
for a ∈ L. Recall that a spectral space is a T0 space X such that:
14
In [26] we define regular opens of a poset to be regular opens in the upset topology on the
poset, in which case ⊑ coincides with the specialization order of the topology generated by P .
Possibility Semantics
1. X is coherent: CO(X), the family of compact open sets of X, is closed under
finite intersections and forms a basis for the topology of X;
2. X has enough points: every completely prime filter in O(X), the lattice of
open sets of X, is of the form O(x) = {U ∈ O(X) | x ∈ U } for some x ∈ X.15
A special case of this representation is Stone’s [160] representation of Boolean
algebras: if L is a BA, then the compact open sets are precisely the clopen sets,
and S(L) is a zero-dimensional compact Hausdorff space or Stone space [109].
The use of prime filters in Stone’s theorems requires the nonconstructive Prime
Filter Theorem. However, as shown in [27], a choice-free representation of BAs is
possible using the following special spectral spaces.
Definition 3.4.4. An upper Vietoris space (UV-space) is a T0 space X such that:
1. CRO(X), the family of compact regular open sets of X, forms a basis and
is closed under finite intersections and the operation U 7→ int(X \ U );
2. every proper filter in CRO(X) is of the form CRO(x) = {U ∈ CRO(X) |
x ∈ U } for some x ∈ X.
Note the similarity in logical form between conditions 1 and 2 of a UV-space and
the conditions of coherence and having enough points of a spectral space. The
choice of the name “upper Vietoris space” is explained in [27]. In brief, it comes
from the “upper Vietoris” variant of the Vietoris hyperspace construction [169].
Assuming the Prime Filter Theorem, every upper Vietoris space as in Definition
3.4.4 is isomorphic to the upper Vietoris hyperspace of a Stone space.
Proposition 3.4.5 ([27]).
1. Every UV-space is a spectral space.
2. For any UV-space, CRO(X) forms a Boolean algebra with ∩ as meet and
U 7→ int(X \ U ) as complement.
3. For any Boolean algebra B, the space U V (B) whose underlying set is
b | a ∈ B}, with a
b dePropFilt(B) and whose topology is generated by {a
fined as in Proposition 3.3.7, is a UV-space.
4. A space X is homeomorphic to U V (CRO(X)) iff X is a UV-space.
15
Equivalently, a spectral space is a space that is coherent and sober [109, § II.1.6], where X
is sober if the map from X to the set of completely prime filters in O(X) is not only a surjection
but a bijection. Sobriety is equivalent to the conjunction of T0 and having enough points.
Wesley H. Holliday
Remark 3.4.6. Though we defined UV-spaces above using the family CRO(X), we
could have equivalently done so using the family CORO(X) of sets (as in [27]) that
are both compact open in X and regular open in the poset (X, ⊑), where ⊑ is the
converse of the specialization order of X,16 or using the family NegClopRO(X)
(as in [26]) defined as in Definition 3.4.1 in terms of ⊑ and the topology of X.
Proposition 3.4.7 ([27, 26]). For any UV-space X,
CRO(X) = CORO(X) = NegClopRO(X).
UV-spaces and filter-descriptive possibility frames as in Definition 3.3.8 are in
one-to-one correspondence via analogues of the T and F maps of Definition 3.4.3.
Proposition 3.4.8. For any UV-space X, if S is the underlying set of X and
⊑ the converse of the specialization order of X, then (S, ⊑, CRO(X)) is a filterdescriptive possibility frame. For any filter-descriptive possibility frame (S, ⊑, P ),
the space (S, τ ) where τ is the topology generated by P is a UV-space.
Proof. For the first claim, (S, ⊑) is a poset, and CRO(X) is a nonempty subset of
RO(S, ⊑) closed under ¬ and ∩, as CRO(X) = CORO(X) by Proposition 3.4.7.
That CORO(X) forms a basis for X, given by part 1 of Definition 3.4.4, implies
the separation property of filter-descriptive possibility frames in Proposition 3.3.8,
and part 2 of Definition 3.4.4 implies the filter realization property. The second
claim follows from Propositions 3.3.8 and 3.4.5.3.
To go beyond representation of BAs to a full duality for BAs, we introduce
morphisms similar to those in Definition 3.3.10. A spectral map [92] between
spectral spaces X and X ′ is a map f : X → X ′ such that f −1 [U ] ∈ CO(X) for
each U ∈ CO(X ′ ), which implies that f is continuous.
Definition 3.4.9. Let X and X ′ be UV-spaces with specialization orders 6 and
6′ , respectively. A UV-map from X to X ′ is a spectral map f : X → X ′ that also
satisfies the p-morphism condition:
if f (x) 6′ y ′ , then ∃y : x 6 y and f (y) = y ′ .
This condition matches the p-morphic ⊑-back condition of Definition 3.3.10.3
after adjusting for the fact that 6 is the converse of ⊑. That f preserves the
specialization order follows from f being continuous.
We can now state a duality between UV-spaces and BAs analogous to the
duality between filter-descriptive possibility frames and BAs (Theorem 3.3.12).
16
In [27] we define regular opens of a poset to be regular opens in the upset topology on the
poset, in which case ⊑ coincides with the specialization order of X.
Possibility Semantics
MacNeille of B
world approach
possibility approach
regular opens of Stone space
RO(B+ , ≤+ ) or regular
opens of UV-space
canonical extension
powerset of Stone space
of B
regular opens of poset
of proper filters
Figure 4: realizations of MacNeille completion and canonical extension
Theorem 3.4.10 ([27]). (ZF) The category UV of UV-spaces with UV-maps is
dually equivalent to the category BA of Boolean algebras with Boolean homomorphisms.
This duality leads to a dictionary for translating between BA concepts and UVconcepts, given in [27], making it possible to prove facts about BAs using point-set
topological reasoning. Here we highlight only one entry in the dictionary. For this
we stress the distinction between RO(X), the family of regular open sets in the
UV-space X, and RO(X), the family of regular open sets in the poset (X, ⊑)
where ⊑ is the converse of the specialization order of X. While RO(X) realizes
the canonical extension of the BA dual to X (recall Proposition 3.3.7.2), RO(X)
realizes the MacNeille completion of the BA dual to X.
Proposition 3.4.11 ([27]). Let B be a BA and X its dual UV-space. Then
RO(X) is (up to isomorphism) the MacNeille completion of B.
Thus, we have seen two ways of realizing the MacNeille completion of a BA A:
RO(B+ , ≤+ ), as in Proposition 3.2.6, or RO(U V (B)), as in Proposition 3.4.11.
Remark 3.4.12. Compare the choice-free possibilistic approaches to realizing the
MacNeille completion and constructive canonical extension of a BA to the classical
methods based on Stone duality, assuming the Boolean Prime Filter Theorem: the
MacNeille completion of a BA B can be realized as RO(X) where X is the Stone
space of B; and the canonical extension of a BA B can be realized as the powerset
of the Stone space of B. Figure 4 summarizes the foregoing points.
4
First-order case
In this section, we give possibility semantics for classical first-order logic, a topic
previously studied by van Benthem [10, 16], Harrison-Trainor [90], and Massas
Wesley H. Holliday
[129].17 Fine [61] also gave what can be considered a kind of possibility semantics
for classical first-order logic with applications to vague languages (see [96, § 8.1]
for comparison). As we briefly show in § 4.4, first-order possibility semantics can
be seen as generalizing an interpretation of the language of first-order set theory
used in forcing (see, e.g., [6]) to arbitrary first-order languages. In the terminology
of topos theory, first-order possibility semantics can be seen as a variant of sheaf
semantics for first-order logic using the Cohen topos [124, p. 318].
After defining the semantics, our main goal is a completeness theorem. The
traditional completeness theorem for first-order logic for uncountable languages
[127], stating that every consistent set of first-order sentences has a Tarskian
model, is not provable in ZF, as it is equivalent in ZF to the Boolean Prime Filter
Theorem [91] (for a proof, see [7, p. 104]). By contrast, we will prove in ZF that for
arbitrary languages, every consistent set of first-order sentences has a possibility
model. This suggest the possibility of a program of “choice-free model theory.”
We assume familiarity with the definition of a first-order language L, including
identity and function symbols. The only detail to note is that we treat constant
symbols as 0-ary function symbols for simplicity (as in, e.g., [158, § 2.4]). Let
Var(L) and Term(L) be the set of variables and set of terms of L, respectively.
We also assume familiarity with first-order logic as a proof system.
4.1
First-order possibility models
We begin with possibility models for a first-order language L.
In addition to a poset (S, ⊑) of possibilities, these models have a set D of
what we call guises of objects. A possibility s ∈ S may settle that two guises
a and b are guises of the same object: a ≍s b.18 On the other hand, a partial
possibility s may not yet settle that they are guises of the same object; borrowing
Frege’s [70] famous example, s may not settle that the object known under the
guise of The Morning Star is the same as the object known under the guise of The
Evening Star. Then by the standard refinability property in possibility semantics
(see Definition 4.1.1.3), there will be a possibility s′ ⊑ s that settles that they
are not guises of the same object: for all s′′ ⊑ s′ , a 6≍s′′ b. Note the distinction
between a 6≍s′ b, which means s′ does not settle that a and b are guises of the
same object, and the stronger condition that for all s′′ ⊑ s′ , a 6≍s′′ b, which means
17
These versions of possibility semantics for first-order logic differ in their details. Van Benthem and Massas consider first-order languages without identity or function symbols, and their
models are “varying domain” models, whereas our models below are “constant domain” models,
like Harrison-Trainor’s. Our models differ from the others in adding “equality” relations.
18
The use of a family of equivalence relations {≍s }s∈S appears in the standard definition of
Kripke models with equality in [73, § 3.5].
Possibility Semantics
s′ settles that a and b are not guises of the same object.
A similar distinction applies to the interpretation of predicate symbols by an
interpretation function V : if (a, b) ∈ V (R, s), then s settles that the pair (a, b)
is in the interpretation of R, which we take to mean that the objects of which
a and b are guises, respectively, stand in the relation to which R corresponds;
if (a, b) 6∈ V (R, s), then s does not settle that (a, b) is in the interpretation of
R; and if for all s′ ⊑ s, (a, b) 6∈ V (R, s′ ), then s settles that (a, b) is not in the
interpretation of R. We comment on the interpretation of function symbols below.
Definition 4.1.1. Let L be a first-order language. A first-order possibility model
for L is a tuple A = (S, ⊑, D, ≍, V ) where:
1. (S, ⊑) is a poset;
2. D is a nonempty set;
3. ≍ is a function assigning to each s ∈ S an equivalence relation ≍s on D
satisfying:
• persistence for ≍: if a ≍s b and s′ ⊑ s, then a ≍s′ b;
• refinability for ≍: if a 6≍s b, then ∃s′ ⊑ s ∀s′′ ⊑ s′ a 6≍s′′ b.
4. V is a function assigning to each pair of an n-ary predicate R of L and s ∈ S
a set V (R, s) ⊆ Dn and to each n-ary function symbol f of L and s ∈ S a
set V (f, s) ⊆ Dn+1 satisfying:
• persistence for R: if a ∈ V (R, s), s′ ⊑ s, and a ≍s′ b, then b ∈ V (R, s′ );19
• refinability for R: if a 6∈ V (R, s), then ∃s′ ⊑ s ∀s′′ ⊑ s′ a 6∈ V (R, s′′ );
• persistence for f : if a ∈ V (f, s), s′ ⊑ s, and a ≍s′ b, then b ∈ V (f, s′ );
• quasi-functionality for f : if (a, b) ∈ V (f, s) and (a, b′ ) ∈ V (f, s), then b ≍s b′ ;
• eventual definedness for f : ∀a ∈ Dn ∃s′ ⊑ s ∃b ∈ D: (a, b) ∈ V (f, s).
We say A has total functions if for each n-ary function symbol f of L, s ∈ S,
and a ∈ Dn , there is some b ∈ D such that (a, b) ∈ V (f, s).
A Tarskian model is a first-order possibility model with total functions in
which S contains only one possibility s, and ≍s is the identity relation.
A pointed model is a pair A, s of a possibility model A and possibility s in A.
19
Here a ≍s′ b means that a1 ≍s′ b1 , . . . , an ≍s′ bn .
Wesley H. Holliday
Example 4.1.2. The following example of a non-Tarskian first-order possibility
model comes from the development of choice-free nonstandard analysis using possibility models by Massas [131]. Let M be a Tarskian model with domain M . The
Fréchet power MF of M is the tuple MF = (F , ⊇, D, ≍, V ) where:
1. F is the set of all proper filters of ℘(ω) extending the Fréchet filter (i.e.,
the set of all cofinite subsets of ω), which we order by reverse inclusion ⊇;
2. D = M ω , i.e., the set of all functions f : ω → M ;
3. for a, b ∈ D and F ∈ F, a ≍F b iff {i ∈ ω | a(i) = b(i)} ∈ F ;
4. for any n-ary predicate R of L, a ∈ Dn , and F ∈ F, a ∈ V (R, F ) iff
{i ∈ ω | M |=g[x:=a(i)] R(x)} ∈ F , where |=g[x:=a(i)] is the standard Tarskian
satisfaction relation with the variables x mapped to a(i).
Then one can show that MF is a first-order possibility model.
Let us note two aspects of our interpretation of an n-ary function symbol f .
First, in traditional semantics, f is interpreted as an n-ary function on the domain
of objects, i.e., an n + 1-ary relation such that if (a, b) and (a, b′ ) are both in the
relation, then b = b′ . By contrast, we ask only that b and b′ be guises of the same
object, i.e., b ≍s b′ , not that they are one and the same guise. For (a, b) ∈ V (f, s)
means that when the function to which f corresponds takes in objects of which
a are guises, then it outputs an object of which b is a guise. Hence if b ≍s b′ ,
we should have (a, b′ ) ∈ V (f, s) as well. Thus, we do not interpret f at s as an
n-ary function on the domain of guises. Instead, our approach is equivalent to
interpreting f at s as an n-ary function on D/≍s ,20 as well as interpreting an
20
There is a mathematically viable alternative version of the semantics that interprets f at s
as an n-ary function on D, which for a given input a ∈ Dn chooses a unique output b from the
equivalence class {b′ ∈ D | b ≍s b′ }. Although directly transforming one of our quasi-functional
models to such a properly functional model by choosing representatives of equivalence classes
may require the Axiom of Choice, the canonical model construction below can be adapted to
show that properly functional models would suffice for completeness. A possible convenience of
the properly functional approach is that one can take the denotation of a term to be a single
element of D (provided one then takes the truth clause for t1 = t2 at s to be that the denotations
of t1 and t2 stand in the ≍s relation), rather than an equivalence class of elements of D, as we do
in Definition 4.1.3. On the other hand, an attraction of the quasi-functional approach is that it
treats relations and function in a uniform manner. Consider the binary relation symbol Square
such that Square(t1 , t2 ) means that t2 is the square of t1 vs. the unary function symbol square
such that square(t1 ) = t2 means that t2 is the square of t1 . If (a, b) is in the interpretation of
the Square relation at s, and b ≍s b′ , then (a, b′ ) should be in the interpretation of Square at
s, since we take (a, b) ∈ V (Square, s) (resp. (a, b′ ) ∈ V (Square, s)) to mean that the objects of
which a, b (resp. a, b′ ) are guises stand in the relevant relation, and b and b′ are guises of the same
object. The quasi-functional approach applies the same idea to the interpretation of square.
Possibility Semantics
n-ary relation symbol R at s as an n-ary relation on D/≍s (see Definition 4.1.5).
However, our conditions on V are somewhat simpler to state when officially taking
the interpretations of symbols at s to be relations on D.
Second, we have allowed the “quasi-function” interpreting a function symbol at
a given partial possibility to be undefined for a given argument, i.e., there is some
a ∈ Dn such that for no b ∈ D do we have (a, b) ∈ V (f, s), as the possibility s has
not yet settled any guise of the output of the function. Allowing such undefinedness is not strictly necessary. We will see that (in ZF) every pointed model has the
same first-order theory as a pointed model with total functions. However, allowing
partiality of functions may be convenient in constructing specific models—though
functions must eventually become defined in order to validate theorems such that
∀x∃yf (x) = y. Of course, it will turn out that in ZF + Boolean Prime Filter Theorem, every pointed model has the same first-order theory as a Tarskian model
with no partiality at all, thanks to the soundness of first-order logic with respect
to possibility models and the standard completeness theorem for first-order logic
(using the Prime Filter Theorem) with respect to Tarskian models.
Next we define the denotation of a term relative to a possibility and variable
assignment. The denotation of a term will be an equivalence class of ≍s or ∅.
Definition 4.1.3. Given a first-order possibility model A = (S, ⊑, D, ≍, V ),
s ∈ S, and variable assignment g : Var(L) → D, we define a function
J KA,s,g : Term(L) → ℘(D) recursively as follows:
1. JxKA,s,g = {a ∈ D | a ≍s g(x)} for x ∈ Var(L);
2. for an n-ary function symbol f and t1 , . . . , tn ∈ Term(L),
Jf (t1 , . . . , tn )KA,s,g = {b ∈ D | ∃a1 , . . . , an : ai ∈ Jti KA,s,g and
(a1 , . . . , an , b) ∈ V (f, s)}.
Given terms t, u and a variable x, let uxt be the result of replacing x in u by t.
Given a variable assignment g, variable x, and a ∈ D, let g[x := a] be the variable
assignment that differs from g at most in assigning a to x.
Lemma 4.1.4. For any model A = (S, ⊑, D, ≍, V ), s ∈ S, variable assignment
g : Var(L) → D, t, u ∈ Term(L), and x ∈ Var(L):
1. JtKA,s,g is an ≍s -equivalence class or ∅;
2. if g(x) ≍s a, then JtKA,s,g = JtKA,s,g[x:=a] ;
3. if a ∈ JtKA,s,g and s′ ⊑ s, then a ∈ JtKA,s′ ,g ;
Wesley H. Holliday
4. there is some s′ ⊑ s such that JtKA,s′ ,g 6= ∅;
5. if A has total functions, then JtKA,s,g 6= ∅;
6. Juxt KA,s,g = JuKA,s,g[x:=a] for any a ∈ JtKA,s,g .
Proof. For part 1, the case for a variable x is immediate from Definition 4.1.3.1.
For terms with function symbols, assuming b ∈ Jf (t1 , . . . , tn )KA,s,g , we claim that
b ≍s b′ iff b′ ∈ Jf (t1 , . . . , tn )KA,s,g . The left-to-right direction follows from persistence for f , while the right-to-left direction follows from quasi-functionality for f .
Part 2 is an obvious induction on t. Part 3 follows from part 1 and persistence
for ≍ and function symbols. Part 4 follows from eventual definedness for function
symbols. Parts 5 and 6 are straightforward inductions on t and u, respectively.
We can also view the interpretation of a relation symbol at a possibility s as
a relation on the set of ≍s -equivalence classes, as follows.
Definition 4.1.5. Given a first-order possibility model A = (S, ⊑, D, ≍, V ), define a function I≍ that assigns to each pair of an n-ary predicate R of L and s ∈ S
a set I≍ (R, s) ⊆ (D/≍s )n by
(ξ1 , . . . , ξn ) ∈ I≍ (R, s) iff ∃a1 , . . . , an : ai ∈ ξi and (a1 , . . . , an ) ∈ V (R, s).
Finally, we define the satisfaction of a formula by a possibility and variable
assignment.
Definition 4.1.6. Given a first-order possibility model A = (S, ⊑, D, ≍, V ) for
L, formula ϕ of L, s ∈ S, and variable assignment g : Var(L) → D, we define the
satisfaction relation A, s g ϕ recursively as follows:
1. A, s g t1 = t2 iff for all s′ ⊑ s, if each Jti KA,s′ ,g is nonempty, then
Jt1 KA,s′ ,g = Jt2 KA,s′ ,g ;
2. A, s g R(t1 , . . . , tn ) iff for all s′ ⊑ s, if each Jti KA,s′ ,g is nonempty, then
(Jt1 KA,s′ ,g , . . . , Jtn KA,s′ ,g ) ∈ I≍ (R, s′ );
3. A, s
g
¬ϕ iff for all s′ ⊑ s, A, s′ 2g ϕ;
4. A, s
g
ϕ ∧ ψ iff A, s
5. A, s
g
∀xϕ iff for all a ∈ D, A, s
g
ϕ and A, s
g
ψ;
g[x:=a]
ϕ.
We then say that:
• a set Γ of formulas is satisfiable in A if there is some possibility s in A and
variable assignment g such that A, s g ϕ for all ϕ ∈ Γ;
Possibility Semantics
• if ϕ is a sentence, i.e., a formula with no free variables, then A, s
A, s g ϕ for some (equivalently, every) variable assignment g;
ϕ if
• two pointed models A, s and A′ , s′ are elementarily equivalent if for all sentences ϕ, A, s ϕ iff A′ , s′ ϕ.
In possibility models with total functions, the definition of satisfaction simplifies as follows.
Lemma 4.1.7. If A = (S, ⊑, D, ≍, V ) has total functions, then for any s ∈ S and
variable assignment g : Var(L) → D:
1. A, s
g t1
2. A, s
g
= t2 iff Jt1 KA,s,g = Jt2 KA,s,g ;
R(t1 , . . . , tn ) iff (Jt1 KA,s,g , . . . , Jtn KA,s,g ) ∈ I≍ (R, s).
Proof. If A has total functions, then Jti KA,s′ ,g is always nonempty by Lemma
4.1.4.5, so the nonemptiness condition drops out of Definition 4.1.6.1-2, and the
quantification over s′ ⊑ s can be dropped thanks to persistence for ≍ and R
(Definition 4.1.1).
Moreover, the following lemma, which is a consequence of Lemma 4.3.5 and
Theorem 4.3.8 below, shows that we can work with possibility models with total
functions without loss of generality.
Lemma 4.1.8. For every pointed first-order possibility model A, s, there is a
pointed first-order possibility model A′ , s′ with total functions such that A, s and
A′ , s′ are elementarily equivalent.
Finally, if our only concern is which formulas are satisfied at a particular possibility s in a possibility model, then we can delete from the model all possibilities
that are not refinements of s (cf. the fact that in modal logic, satisfaction is
invariant under taking generated submodels [30, Prop. 2.6]).
Lemma 4.1.9. Given a first-order possibility model A with A = (S, ⊑, D, ≍, V )
and s ∈ S, let As be the obvious restriction of A to the set {s′ ∈ S | s′ ⊑ s} of
possibilities, called the generated submodel of A generated by s. Then A, s and
As , s are elementarily equivalent.
Proof sketch. An easy induction on the structure of formulas ϕ, noting that the
definition of truth for ϕ at s only quantifies over states in {s′ ∈ S | s′ ⊑ s}.
Wesley H. Holliday
4.2
Soundness of first-order logic with respect to possibility semantics
In this section, we work up to a first main result concerning first-order possibility
models: the soundness of first-order logic with respect to the semantics.
To that end, the first key point is that the set of possibilities satisfying a
given formula always belongs to the Boolean algebra of regular open subset of the
underlying poset of the possibility model.21
Lemma 4.2.1. For any first-order possibility model A = (S, ⊑, D, ≍, V ) for L,
variable assignment g : Var → D, and formula ϕ of L, define
kϕkA,g = {s ∈ S | A, s
g
ϕ}.
Then for all formulas ϕ, ψ and variables x:
kϕkA,g
∈ RO(S, ⊑)
k¬ϕkA,g = ¬kϕkA,g
kϕ ∧ ψkA,g = kϕkA,g ∧ kϕkA,g
k∀xϕkA,g =
^
{kϕkA,g[x:=a] | a ∈ D}.
Proof. We prove only that kϕkA,g ∈ RO(S, ⊑), by induction on ϕ. Persistence for
the semantic values of t = t′ and R(t1 , . . . , tn ) is immediate from Definition 4.1.6.
For refinability for t1 = t2 , suppose A, s 2g t1 = t2 , so there is an s′ ⊑ s such
that each Jti KA,s′ ,g is nonempty and Jt1 KA,s′ ,g 6= Jt2 KA,s′ ,g . It follows by Lemma
4.1.4.1 that there is some a ∈ Jt1 KA,s′ ,g and b ∈ Jt2 KA,s′ ,g such that a 6≍s′ b. Then
by refinability for ≍, there is an s′′ ⊑ s′ such that for all s′′′ ⊑ s′′ , a 6≍s′′′ b. It
follows by Lemma 4.1.4.3 that for all s′′′ ⊑ s′′ , ∅ 6= Jt1 KA,s′′′ ,g 6= Jt2 KA,s′′′ ,g 6= ∅
and hence A, s′′′ 2g t1 = t2 . By transitivity of ⊑, s′′ ⊑ s′ ⊑ s implies s′′ ⊑ s.
Thus, assuming A, s 2g t1 = t2 , we have shown that there is an s′′ ⊑ s such that
for all s′′′ ⊑ s′′ , A, s′′′ 2g t1 = t2 , as desired.
Similarly, for R(t1 , . . . , tn ), suppose A, s 2g R(t1 , . . . , tn ), so there is an s′ ⊑ s
such that each Jti KA,s′ ,g is nonempty and (Jt1 KA,s′ ,g , . . . , Jtn KA,s′ ,g ) 6∈ I≍ (R, s′ ).
It follows by Definition 4.1.5 that there are a1 , . . . , an such that ai ∈ Jti KA,s′ ,g
and (a1 , . . . , an ) 6∈ V (R, s). Then by refinability for R, there is an s′′ ⊑ s′
such that for all s′′′ ⊑ s′′ , (a1 , . . . , an ) 6∈ V (R, s′′′ ). By transitivity of ⊑, s′′′ ⊑
s′′ ⊑ s′ implies s′′′ ⊑ s′ . Then ai ∈ Jti KA,s′ ,g implies ai ∈ Jti KA,s′′′ ,g by Lemma
4.1.4.3, whence (a1 , . . . , an ) 6∈ V (R, s′′′ ) implies that there are no b1 , . . . , bn such
21
For the last part of Lemma 4.2.1 concerning ∀xϕ, cf. the algebraic interpretation of quantifiers [146, 153].
Possibility Semantics
that bi ∈ Jti KA,s′′′ ,g and (b1 , . . . , bn ) 6∈ V (R, s′′′ ) by persistence for R. Thus,
(Jt1 KA,s′′′ ,g , . . . , Jtn KA,s′′′ ,g ) 6∈ I≍ (R, s′′′ ) and hence A, s′′′ 2g R(t1 , . . . , tn ). By transitivity of ⊑ again, we have s′′ ⊑ s. Thus, assuming A, s 2g R(t1 , . . . , tn ), we have
shown that there is an s′′ ⊑ s such that for all s′′′ ⊑ s′′ , A, s′′′ 2g R(t1 , . . . , tn ).
The cases for ¬, ∧, and ∀x are straightforward.
Given the standard definitions of ∨, →, and ↔ in terms of ¬ and ∧ and of ∃
in terms of ¬ and ∀, we have the following derived semantic clauses.
Lemma 4.2.2. Given a first-order possibility model A = (S, ⊑, D, ≍, V ) for L,
formula ϕ of L, s ∈ S, and variable assignment g : Var(L) → D:
1. A, s
g
ϕ ∨ ψ iff for all s′ ⊑ s there is s′′ ⊑ s′ : A, s′′
2. A, s
g
ϕ → ψ iff for all s′ ⊑ s, if A, s′
3. A, s
g
ϕ ↔ ψ iff for all s′ ⊑ s, A, s′
4. A, s
g
∃xϕ iff for all s′ ⊑ s there is s′′ ⊑ s′ and a ∈ D: A, s′′
g
g
g
ϕ or A, s′′
ϕ then A, s′
ϕ iff A, s′
g
g
g
ψ.
ψ;
ψ;
g[x:=a]
ϕ.
Proof. We define ϕ ∨ ψ as ¬(¬ϕ ∧ ¬ψ). By the semantic clauses for ¬ and ∧, it is
easy to see that A, s g ¬(¬ϕ ∧ ¬ψ) iff for all s′ ⊑ s there is s′′ ⊑ s′ : A, s′′ g ϕ
or A, s′′ g ψ. The proof for ∃xϕ as ¬∀x¬ϕ is just as easy. The proof for ϕ → ψ
as ¬(ϕ ∧ ¬ψ) requires slightly more. First, observe that A, s g ¬(ϕ ∧ ¬ψ) iff
for all s′ ⊑ s, if A, s′
g
ϕ, then there is s′′ ⊑ s′ : A, s′′
g
ϕ.
(4)
In fact (4) is equivalent to
for all u ⊑ s, if A, u
g
ϕ, then A, u
g
ψ.
(5)
The implication from (5) to (4) is obvious. From (4) to (5), suppose u ⊑ s,
A, u g ϕ, and A, u 6 g ψ. As {v ∈ S | A, v g ϕ} ∈ RO(S, ⊑) by Lemma 4.2.1,
A, u 6 g ψ implies that there is a u′ ⊑ u such that for all u′′ ⊑ u′ , A, u′′ 6 g ψ.
Then as A, u g ϕ and u′ ⊑ u, we have A, u′ g ϕ. But then (4) is false with
s′ := u′ . This completes the proof for →, from which the ↔ case follows.
The next lemma is used to prove the validity of the key first-order axiom
∀xϕ → ϕxt when t is substitutable for x in ϕ and ϕxt is the result of replacing all
free occurrences of x in ϕ by t.
Lemma 4.2.3 (Substitution Lemma). For any model A = (S, ⊑, D, ≍, V ) for L,
s ∈ S, variable assignment g : Var → D, formula ϕ of L, and x ∈ Var(L):
1. if A, s
g
ϕ and g(x) ≍s a, then A, s
g[x:=a]
ϕ;
Wesley H. Holliday
and for any t ∈ Term(L) substitutable for x in ϕ, the following are equivalent:
2. A, s
g
ϕxt ;
3. for all s′ ⊑ s, A, s′
g[x:=a]
ϕ for all a ∈ JtKA,s′ ,g .
If A has total functions, then 3 may be replaced with:
3′ . A, s
g[x:=a]
ϕ for all a ∈ JtKA,s,g .
Proof. The proof of part 1 is an easy induction on ϕ using Lemma 4.1.4.2. The
equivalence of 3 and 3′ under the assumption of total functions follows from parts
1, 3, and 5 of Lemma 4.1.4.
For the equivalence of 2 and 3, the proof is by induction on ϕ using Lemma
4.1.4.6 in the base cases. For the inductive step, we give only the ¬ϕ case, which
is the most involved. We have the following equivalences:
A, s
g
(¬ϕ)xt ;
⇔ (by the recursive definition of substitution) A, s
g
¬ϕxt ;
⇔ (by the semantics of ¬) for all s∗ ⊑ s, A, s∗ 2g ϕxt ;
⇔ (by the inductive hypothesis) for all s∗ ⊑ s there is an s∗∗ ⊑ s∗ such that
A, s∗∗ 2g[x:=b] ϕ for some b ∈ JtKA,s∗∗ ,g ;
⇔ for all s′ ⊑ s, A, s′
g[x:=a]
¬ϕ for all a ∈ JtKA,s′ ,g .
For the last equivalence, assume the right-hand side. Suppose s∗ ⊑ s. Then by
Lemma 4.1.44, there is an s∗∗ ⊑ s∗ such that JtKA,s∗∗ ,g 6= ∅. By the right-hand
side, we have A, s∗∗ g[x:=a] ¬ϕ for all a ∈ JtKA,s∗∗ ,g . It follows that A, s∗∗ 2g[x:=b] ϕ
for some b ∈ JtKA,s∗∗ ,g by the semantics of ¬ and the fact that JtKA,s∗∗ ,g 6= ∅.
Conversely, assume the left-hand side. Suppose s′ ⊑ s and pick any a ∈ JtKA,s′ ,g .
To show that A, s′ g[x:=a] ¬ϕ, consider any s′∗ ⊑ s′ , so s′∗ ⊑ s by transitivity of
⊑. Then by the left-hand side, there is an s′∗∗ ⊑ s′∗ such that A, s′∗∗ 2g[x:=b] ϕ
for some b ∈ JtKA,s′∗∗ ,g . By Lemma 4.1.4.3, a ∈ JtKA,s′ ,g and s′∗∗ ⊑ s′ together
imply a ∈ JtKA,s′∗∗ ,g , so A, s′∗∗ 2g[x:=b] ϕ implies A, s′∗∗ 2g[x:=a] ϕ by part 1. Given
s′∗∗ ⊑ s′∗ , it follows by persistence for ϕ (Lemma 4.2.1) that A, s′∗ 2g[x:=a] ϕ. Thus,
A, s′ g[x:=a] ¬ϕ, establishing the right-hand side.
Lemmas 4.2.1–4.2.3 are the key parts of the proof that first-order logic is sound
with respect to possibility semantics. Given a set Γ of formulas and formula ϕ,
let Γ ⊢ ϕ mean that ϕ is provable in first-order logic from assumptions in Γ.
Possibility Semantics
Theorem 4.2.4 (Soundness). For any set Γ of formulas and formula ϕ of L, if
Γ ⊢ ϕ, then for every pointed model A, s and variable assignment g, if A, s g ψ
for all ψ ∈ Γ, then A, s g ϕ.
Proof sketch. Consider a Hilbert-style proof system for first-order logic as in [54].
Use Lemma 4.2.1 to verify that all first-order substitution instances of tautologies
are valid, Lemma 4.2.3 to verify that ∀xϕ → ϕxt is valid when t is substitutable
for x in ϕ, and Lemma 4.2.2.2 to verify that the other axioms involving ∀ and →
are valid and that modus ponens preserves validity.
4.3
Completeness of first-order logic with respect to possibility
semantics
In this section, we prove the completeness of first-order logic—for an arbitrary
first-order language L—with respect to first-order possibility semantics.
We first recall how to extend any consistent set of first-order formulas to one
with witnesses for existential formulas—a standard construction not specific to
possibility semantics.
Definition 4.3.1. Given a set Γ of formulas of L, let
CnL (Γ) = {ϕ a formula of L | Γ ⊢ ϕ}.
Γ is deductively closed if Γ = CnL (Γ).
Γ is Henkinized if for every formula of L of the form ∃xϕ, there is a constant
symbol c of L such that ∃xϕ → ϕxc ∈ Γ.
There is sometimes confusion about whether extending a consistent set to
a Henkinized one requires choice principles or transfinite recursion. In fact, no
matter the cardinality of L, no choice or transfinite recursion is needed.
Definition 4.3.2. Given any first-order language L, we define a countable sequence of first-order languages as follows:
L0 = L
Ln+1 = extension of Ln with new constant c∃xϕ for each formula ∃xϕ of Ln
Lω =
[
Ln .
n∈ω
Lemma 4.3.3. (ZF) Let Γ be a consistent set of formulas of L. Then
H(Γ) = CnLω (Γ ∪ {∃xϕ → ϕxc∃xϕ | ∃xϕ a formula of Lω })
is a consistent, deductively closed, and Henkinized set of formulas of Lω .
Wesley H. Holliday
For proofs of Lemma 4.3.3, see, e.g., [158, pp. 46-7], [53, pp. 82-4], [167, pp. 98-9].
Now we begin the specifically possibility-semantic part of the completeness
proof, by defining a single first-order possibility model in which all consistent sets
of formulas are satisfiable.
Definition 4.3.4. The canonical model for Lω is the tuple ALω = (S, ⊑, D, ≍, V )
where:
1. S is the set of all consistent, deductively closed, and Henkinized sets of
formulas of Lω ;
2. Γ′ ⊑ Γ iff Γ′ ⊇ Γ;
3. D is the set of terms of Lω ;
4. t ≍Γ t′ iff t = t′ ∈ Γ;
5. for any n-ary predicate symbol R and Γ ∈ S,
V (R, Γ) = {(t1 , . . . , tn ) | R(t1 , . . . tn ) ∈ Γ};
6. for any n-ary function symbol f of Lω and Γ ∈ S,
V (f, Γ) = {(t1 , . . . , tn+1 ) | f (t1 , . . . , tn ) = tn+1 ∈ Γ}.
The canonical model for L, AL , is the reduct of ALω to L, i.e., the model exactly
like ALω except it does not interpret constants that are in Lω but not L.
In Definition 4.3.4.1, one may replace ‘Henkinized’ with ‘canonically Henkinized’, where a set of formulas of Lω is canonically Henkinized if for every formula
of Lω of the form ∃xϕ, we have ∃xϕ → ϕxc∃xϕ ∈ Γ for c∃xϕ in particular, where
c∃xϕ is the constant introduced for ∃xϕ in Definition 4.3.2.
Lemma 4.3.5. The canonical model for Lω (resp. L) is a first-order possibility
model for Lω (resp. L) with total functions.
Proof. Conditions 1 and 2 of Definition 4.1.1 are immediate. The other conditions
correspond to the following easily verified syntactic facts:
• that ≍Γ is an equivalence relation follows from the facts that ⊢ t = t,
⊢ t = t′ → t′ = t, and ⊢ (t = t′ ∧ t′ = t′′ ) → t = t′′ ;
• persistence for ≍: if a = b ∈ Γ and Γ′ ⊑ Γ, so Γ′ ⊇ Γ, then a = b ∈ Γ′ ;
Possibility Semantics
• refinability for ≍: if a = b 6∈ Γ, then there is a Γ′ ⊑ Γ such that ¬a = b ∈ Γ′ ,
namely Γ′ = CnLω (Γ ∪ {¬a = b}), and hence a = b 6∈ Γ′′ for all Γ′′ ⊑ Γ′ ;
• persistence for R: if R(t1 , . . . , tn ) ∈ Γ, Γ′ ⊑ Γ, and for each i,
ti = t′i ∈ Γ′ , then R(t′1 , . . . , t′n ) ∈ Γ′ ;
• refinability for R: if R(t1 , . . . , tn ) 6∈ Γ, then there is a Γ′ ⊑ Γ such that
¬R(t1 , . . . , tn ) ∈ Γ′ and hence R(t1 , . . . , tn ) 6∈ Γ′′ for all Γ′′ ⊑ Γ′ ;
• persistence for f : if f (t1 , . . . , tn ) = tn+1 ∈ Γ, Γ′ ⊑ Γ, and for each i,
ti = t′i ∈ Γ′ , then f (t′1 , . . . , t′n ) = t′n+1 ∈ Γ′ .
• quasi-functionality for f : if f (t1 , . . . , tn ) = tn+1 ∈ Γ, then
f (t1 , . . . , tn ) = t′ ∈ Γ implies tn+1 = t′ ∈ Γ.
Finally, that the model has total functions follows from the fact that for every term f (t1 , . . . , tn ) and Γ ∈ S, we have f (t1 , . . . , tn ) = f (t1 , . . . , tn ) ∈ Γ, so
(t1 , . . . , tn , f (t1 , . . . , tn )) ∈ V (f, Γ).
For the next two lemmas, let g be the variable assignment for ALω such that
g(x) = x.
Lemma 4.3.6 (Term Lemma). For every term t of Lω and Γ ∈ S, t ∈ JtKALω ,Γ,g .
Proof sketch. By induction on the structure of t, using the semantics of terms in
Definition 4.1.3.2 and interpretation of function symbols in Definition 4.3.4.6.
Lemma 4.3.7 (Truth Lemma). For every formula ϕ of Lω and Γ ∈ S,
ALω , Γ
g
ϕ iff ϕ ∈ Γ.
Proof. As usual, the proof is by induction on the complexity of ϕ, and the most
interesting part is the proof that ALω , Γ g ∀xϕ implies ∀xϕ ∈ Γ, which uses that
Γ is Henkinized, so there is a constant c such that ∃x¬ϕ → ¬ϕxc ∈ Γ:
ALω , Γ
g
∀xϕ ⇒ ALω , Γ
⇒ ALω , Γ
g[x:=c] ϕ by Definition 4.1.6.5
x
g ϕc by Lemmas 4.2.3 and 4.3.6
⇒ ϕxc ∈ Γ by the inductive hypothesis
⇒ ∀xϕ ∈ Γ since ϕxc → ∀xϕ ∈ Γ.
Theorem 4.3.8 (Strong Completeness). (ZF) Every consistent set Γ of formulas
of L is satisfiable in the canonical first-order possibility model for L.
Wesley H. Holliday
Proof. By Lemma 4.3.7, Γ is satisfied at the possibility H(Γ) (recall Lemma 4.3.3)
in ALω and hence at the same possibility in the reduct AL .
Completeness is only the first step in a model theory for first-order languages
based on possibility models. Using his definition of first-order possibility models,
van Benthem [10, 16] defines several operations on models: generated submodels,
disjoint unions, zigzag images, filter products, and filter bases. He then proves
a Keisler-type definability result, stating that a class of possibility models is definable by a set of first-order sentences iff the class is closed under the listed
operations. His notion of filter product is meant to play the role in possibility semantics of ultraproducts in traditional semantics. While taking a product {Ai }i∈I
for an arbitrary I requires the Axiom of Choice, one can handle a countable I
quasiconstructively, as Dependent Choice implies Countable Choice. It remains
to be seen how other concepts and results of traditional first-order model theory
may be developed for possibility models.
4.4
First-order possibility models and forcing
In this section, we briefly explain how special first-order possibilities models are
essentially already implicit in forcing in set theory (cf. [65]). We follow the presentation of forcing using Boolean-valued models in [6].
Assume there is a standard transitive Tarskian model M of ZFC. Suppose
(S, ⊑) is a poset in M , which we will use as the underlying poset of a first-order
possibility model. Next we will add a set D of guises of sets. The basic idea is
this: for a partial possibility s in S and guises a and b of sets, it may be that s
does not settle that
the set of which a is a guise is a member of the set of which b is a guise,
but also s does not settle that this is not the case. Given this idea, we take the
guise b to be a function that outputs, for a given guise a in its domain, a regular
open set b(a) of possibilities such that every s ∈ b(a) settles that the set of which
a is a guise is a member of the set of which b is a guise. Formally, let B be the
BA RO(S, ⊑) in M , and define the following sequence of sets by recursion on α:
(B)
(B)
• Mα is the set of all functions in M whose domain is a subset of Mξ
some ξ < α and whose codomain is B;
• M (B) =
S
α
(B)
Mα .
for
Possibility Semantics
Let the set D of guises be M (B) . Let LD be the language of first-order set theory,
with ∈˙ the membership symbol, expanded with a constant ca for every a ∈ D.
˙ ·) such that
Next we define by a joint recursion22 the functions ≍ and V (∈,
for all possibilities s ∈ S and guises a, b ∈ D:
˙ s′ ),
• a ≍s b iff for all c ∈ dom(a) and s′ ⊑ s, if s′ ∈ a(c), then (c, b) ∈ V (∈,
′
′
′
˙ s );
and for all c ∈ dom(b) and s ⊑ s, if s ∈ b(c), then (c, a) ∈ V (∈,
˙ s) iff ∀s′ ⊑ s ∃s′′ ⊑ s′ ∃c ∈ dom(b): s′′ ∈ b(c) and a ≍s′′ c.
• (a, b) ∈ V (∈,
Finally, we interpret the constants in the obvious way:
• b ∈ V (ca , s) iff a ≍s b.
Proposition 4.4.1. The tuple (S, ⊑, D, ≍, V ) constructed above is a first-order
possibility model for LD with total functions.
Proof. First observe that for each s ∈ S, ≍s is an equivalence relation.
Persistence for ≍ is immediate from the definition of ≍ due to the quantifica˙ suppose (a, b) ∈ V (∈,
˙ s), u ⊑ s, a ≍u a∗ ,
tion over all s′ ⊑ s. For persistence for ∈,
˙ s), we have (i) ∀s′ ⊑ s ∃s′′ ⊑ s′ ∃c ∈ dom(b):
and b ≍u b∗ . From (a, b) ∈ V (∈,
′′
˙ u), let u′ ⊑ u. Then u′ ⊑ s, so by
s ∈ b(c) and a ≍s′′ c. To show (a∗ , b∗ ) ∈ V (∈,
′′
′
′′
(i), there are u ⊑ u and c ∈ dom(b) with u ∈ b(c) and a ≍u′′ c. Since b ≍u b∗
and u′′ ⊑ u, we have b ≍u′′ b∗ by persistence for ≍, which with u′′ ∈ b(c) implies
˙ u′′ ). By definition of V (∈,
˙ u′′ ), it follows that for some u′′′ ⊑ u′′ and
(c, b∗ ) ∈ V (∈,
d ∈ dom(b∗ ), we have u′′′ ∈ b∗ (d) and c ≍u′′′ d. Since a ≍u a∗ , a ≍u′′ c, c ≍u′′′ d,
and u′′′ ⊑ u′′ ⊑ u, we have a∗ ≍u′′′ d by persistence for ≍ and the fact that ≍u′′′ is
an equivalence relation. Thus, we have shown that for all u′ ⊑ u there is a u′′′ ⊑ u′
˙ u).
and d ∈ dom(b∗ ) such that u′′′ ∈ b∗ (d) and a∗ ≍u′′′ d. Hence (a∗ , b∗ ) ∈ V (∈,
˙ s) due to the ∀s′ ⊑ s
Refinability for ∈˙ is immediate from the definition of V (∈,
∃s′′ ⊑ s′ quantification pattern. For refinability for ≍, suppose a 6≍s b. Without
loss of generality, suppose there is a c ∈ dom(a) and s′ ⊑ s such that s′ ∈ a(c)
˙ s′ ). Then by refinability for ∈,
˙ there is an s′′ ⊑ s′ such that for
but (c, b) 6∈ V (∈,
˙ s′′′ ). From s′ ∈ a(c) and s′′′ ⊑ s′ , we have s′′′ ∈ a(c),
all s′′′ ⊑ s′′ , (c, b) 6∈ V (∈,
˙ s′′′ ) implies a 6≍s′′′ b. Thus, we have an s′′ ⊑ s such that
which with (c, b) 6∈ V (∈,
′′′
′′
for all s ⊑ s , a 6≍s′′′ b, which establishes refinability for ≍.
Persistence, quasi-functionality, and totality for constant symbols follows from
the properties of ≍.
One may now verify that possibility semantics for LD using (S, ⊑, D, ≍, V )
results in the same semantic values for formulas in the Boolean algebra RO(S, ⊑)
(recall Lemma 4.2.1) as the algebraic semantics for LD in [6, Ch. 1].
22
To make this rigorous in terms of recursion on an appropriate well-founded relation, see [6,
p. 23].
Wesley H. Holliday
5
Modal case
In this section, we cover possibility semantics for modal logic. We assume some
previous familiarity with possible world semantics using Kripke frames [37, 30]
and neighborhood frames [140], to which we will compare possibility semantics.
We will consider four ways of interpreting the modal language using possibility
frames:
1. (§ 5.1) Interpret ✷ as the universal modality in possibility frames. When
combined with the device of propositional quantification, this simple semantics already allows us to characterize naturally occurring logics that cannot
be characterized by Kripke frames.
2. (§ 5.2) Add neighborhood functions to possibility frames to interpret arbitrary congruential modalities. This allow us to characterize simple modal
logics (without propositional quantification) that cannot be characterized
by possible world frames with neighborhood functions.
3. (§ 5.3) Add accessibility relations to possibility frames to interpret normal
modalities. This was Humberstone’s [104] original setting, though our definition of relational possibility frames is more general. Again this allows
us to characterize modal logics (without propositional quantification) that
cannot be characterized by Kripke frames or neighborhood world frames.
4. (§ 5.4) Add “accessibility functions” to possibility frames to interpret normal
modalities: a modal formula ✷ϕ is true at a possibility x iff ϕ is true at the
functionally determined possibility f (x). While this functional semantics is
severely limited when paired with possible world frames, where f (x) must
be a complete world, it is quite general when paired with possibility frames.
After covering these four ways of interpreting modalities in the context of
propositional modal logic, in § 5.5 we briefly consider first-order modal logic,
extending the first-order possibility frames of § 4 with accessibility relations as in
§ 5.3. We show there are philosophically controversial first-order modal inferences
valid over possible world frames but invalid over possibility frames.
5.1
Universal modality and propositional quantification
The language L of unimodal propositional logic is defined by the grammar
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷ϕ
Possibility Semantics
where p belongs to a countably infinite set Prop of propositional variables. We
define ∨, →, and ↔ in terms of ¬ and ∧ as usual, and
✸ϕ := ¬✷¬ϕ.
A simple possibility semantics for this language using posets (S, ⊑) with ✷ interpreted as the universal modality (see Definition 5.1.1 below) provides a semantics
for the logic S5, the smallest normal modal logic (see Definition 5.3.1) containing
✷p → p, ✷p → ✷✷p, ¬✷p → ✷¬✷p,
which is a standard logic for the “necessity” interpretation of ✷ in philosophy and
the “knowledge” interpretation of ✷ in computer science and game theory. As
S5 is already complete with respect to sets of worlds with ✷ interpreted as the
universal modality, so far there is no gain concerning completeness. However, as
we will see, if we enrich the language sufficiently, then there are extensions of S5
that are incomplete with respect to a semantics based on worlds but complete
with respect to possibility semantics.
Consider the language LΠ that extends L with propositional quantifiers:
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷ϕ | ∀pϕ
where p ∈ Prop. We define ∃pϕ := ¬∀p¬ϕ. From a syntactic perspective, perhaps
the most natural extension of S5 with principles for propositional quantification
is the logic S5Π studied by Bull [35] and Fine [60], which extends the axioms and
rules of S5 with the following axioms and rule for the propositional quantifiers:
• Universal distribution axiom: ∀p(ϕ → ψ) → (∀pϕ → ∀pψ).
• Universal instantiation axiom: ∀pϕ → ϕpψ where ψ is substitutable for p
in ϕ, and ϕpψ is the result of replacing all free occurrences of p in ϕ by ψ
(defining substitutability and free occurences of propositional variables just
as we do for individual variables in first-order logic [54]).
• Vacuous quantification axiom: ϕ → ∀pϕ where p is not free in ϕ.
• Rule of universal generalization: if ϕ is a theorem, then ∀pϕ is a theorem.
The set of theorems of S5Π is the smallest set of sentences of LΠ containing the
stated axioms and closed under the stated rules.
A simple possibility semantics for the logic S5Π is as follows.
Definition 5.1.1. For a full possibility frame F = (S, ⊑, RO(S, ⊑)), a possibility
model based on F is a pair M = (F, π) such that π : Prop → RO(S, ⊑). For
ϕ ∈ LΠ and x ∈ S, we define the forcing relation M, x ϕ recursively as follows:
Wesley H. Holliday
1. M, x
p iff x ∈ π(p);
2. M, x
¬ϕ iff for all x′ ⊑ x, M, x′ 1 ϕ;
3. M, x
(ϕ ∧ ψ) iff M, x
4. M, x
✷ϕ iff for all y ∈ S, M, y
5. M, x
∀pϕ iff for all M′ ∼p M, M′ , x
ϕ and M, x
ψ;
ϕ;
ϕ,
where M′ ∼p M means that M′ is a possibility model based on the same F as
M whose valuation π ′ differs from the valuation π in M at most at p.
A formula ϕ is valid on F if for every model M based on F and every x ∈ S,
M, x ϕ; and ϕ is valid on a class K of full possibility frames if it is valid on
every F in K. The logic of K is the set of all formulas valid on K.
Remark 5.1.2. Algebraically, we are interpreting LΠ in a complete Boolean algebra, namely RO(S, ⊑), taking the semantic value of ✷ϕ to be 1 if the value of ϕ
is 1, and 0 otherwise, and taking the semantic value of ∀pϕ to be the meet of all
values of ϕ as we vary the value of p to be any element of the algebra (cf. [97]).
Remark 5.1.3. We can also define models based on arbitrary—not only full—
possibility frames. For a frame F = (S, ⊑, P ), a possibility model based on F is a
pair M = (F, π) with π : Prop → P . However, for interpreting LΠ, we restrict
attention to full possibility frames so that P = RO(S, ⊑) and hence the semantic
value of ∀pϕ belongs to P by Remark 5.1.2. This is used to prove the validity of
the universal instantiation axiom for ψ containing propositional quantifiers.
The derived semantic clauses for ∨, →, ↔, and ∃ are analogous to those in
Lemma 4.2.2. In addition, we have
M, x
✸ϕ iff for some y ∈ S, M, y
ϕ.
Recall that a world frame is a possibility frame in which ⊑ is the identity
relation (Definition 3.3.1).
Proposition 5.1.4. S5Π is incomplete with respect to the class of all full world
frames, as
∃q(q ∧ ∀p(p → ✷(q → p)))
(W)
is valid on every full world frame but is not a theorem of S5Π.
Proof. Suppose M = (F, π) is a model based on a world frame F, and w is a world
in F. Since F is a world frame, we have M, w (W) iff there is some M′ ∼q M
Possibility Semantics
such that M′ , w q ∧ ∀p(p → ✷(q → p)). To see that the right-hand side holds,
let M′ = (F, π ′ ) be such that π ′ differs from π only in that π ′ (q) = {w}.
To see that (W) is not a theorem of S5Π, first it is easy to check that S5Π
is valid on any full possibility frame. Second, we observe that (W) is refuted by
the full possibility frame based on the full infinite binary tree in Examples 3.1.1
and 3.3.3. For any possibility x and formula ϕ, we have M, x ∃qϕ iff for all
x′ ⊑ x there is x′′ ⊑ x′ and M′ ∼q M such that M′ , x′′ ϕ. But for any x′ ⊑ x,
x′′ ⊑ x′ , and regular open set U chosen to interpret q in M′ , we claim that
M′ , x′′ 1 q ∧ ∀p(p → ✷(q → p)).
Assume x′′ ∈ U , so the first conjunct is true. Where y is a child of x′′ , interpret p
as the regular open set ↓y ( U . Then p → ✷(q → p) is not true at x′′ , since the
antecedent is true at y ⊑ x′′ but the consequent is not true at any possibility.
Stated algebraically, (W) is valid on any complete and atomic Boolean algebra.
Remark 5.1.5. More generally, if instead of interpreting LΠ in complete Boolean
algebras as in Remark 5.1.2, we interpret LΠ in complete Boolean algebra expansions, defined in the next section (Definition 5.2.1), then (W) is valid on any
complete and atomic Boolean algebra expansion in which ✷⊤ is valid.
Although S5Π is incomplete with respect to possible world semantics, it is
complete with respect to possibility semantics in virtue of its completeness with
respect to complete Boolean algebras, shown in [97], and the fact that we can
transform any complete Boolean algebra refuting a formula into a possibility frame
refuting the formula using Theorem 3.2.3.2.
Theorem 5.1.6 ([97]). S5Π is the logic of the class of all full possibility frames.
Theorem 5.1.7 ([45]). There are infinitely many normal extensions L of S5Π
such that L is the logic of a class of full possibility frames but not of any class of
full world frames.
Below we will see that when we consider modal logics weaker than S5, there
are many open problems concerning propositional quantification.
Remark 5.1.8. The approach to propositional quantification in the references
above relies on the completeness of the Boolean algebra of propositions for the
interpretation of ∀ and ∃. For a way of interpreting propositional quantification
in possibly incomplete algebras, see [100].
Wesley H. Holliday
5.2
Neighborhood frames
To go beyond semantics for the universal modality and cover all manner of other
modalities, in this section we introduce the possibility semantic version of a very
general style of modal semantics known as neighborhood semantics [140], due to
Scott [152] and Montague [137]. We give neighborhood possibility semantics for
a language containing a family of modalities indexed by some nonempty set I.
The grammar of the polymodal language L(I) is given by
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷i ϕ
where p ∈ Prop and i ∈ I. L(I)Π adds the clause for ∀pϕ. When I is a singleton
set, identify L from § 5.1 with L(I).
The language L(I) has a straightforward interpretation in Boolean algebras
equipped with functions interpreting the modalities.
Definition 5.2.1. A Boolean algebra expansion (BAE) is a pair B = (B, {fi }i∈I )
where B is a Boolean algebra and fi : B → B. A valuation on B is a function
θ : Prop → B, which extends to an L(I)-valuation θ̃ : L(I) → B by: θ̃(p) = θ(p);
θ̃(¬ϕ) = ¬θ̃(ϕ); θ̃(ϕ ∧ ψ) = θ̃(ϕ) ∧ θ̃(ψ); and θ̃(✷i ϕ) = fi (θ̃). We say B validates
a formula ϕ if for every valuation θ on B, θ̃(ϕ) = 1B where 1B is the top element
of B; and a class of BAEs validates ϕ if every BAE in the class validates ϕ.
The set of formulas validated by any class of BAEs is a congruential modal logic.
Definition 5.2.2. A congruential modal logic for L(I) is a set L of formulas
satisfying the following conditions:
1. L contains every tautology of classical propositional logic;
2. rule of uniform substitution: if ϕ ∈ L and ϕ′ is obtained from ϕ by uniformly
substituting formulas for propositional variables, then ϕ′ ∈ L;
3. rule of modus ponens: if ϕ → ψ ∈ L and ϕ ∈ L, then ψ ∈ L;
4. congruence rule: if ϕ ↔ ψ ∈ L, then ✷i ϕ ↔ ✷i ψ ∈ L.
5.2.1
Basic frames
The first step toward neighborhood possibility semantics is to add to a poset a
collection of neighborhood functions for interpreting the modalities ✷i . Recall
that in possible world semantics, a neighborhood frame is a pair (W, {Ni }i∈I )
where W is a nonempty set and Ni is a function assigning to each w ∈ W a set
Possibility Semantics
Ni (w) of propositions, i.e., of subsets of W (see [140]). The modal operation ✷i
is defined as follows: given a proposition U , a world w belongs to the proposition
✷i U iff U belongs to N (w). Now we will replace W by a poset of possibilities
and require that Ni assigns to each possibility a set of propositions in the sense
of possibility semantics, i.e., regular open subsets of the poset.
Definition 5.2.3. A neighborhood possibility foundation is a triple
F = (S, ⊑, {Ni }i∈I ) such that:
1. (S, ⊑) is a poset;
2. Ni : S → ℘(RO(S, ⊑)).
For i ∈ I and U ∈ RO(S, ⊑), we define
✷Ni U = {x ∈ S | U ∈ Ni (x)}.
These structures are not yet neighborhood possibility frames, because without
some interaction conditions relating ⊑ and Ni , there is no guarantee that ✷Ni U ∈
RO(S, ⊑). The following conditions are necessary and sufficient to guarantee that
for all U ∈ RO(S, ⊑), we have ✷Ni U ∈ RO(S, ⊑):
• Ni -persistence: if x′ ⊑ x, then Ni (x′ ) ⊇ Ni (x);
• Ni -refinability: if U 6∈ Ni (x), then ∃x′ ⊑ x ∀x′′ ⊑ x′ U 6∈ Ni (x′′ ).
Proposition 5.2.4. For any foundation F = (S, ⊑, {Ni }i∈I ), the following are
equivalent:
1. RO(S, ⊑) is closed under ✷Ni ;
2. F satisfies Ni -persistence and Ni -refinability for each i ∈ I.
Proof. The claim that for all U ∈ RO(S, ⊑), ✷Ni U satisfies persistence and refinability (recall § 3.2) is clearly equivalent to Ni -persistence and Ni -refinability.
Now if we only wish to represent complete BAEs, i.e., BAEs whose underlying
Boolean algebras are complete, then there is no need for the distinguished family
P of admissible sets in a possibility frame (recall Definition 3.3.1), so it suffices to
work with the following “basic” frames that simply expand posets with appropriate
neighborhood functions.
Wesley H. Holliday
Definition 5.2.5. A basic neighborhood possibility frame is a neighborhood possibility foundation F = (S, ⊑, {Ni }i∈I ) satisfying, for each i ∈ I, Ni -persistence
and Ni -refinability.
A basic neighborhood world frame is a neighborhood possibility foundation in
which ⊑ is the identity relation.
Proposition 5.2.6. For any basic neighborhood possibility frame F , the pair
F b = (RO(S, ⊑), {✷Ni }i∈I ) is a complete BAE.
Proof. Follows from Theorem 3.2.3.1 and Proposition 5.2.4.
Remark 5.2.7. Basic neighborhood world frames may be identified with the standard “neighborhood frames” of possible world semantics mentioned at the beginning of this section. The key point about these world frames is that they can
realize only atomic algebras as F b .
Example 5.2.8. This example is inspired by the theory of imprecise probability
[33]. Fix a measurable space (W, Σ), i.e., W is a nonempty set and Σ is an algebra
of subsets of W closed under countable unions. A set P of probability measures on
(W, Σ) is convex if for all µ1 , µ2 ∈ P and α ∈ [0, 1], we have αµ1 + (1 − α)µ2 ∈ P,
where αµ1 + (1 − α)µ2 is the probability measure defined for each A ∈ Σ by
(αµ1 +(1−α)µ2 )(A) = αµ1 (A)+(1−α)µ2 (A). We call a convex set P of measures
everywhere imprecise if for all A ∈ Σ, there are µ, µ′ ∈ P with µ(A) 6= µ′ (A). Let
S be the set of all pairs (w, P) where w ∈ W and P is an everywhere imprecise
convex set of measures on (W, Σ). We think of w as the state of the world and P
as an imprecise representation of an agent’s uncertainty, which could be further
refined, though never to the point of providing precise subjective probabilities.
Thus, let (w′ , P ′ ) ⊑ (w, P) iff w′ = w and P ′ ⊆ P. Hence (S, ⊑) is a poset. For
A ∈ Σ, let ψ(A) = {(w, P) ∈ S | w ∈ A}. Observe that ψ is a Boolean embedding
of Σ into RO(S, ⊑). For each r ∈ [0, 1], we define a neighborhood function by
N≥r (w, P) = {ψ(A) | for all µ ∈ P, µ(A) ≥ r}.
We claim that N≥r satisfies the two conditions of Proposition 5.2.4.2:
• N≥r -persistence: suppose (w, P ′ ) ⊑ (w, P) and ψ(A) ∈ N≥r (w, P). Hence
for all µ ∈ P, we have µ(A) ≥ r. Since (w, P ′ ) ⊑ (w, P), we have P ′ ⊆ P,
so for all µ ∈ P ′ , we have µ(A) ≥ r. Thus, ψ(A) ∈ N≥r (w, P ′ ).
• N≥r -refinability: suppose ψ(A) 6∈ N≥r (w, P), so there is some µ ∈ P such
that µ(A) < r. Then we claim that the set P ′ = {ν ∈ P | ν(A) < r} is
an everywhere imprecise convex set of probability measures. For convexity,
Possibility Semantics
if ν1 , ν2 ∈ P ′ , then αν1 (A) + (1 − α)ν2 (A) < αr + (1 − α)r = r, so that
(αν1 +(1−α)ν2 )(A) ∈ P ′ . For everywhere imprecision, since P is everywhere
imprecise, for each B ∈ Σ, there are µ1 , µ2 ∈ P with µ1 (B) 6= µ2 (B). Hence
for some i ∈ {1, 2}, µ(B) 6= µi (B). Since µ(A) < r, taking α sufficiently
large but less than 1, αµ(A) + (1 − α)µi (A) < r. Thus, αµ + (1 − α)µi ∈ P ′ .
Moreover, µi (B) 6= µ(B) and α < 1 together imply that µ(B) 6= αµ(B)+(1−
α)µi (B) = (αµ + (1 − α)µi )(B). It follows that P ′ is everywhere imprecise.
Thus, (w, P ′ ) ∈ S and (w, P ′ ) ⊑ (w, P). Moreover, for all (w, P ′′ ) ⊑ (w, P ′ ),
(w, P ′′ ) 6∈ N≥r (w, P ′′ ). This establishes N≥r -refinability.
Hence (S, ⊑, {N≥r }r∈[0,1] ) is a basic neighborhood possibility frame. The modality
✷≥r associated with N≥r , applied to the image ψ(A) of an event A ∈ Σ, expresses
the proposition that the probability of A is settled to be at least r.23 Similarly,
we can define neighborhood functions N≤r and modalities ✷≤r for expressing that
the probability of A is settled to be at most r. For logics with such modalities
interpreted using possible world semantics, see [44, § 4.1].
For a converse of Proposition 5.2.6, we can go from any complete BAE to a
basic neighborhood possibility frame by extending Theorems 3.2.3.2.
Theorem 5.2.9. For any complete BAE B = (B, {fi }i∈I ), define
Bn = (B+ , ≤+ , {Ni }i∈I )
where (B+ , ≤+ ) is as in Theorem 3.2.3.2 and for x ∈ B+ and i ∈ I,
_
Ni (x) = {U ∈ RO(B+ , ≤+ ) | x ≤+ fi (
U )}.
Then:
1. Bn is a basic neighborhood possibility frame;
2. (Bn )b is isomorphic to B;
3. a basic frame F is isomorphic to (F b )n iff the underlying poset of F is
obtained from a complete Boolean lattice by deleting its bottom element.
Proof. For part 1, Ni -persistence is immediate from the definition of Ni . For
W
W
Ni -refinability, if U 6∈ Ni (x), so x 6≤+ fi ( U ), then where x′ = x ∧ ¬fi ( U ),
W
we have that for all x′′ ≤+ x′ , x′′ 6≤+ fi ( U ), so U 6∈ Ni (x′′ ). The proof of part
23
This model does not attribute to the agent subjective probabilities for higher-order propositions about her own uncertainty, i.e., propositions in RO(S, ⊑) not of the form ψ(A) for A ∈ Σ,
but richer models representing uncertainty about one’s own uncertainty can also be defined.
Wesley H. Holliday
2 extends that of Theorem 3.2.3.2, using the map ϕ : B → RO(B+ , ≤+ ) given by
ϕ(b) = ↓+ b := {b′ ∈ B+ | b′ ≤+ b}. Observe that
↓+ fi (b) = {b′ ∈ B+ | b′ ≤+ fi (b)}
_
= {b′ ∈ B+ | b′ ≤+ fi (
↓+ b)}
= {b′ ∈ B+ | ↓+ b ∈ Ni (b′ )}
= ✷Ni ↓+ b.
For part 3, the left-to-right direction is by the definition of (·)n . For the rightto-left direction, the poset reduct of F = (S, ⊑, {Ni }i∈I ) is isomorphic to the
poset reduct of (F b )n by Theorem 3.2.3.2. The isomorphism ψ : S → RO(S, ⊑)+
is given by ψ(x) = ↓x := {x′ ∈ S | x′ ⊑ x}. Finally, we claim that for all
(F b )n
U ∈ RO(S, ⊑), we have U ∈ NiF (x) iff ψ[U ] ∈ Ni
b
W
iff ↓x ⊆ ✷i U iff ↓x ≤F+ ✷i (
Fb
(ψ(x)). Indeed, U ∈ NiF (x)
(F b )
ψ[U ]) iff ψ[U ] ∈ Ni
n
(ψ(x)).
When constructing neighborhood frames, we will typically not build frames
satisfying the condition of part 3. The whole point of using neighborhood possibility frames, as opposed to just working with complete BAEs, is that we can use
simple posets—such as trees—equipped with neighborhood functions to realize a
BAE that may be less intuitive to define algebraically and reason about directly.
If we already know how to realize a complete Boolean algebra B as the regular
open sets of a nice poset, then we can apply the following.
Proposition 5.2.10. Suppose B is a complete Boolean algebra isomorphic to
RO(S, ⊑) for some poset (S, ⊑). Then for any BAE B = (B, {fi }i∈I ), there is a
basic neighborhood possibility frame B(S,⊑) = (S, ⊑, {Ni }i∈I ) such that (B(S,⊑) )b
is isomorphic to B.
Proof. We use an isomorphism σ from RO(S, ⊑) to B to define neighborhood
functions on S: for any x ∈ S and Z ∈ RO(S, ⊑), set
Z ∈ Ni (x) iff x ∈ σ −1 (fi (σ(Z))).
Then it is easy to check that Ni -persistence and Ni -refinability hold, and
(S, ⊑, {Ni }i∈I )b is isomorphic to (B, {fi }i∈I ).
Remark 5.2.11. Theorem 5.2.9 can be extended to an obvious categorical duality
when morphisms are defined (see Definition 5.2.37), but we omit the details. See
§ 5.3.4 for a more interesting duality for relational possibility frames.
The term “neighborhood frame” comes from the topological notion of a neighborhood system on a set S, which is a function N : S → ℘(℘(S)) satisfying the
following conditions for all x ∈ S and U, V ⊆ S [111, p. 56]:
Possibility Semantics
1. filter condition: N (x) 6= ∅, and U ∩ V ∈ N (x) iff U ∈ N (x) and V ∈ N (x);
2. neighborhood of a point condition: if U ∈ N (x), then x ∈ U ;
3. open neighborhood condition: for each U ∈ N (x), there is a V ∈ N (x) such
that V ⊆ U and for all y ∈ V , V ∈ N (y).
Recall the correspondence between neighborhood systems on a set and topological
spaces on a set: given a neighborhood system, define an open set to be a set
which is a neighborhood of each of its points; given a topological space, define a
neighborhood of a point to be a superset of an open set containing the point.
Definition 5.2.12. A neighborhood possibility system is a basic neighborhood
possibility frame F = (S, ⊑, N ) such that N satisfies conditions 1, 2, and 3 of a
neighborhood system for every U, V ∈ RO(S, ⊑).
Recall that an interior algebra [134, 146, 31, 55] is a pair (B, ✷) where B is a
Boolean algebra and ✷ is an operation on B such that for all a, b ∈ B:
✷a ∧ ✷b = ✷(a ∧ b), ✷a ≤ a, ✷a ≤ ✷✷a.
Recall the correspondence between topological spaces and complete and atomic
interior algebras, i.e., whose underlying BA B is complete and atomic: given a
topological space (S, τ ), we form a complete and atomic interior algebra (℘(S), int)
where int is the interior operation in the space (S, τ ); given a complete and atomic
interior algebra, we represent the BA as a powerset and define a topology on the
underlying set by taking the opens to be the fixpoints of the interior operation ✷.
By contrast, neighborhood possibility systems correspond to complete interior
algebras, i.e., whose underlying BA B is complete but not necessarily atomic.
Proposition 5.2.13. For any basic neighborhood possibility frame
F = (S, ⊑, N ), the following are equivalent:
1. F is a neighborhood possibility system;
2. F b is a complete interior algebra.
Moreover, for any complete interior algebra B = (B, ✷), Bn is a neighborhood
possibility system.
Proof sketch. It is easy to check that F (resp. Bn ) satisfying the first (resp. second,
third) condition of neighborhood possibility systems corresponds to F b (resp. B)
satisfying the first (resp. second third) defining equation of interior algebras.
Wesley H. Holliday
Thus, neighborhood possibility systems can be regarded as a generalization of
neighborhood systems and hence as a generalization of topological spaces (for
investigation of interior algebras that are not necessarily atomic, see [139]). This
generalization of topological spaces can be compared with another generalization
of topological spaces from pointfree topology [144]: locales, defined as complete
lattices in which binary meet distributes over arbitrary joins (see § 6.3 for another
characterization of locales as complete Heyting algebras). The set {a ∈ B | a =
✷a} of “open elements” of any complete interior algebra (B, ✷) forms a locale
[135], and every locale can be represented in this way,24 it follows that the set
{U ∈ RO(S, ⊑) | U = ✷N U }
of “open sets” of any neighborhood possibility system forms a locale, and every
locale can be represented in this way.
Theorem 5.2.14. A lattice L is a locale iff L is isomorphic to the lattice of open
sets of a neighborhood possibility system.
In order to discuss logics that are incomplete with respect to neighborhood
world semantics but complete with respect to neighborhood possibility semantics,
let us give the official formal semantics for L(I).
Definition 5.2.15. Given a basic neighborhood frame F = (S, ⊑, {Ni }i∈I ), a
neighborhood possibility model based on F is a pair M = (F, π) where π : Prop →
RO(S, ⊑). For ϕ ∈ L(I)Π and x ∈ S, we define M, x ϕ and JϕKM = {y ∈ S |
M, y ϕ} as in Definition 5.1.1 but with the following new clause for ✷i :
• M, x
✷i ϕ iff JϕKM ∈ Ni (x).
A formula ϕ is valid on the frame F if for every model M based on F and every
x ∈ S, M, x ϕ; and ϕ is valid on a class K of frames if it is valid on every frame
in K. The L(I)-logic (resp. L(I)Π-logic) of K is the set of all formulas valid on K.
Proposition 5.2.16. The L(I)-logic of any class of basic neighborhood possibility
frames is a congruential modal logic.
Proof. Follows from Proposition 5.2.6.
24
It is stated in [73, p. 318] as open whether every locale can be represented as the locale
of open elements of a complete interior algebra. However, as Guram Bezhanishvili (personal
communication)
observed, if L is a locale, then Funayama’s Theorem (see, e.g., [22]) yields a
W
localic (∧, )-embedding of L into a complete Boolean algebra B. Then the image of L is a
relatively complete sublattice of B, so it defines an interior operation ✷ on B whose algebra
of fixpoints is isomorphic to L. Thus, L is isomorphic to the algebra of open elements of the
complete interior algebra (B, ✷).
Possibility Semantics
As in § 5.1, we can easily detect the difference between world frames and possibility frames using propositional quantification, but now below S5 using neighborhood functions. Consider, for example, the standard logic of introspective belief,
KD45, which is the smallest normal modal logic containing the axioms
¬✷⊥, ✷p → ✷✷p, ¬✷p → ✷¬✷p.
KD45 is valid on a basic neighborhood possibility frame F = (S, ⊑, N ) iff N
satisfies the filter condition and open neighborhood conditions of a neighborhood
possibility system, plus the following for all x ∈ S:
• proper condition: ∅ 6∈ N (x);
• for each U 6∈ N (x), there is a V ∈ N (x) such that for all y ∈ V , U 6∈ N (y).
In fact, we will see below that simpler frames suffice for completeness.
Definition 5.2.17. A uniform normal neighborhood possibility frame is a basic
neighborhood possibility frame F = (S, ⊑, N ) such that N satisfies the filter
condition and proper condition, and for all x, y ∈ S, N (x) = N (y).
Let KD4∀ 5Π [46] be the extension of KD45 with the Π principles for propositional quantification from § 5.1, plus the 4∀ axiom:
∀p✷ϕ → ✷∀p✷ϕ.
KD4∀ 5Π is not complete with respect to any class of basic neighborhood world
frames for the same reason that S5Π was not: the formula (W) from § 5.1 is valid
on every such world frame validating KD45 (in fact, every such world frame validating ✷⊤), but it is not a theorem of S5Π, which extends KD4∀ 5Π. By contrast,
Ding [46] proved that KD4∀ 5Π is complete with respect to complete proper filter
algebras, i.e., complete BAs equipped with a proper filter for interpreting ✷ (i.e.,
✷a = 1 if a is in the filter, and otherwise ✷a = 0), as well as sound with respect
to all complete KD45 BAEs. Combining this with Proposition 5.2.6 and Theorem
5.2.9.1-2, we obtain the following.
Theorem 5.2.18 ([46]). KD4∀ 5Π is the logic of all basic neighborhood possibility
frames validating KD45 (resp. uniform normal neighborhood possibility frames),
but KD4∀ 5Π is not the logic of any class of basic neighborhood world frames.
It is open whether the L(I)Π logics of other classes of basic neighborhood
possibility frames are recursively axiomatizable. Let us mention a few salient
instances of the question.
Wesley H. Holliday
Question 5.2.19. Is the LΠ-logic of all neighborhood possibility systems (i.e.,
of complete interior algebras) recursively axiomatizable or enumerable? The logic
of all basic neighborhood possibility frames? The logic of those normal frames in
which each N (x) is a proper filter? The logics of other classes?
Let us now consider an example of a congruential modal logic without propositional quantifiers that is the logic of a class of basic neighborhood possibility
frames but not of any class of basic neighborhood world frames. Consider a bimodal language with modalities we will write as ✷ and Q (instead of ✷1 and ✷2 ).
Let S be the smallest congruential logic for this language containing ✷⊤ and
p → ✸(p ∧ Qp) ∧ ✸(p ∧ ¬Qp) .
(Split)
Remark 5.2.20. As shown in [47, § 7], there is a natural arithmetic interpretation
of (Split) where ✸ is interpreted as consistency in Peano arithmetic and Q is
interpreted as an arithmetic predicate defined by Shavrukov and Visser [154].
Let ES be the smallest congruential logic extending S with the following axioms:
✷p → p, ✷p → ✷✷p, ¬✷p → ✷¬✷p, ✷(p ↔ q) → ✷(Qp ↔ Qq).
Let EST be the smallest congruential modal logic extending ES by the T axiom
for Q: Qp → p. Then EST is sound with respect to the arithmetic interpretation
mentioned in Remark 5.2.20. Yet basic neighborhood world frames cannot give
semantics for these logics.
Proposition 5.2.21. S is not valid on any basic neighborhood world frame.
Proof. Suppose F = hW, N✷ , NQ i validates S. Define a model M = (F, π) such
that for some w ∈ W , π(p) = {w}, so M, w p. Then since F validates (Split),
we have M, w ✸(p ∧ Qp) ∧ ✸(p ∧ ¬Qp), i.e.,
J¬(p ∧ Qp)KM 6∈ N✷ (w) and J¬(p ∧ ¬Qp)KM 6∈ N✷ (w).
Since π(p) is a singleton set, either Jp ∧ QpKM = ∅ or Jp ∧ ¬QpKM = ∅, so
J¬(p ∧ Qp)KM = W or J¬(p ∧ ¬Qp)KM = W . Combining the previous two steps,
we have W 6∈ N (w), which contradicts the fact that F validates ✷⊤.
In fact, it is not hard to check the following stronger result.
Proposition 5.2.22 ([47]). S is not valid on any BAE whose underlying BA is
atomic.
Possibility Semantics
By contrast, we have the following positive result when moving from basic
neighborhood world frames to basic neighborhood possibility frames.
Proposition 5.2.23. EST can be validated by a basic neighborhood possibility
frame.
Proof sketch. Consider from Examples 3.1.1-3.3.3 the full infinite binary tree regarded as a poset (S, ⊑). For x ∈ S, let N✷ (x) = {S}. Let Par(x) be the parent
of x in the tree and x0 and x1 the two extensions of x by 0 and 1, respectively.
Define NQ inductively by:
NQ (ǫ) = ∅;
NQ (x0) = NQ (x) ∪ {U ∈ RO(S, ⊑) | x ∈ U, Par(x) 6∈ U };
NQ (x1) = NQ (x).
As shown in [47], (S, ⊑, N✷ , NQ ) is a basic neighborhood possibility frame that
validates EST.
In fact, these logics are not only consistent but also possibility complete.
Theorem 5.2.24 ([47]). Each of the world-inconsistent logics S, ES, and EST is
complete with respect to the class of basic neighborhood possibility frames that
validate the logic.
We conclude with two open problems suggested by the preceding theorem.
Question 5.2.25. Is there a consistent congruential modal logic that is not valid
on any basic neighborhood world frame, or at least not the logic of any class
of such frames, axiomatized with only one modality, one propositional variable,
and modal depth two [47]? What other logics are incomplete with respect to
basic neighborhood world frames but complete with respect to basic neighborhood
possibility frames?
5.2.2
Tree completeness
While Theorem 5.2.9 tells us that any congruential modal logic complete with
respect to complete BAEs is automatically complete with respect to basic neighborhood possibility frames, more interesting would be completeness with respect
to basic neighborhood possibility frames based on special posets such as trees.
Definition 5.2.26. A congruential modal logic L is neighborhood tree complete if
there is a class of basic neighborhood possibility frames, whose underlying posets
are trees, such that L is the logic of the class.
Wesley H. Holliday
The following observation gives us tree completeness for many standard logics.
Proposition 5.2.27 ([47]). If a congruential modal logic L is the logic of a class
of BAEs based on the MacNeille completion of the countable atomless Boolean
algebra, then L is the logic of a class of basic neighborhood possibility frames
based on the tree 2<ω .
Proof. As in Example 3.3.3, RO(2<ω ) is isomorphic to the MacNeille completion
of the countable atomless Boolean algebra. Now apply Proposition 5.2.10.
Taking the MacNeille completion of the Lindenbaum algebra of a logic is a
standard technique for proving the completeness of a logic with respect to complete algebras. In the typical case, the Boolean reduct of the Lindenbaum algebra
is (up to isomorphism) the countable atomless Boolean algebra. Thus, Theorem
5.2.27 assures us that when this standard technique is applicable, we automatically have neighborhood tree completeness as well. For example, given the proof
of Theorem 5.2.24 in [47], Proposition 5.2.27 yields the following.
Theorem 5.2.28 ([47]). Each of the world-inconsistent logics S, ES, and EST is
neighborhood tree complete.
5.2.3
General frames
As basic neighborhood possibility frames can only be used to represent complete
BAEs, it is natural to introduce a more general notion of neighborhood possibility
frames that can be used to represent arbitrary BAEs, as we do below. This is a
straightforward development of the theory of possibility frames in § 3.3.2, but we
state the main definitions and results for the sake of completeness.
Remark 5.2.29. In possible world semantics, Došen’s [49] general frames, equivalent to our general neighborhood world frames below, already allow the representation of arbitrary BAEs and indeed provide a category of frames dually
equivalent to that of BAEs (morphisms are discussed in Definition 5.2.37 below).
Once again an advantage of general possibility frames is that they provide such
representation and categorical duality theorems choice free.
Definition 5.2.30. A (general) neighborhood possibility frame is a tuple
F = (S, ⊑, P, {Ni }i∈I ) such that:
1. (S, ⊑, P ) is a possibility frame as in Definition 3.3.1;
2. Ni : S → ℘(P );
3. for all X ∈ P , ✷Ni X ∈ P .
Possibility Semantics
When P = RO(S, ⊑), we say that F is a full neighborhood possibility frame.
When ⊑ is the identity relation, we say that F is a neighborhood world frame.
Example 5.2.31. Recall from Example 5.2.8 the basic neighborhood possibility
frame (S, ⊑, {N≥r }r∈[0,1] ) for a given measurable space (W, Σ). For A ⊆ W , let
χ(A) = {(w, P) ∈ S | w ∈ A}, and observe that χ is a Boolean embedding of ℘(W )
into RO(S, ⊑). Thus, each element of ℘(W ) yields a proposition in RO(S, ⊑).
Yet if ℘(W ) 6= Σ, then not every element of ℘(W ) is an admissible proposition
in the sense of belonging to Σ. Thus, we may wish to restrict what counts as a
proposition in our neighborhood frame accordingly: let P be the smallest subset
of RO(S, ⊑) that includes {χ(A) | A ∈ Σ} and is closed under ∩, ¬, and ✷N≥ r
for each r ∈ [0, 1]. Then (S, ⊑, {N≥ r}r∈[0,1] , P ) is a general possibility frame. In
this frame, we have a proposition corresponding to each proposition in Σ, plus
propositions generated using the ✷N≥ r modalities and Boolean operations.
One may identify full frames as in Definition 5.2.30 and basic frames as in
Definition 5.2.5, by the following consequence of Proposition 5.2.4.
Proposition 5.2.32. The map (·)♯ sending each basic frame
F = (S, ⊑, {Ni }i∈I )
to its associated full frame
F ♯ = (S, ⊑, RO(S, ⊑), {Ni }i∈I )
is a bijection between the class of basic neighborhood possibility frames (resp. basic neighborhood world frames) and the class of full neighborhood possibility
frames (resp. full neighborhood world frames).
Remark 5.2.33. Thus, we may identify full neighborhood world frames
(W, =, ℘(W ), {Ni }i∈I ) with basic neighborhood world frames and hence the usual
neighborhood frames (W, {Ni }i∈I ) of possible world semantics as in Remark 5.2.7.
Each general frame gives rise to a BAE in the obvious way.
Proposition 5.2.34. Given F = (S, ⊑, P, {Ni }i∈I ), a neighborhood possibility
frame, define F b = (BA(P ), {✷Ni }i∈I ) where BA(P ) is the Boolean subalgebra of
RO(S, ⊑) based on P . Then F b is a BAE.
Thus, we can give formal semantics for L(I) and congruential modal logics using general neighborhood possibility frames, where models interpret propositional
variables as elements of P instead of arbitrary elements of RO(S, ⊑).
We can go in the converse direction from BAEs to frames as follows.
Wesley H. Holliday
Proposition 5.2.35. Given any BAE B = (B, {fi }i∈I ), define
Bf = (Bf , {Ni }i∈I ) and Bg = (Bg , {Ni }i∈I )
where Bf and Bg are as in Proposition 3.3.7, and for F ∈ PropFilt(B),
Then:
b | ✷i a ∈ F }.
Ni (F ) = {a
1. Bf and Bg are neighborhood possibility frames;
2. (Bf )b is (up to isomorphism) the canonical extension of B;
3. (Bg )b is isomorphic to B.
Proof sketch. The proof is an easy extension of the proof of Proposition 3.3.7
taking into account ✷i and Ni .
Frames in the image of the (·)g map can be characterized as in Proposition 3.3.8.
Proposition 5.2.36. Let F = (S, ⊑, P, {Ni }i∈I ) be a neighborhood possibility
frame. Then F is isomorphic to (F b )g iff (S, ⊑, P ) is a filter-descriptive possibility
frame (as in Proposition 3.3.8). We also call such an F filter-descriptive.
To obtain a full categorical duality between filter-descriptive neighborhood
frames and BAEs, we need morphisms that combine conditions from the standard
p-morphisms in modal logic and neighborhood morphisms as in [49].
Definition 5.2.37. Given neighborhood possibility frames F = (S, ⊑, P, {Ni }i∈I )
and F ′ = (S ′ , ⊑′ , P ′ , {Ni′ }i∈I ), a neighborhood p-morphism from F to F ′ is a map
h : S → S ′ satisfying the following conditions for all x, y ∈ S and y ′ ∈ S ′ :
1. for all Z ′ ∈ P ′ , h−1 [Z ′ ] ∈ P ;
2. ⊑-forth: if y ⊑ x, then h(y) ⊑ h(x);
3. ⊑-back: if y ′ ⊑′ h(x), then ∃y: y ⊑ x and h(y) = y ′ ;
4. for all Z ′ ∈ P ′ and i ∈ I, h−1 [Z ′ ] ∈ Ni (x) iff Z ′ ∈ Ni′ (h(x)).
Proposition 5.2.38.
1. Filter-descriptive possibility frames together with neighborhood
p-morphisms form a category, FiltNeighPoss.
Possibility Semantics
2. BAEs with BAE-homomorphisms form a category, BAE.
Proof sketch. For part 1, it is easy to check that the composition of two neighborhood p-morphisms is a neighborhood p-morphism (cf. [96] for the analogous proof
for p-morphisms between relational possibility frames and [49] for p-morphisms
between neighborhood world frames). Part 2 is well known (cf. [49]).
Theorem 5.2.39. (ZF) FiltNeighPoss is dually equivalent to BAE.
Proof sketch. The proof is an easy extension of the proof of Theorem 3.3.12 taking
into account the neighborhood functions and congruential modalities.
5.3
Relational frames
In this section, we narrow our focus from semantics for arbitrary congruential
modal logics, as in § 5.2, to semantics for normal modal logics in particular.
Definition 5.3.1. A normal modal logic is a congruential modal logic L as in
Definition 5.2.2 satisfying the following conditions:
1. (✷p ∧ ✷q) ↔ ✷(p ∧ q) ∈ L;
2. necessitation rule: if ϕ ∈ L, then ✷i ϕ ∈ L.
If we restrict attention to normal modal logics, we can use a possibility-semantic
version of a very intuitive style of modal semantics known as relational semantics
(cf. [30]). In this section—the longest of the chapter—we cover a number of
aspects of relational possibility semantics:
In § 5.3.1, we define relational possibility frames and discuss the key issue
of the interaction between a modal accessibility relation R and the refinement
relation ⊑. Every Kripke frame gives rise to a possibility frame (by taking ⊑ to
be the identity relation on worlds), but the converse does not hold, since possibility
frames can give rise to non-atomic modal algebras.
In § 5.3.2, we briefly focus on models—frames with a fixed valuation of propositional variables—and show how possibility models can be turned into Kripke
models. We then return to our focus on frames for the rest of the section.
In §§ 5.3.3-5.3.4, we relate relational possibility frames to Boolean algebras
with ✷i operations that distribute over arbitrary, not only finite, conjunctions
(meets). The moral is that from an algebraic point of view, the essence of relational semantics for modal logic in its most basic form (i.e., before the step to
general frames or topological frames that can realize all modal algebras) is the
distribution of ✷i over arbitrary conjunctions—or dually, the distribution of ✸i
Wesley H. Holliday
over arbitrary disjunctions—not the atomicity that comes from assuming that
accessibility relations relate possible worlds.
In §§ 5.3.5, we briefly introduce possibility semantics for quasi-normal modal
logics, a generalization of normal modal logics not requiring the necessitation rule
(see Definition 5.3.1), which have important applications in provability logic [32].
Provability logic is also the theme of §§ 5.3.6-5.3.7. In § 5.3.6, we use examples
inspired by provability logic to illustrate the greater generality of full relational
possibility frames over Kripke frames, by identifying provability-logical principles
that cannot be validated by the latter frames but can by the former frames. Then
in § 5.3.7, we use possibility semantics to deepen a well-known Kripke incompleteness result for bimodal provability logic, showing that this incompleteness
does not depend on the atomicity of the dual algebras of Kripke frames and arises
only from the assumption that ✷i distributes over arbitrary conjunctions. Thus,
possibility semantics can be used both to overcome some incompleteness results
(§ 5.3.6) and to strengthen other incompleteness results (§ 5.3.7).
Finally, in § 5.3.8, we briefly sketch the theory of filter-descriptive relational
possibility frames, the relational analogue of the filter-descriptive neighborhood
possibility frames of § 5.2.3, extending the theory of filter-descriptive (non-modal)
possibility frames from § 3.3.2. These filter-descriptive relational frames provide
a fully general, choice-free relational semantics for normal modal logics.
Before introducing relational possibility frames for normal modal logics in
§ 5.3.1, we first recall the relevant algebras for such logics.
Definition 5.3.2. A Boolean algebra with (unary) operators (BAOs) is a BAE
B = (B, {✷i }i∈I ) as in Definition 5.2.1 in which each ✷i distributes over all finite
meets.
Remark 5.3.3. The term ‘operator’ usually refers to the ✸i operation that distributes over all finite joins, whereas the ✷i operation that distributes over all
finite meets is called the ‘dual operator’. It turns out to be convenient for us to
take the ✷i operation as primitive and ✸i defined by ✸i a := ¬✷i ¬a.
5.3.1
Accessibility relations
The restriction to normal modal logics allows us to replace neighborhood functions Ni with more easily visualized accessibility relations Ri . As in § 5.2.1, we
could first define relational possibility foundations, then basic relational possibility frames, and finally general relational possibility frames—but instead we will
cut straight to the most general notion.
Definition 5.3.4. A (general) relational possibility frame is a tuple
F = (S, ⊑, P, {Ri }i∈I ) such that:
Possibility Semantics
1. (S, ⊑, P ) is a possibility frame as in Definition 3.3.1;
2. Ri is a binary relation on S;
3. for all Z ∈ P , we have ✷i Z ∈ P , where
✷i Z = {x ∈ S | Ri (x) ⊆ Z} with Ri (x) = {y ∈ S | xRi y}.
When P = RO(S, ⊑), we say that F is a full relational possibility frame. When
⊑ is the identity relation, we say that F is a relational world frame.
A possibility model based on F is a pair M = (F, π) where π : Prop → P . For
ϕ ∈ L(I) and x ∈ S, we define M, x ϕ recursively as in Definition 5.1.1 but
with the following new clause for ✷i :
• M, x
✷i ϕ iff for all y ∈ S, if xRi y then M, y
ϕ.
Validity of a formula on a frame or class of frames is defined as in Definition 5.1.1,
as is the logic of a class of frames.
Proposition 5.3.5. For a relational possibility frame F, define the algebra F b =
(BA(P ), {✷i }i∈I ) where BA(P ) is the Boolean subalgebra of RO(S, ⊑) based on
P . Then F b is a BAO.
Proof. Immediate from Definitions 5.3.2 and 5.3.4, using the fact that RO(S, ⊑)
is a Boolean algebra (Theorem 3.2.3.1).
Remark 5.3.6. Full relational world frames F = (W, =, ℘(W ), {Ri }i∈I ) may be
regarded as the standard Kripke frames [116] from possible world semantics. The
key difference between full world frames and full possibility frames is that the
former can realize only atomic algebras as F b . In this chapter, we focus mainly on
this difference at the level of frames and algebras. For relations between possibility
models and world models, see § 5.3.2 and [89].
The following lemma gives convenient sufficient conditions for RO(S, ⊑) to be
closed under ✷i . The first two are familiar from semantics for intuitionistic modal
logic [170]. We give necessary and sufficient conditions in Appendix A.
Lemma 5.3.7 ([96]). For a poset (S, ⊑) and family {Ri }i∈I of binary relations
on S, suppose the following conditions hold for each i ∈ I:
• up-R – if x′ ⊑ x and x′ Ri y ′ , then xRi y ′ (see Figure 5);
• R-down – if xRi y and y ′ ⊑ y, then xRi y ′ (see Figure 6);
Wesley H. Holliday
• R-refinability – if xRi y, then ∃x′ ⊑ x ∀x′′ ⊑ x′ ∃y ′ ⊑ y: x′′ Ri y ′ (see
Figure 7).
Then:
1. RO(S, ⊑) is closed under ✷i ;
2. (S, ⊑, RO(S, ⊑), {Ri }i∈I ) is a full relational possibility frame;
3. (S, ⊑, P, {Ri }i∈I ) is a relational possibility frame for any P ⊆ RO(S, ⊑)
closed under ¬ and ∩.
x
x
y′
x′
⇒
x′
y′
Figure 5: the up-R condition. Given x′ Ri y ′ , we may go up in the first coordinate
to any x above x′ to obtain xRi y ′ .
y
x
⇒
x
y′
y
y′
Figure 6: the R-down condition. Given xRi y, we may go down in the second
coordinate to any y ′ below y to obtain xRi y ′ .
x
y
⇒
x
y
a. ∃
c. ∃
x′
b. ∀
x′′
y′
Figure 7: the R-refinability condition. The quantifiers in the diagram correspond to the quantifiers in the definition of R-refinability in Lemma 5.3.7.
Possibility Semantics
Note our naming convention: we use the name ‘up-R’ instead of ‘R-up’
because we are going up in the position before ‘R’: from x′ Ri y ′ to xRi y ′ where x
is above x′ ; by contrast, we use the name ‘R-down’ because we are going down
in the position after ‘R’: from xRi y to xRi y ′ where y ′ is below y.
A key fact about R-down is that it allows us to simplify the interpretation of
✸. Defining ✸i Z as ¬✷i ¬Z, we have
✸i Z = {x ∈ S | ∀x′ ⊑ x ∃y ′ ∈ Ri (x′ ) ∃y ′′ ⊑ y ′ : y ′′ ∈ Z}.
(6)
However, with R-down we achieve the following simplication.
Lemma 5.3.8. Let F = (S, ⊑, P, {Ri }i∈I ) be a possibility frame satisfying Rdown. Then for all Z ∈ P :
✸i Z = {x ∈ S | ∀x′ ⊑ x ∃z ′ ∈ Ri (x′ ) : z ′ ∈ Z}.
Proof. Let z ′ be y ′′ in (6). By R-down, together y ′ ∈ Ri (x′ ) and y ′′ ⊑ y ′ imply
y ′′ ∈ Ri (x′ ).
Example 5.3.9. We begin with a simple example of a finite possibility frame.
Although finite frames do not generate any new algebras compared to possible
world frames, the frames themselves may be interesting structures. We give a
temporal frame inspired by Aristotle’s Sea Battle argument (see, e.g., [66, p. 35]):
present
x
y
x′
y′
sea battle
no sea battle
Solid arrows represent the refinement relation; dashed arrows represent the future
accessibility relation Rf used to interpret future modalities ✷f and ✸f (henceforth and sometime in the future, respectively); dotted arrows represent the past
accessibility relation Rp used to interpret past modalities ✷p and ✸p (hitherto and
sometime in the past, respectively). It is easy to check that up-R, R-down, and
R-refinability hold for Rf and Rp . For example, since x ⊑ present and xRf x′ ,
up-R requires that presentRf x′ , which indeed holds. The present moment is represented by the possibility labeled ‘present’. There are two possible refinements
of the present, x and y, but we suppose that neither is currently realized. There
Wesley H. Holliday
are two associated future possibilities for what happens tomorrow: one (x′ ) in
which there is sea battle, and one (y ′ ) in which there is no sea battle.
Define a model M based on this frame where a propositional variable s, expressing that there is a sea battle, is true only at x′ . As we are considering
possibility semantics for classical logic, either there will be a sea battle tomorrow
or there won’t be: M, present
✸f s ∨ ¬✸f s. However, the present does not
settle that there will be sea battle tomorrow, and the present does not settle that
there won’t be a sea battle tomorrow: M, present 1 ✸f s, since y ⊑ present and
M, y
¬✸f s, and M, present 1 ¬✸f s, since x ⊑ present and M, x
✸f s.
Thus, the future is presently open. Yet if there is a sea battle, so x′ is realized,
then the past will turn out to be x, in which there would be a future sea battle,
whereas if there is no sea battle, so y ′ is realized, then the past will turn out to be
y, in which there would be no future sea battle. Come tomorrow, we might say,
“the past is not what it used to be.” Arguably this picture resolves some puzzles
about retrospective assessment of statements of future contingents (cf. [125]).
Example 5.3.10. For another temporal example—but now with infinitely divisible time, as opposed to the discrete time of Example 5.3.9—recall from Examples
3.1.2 and 3.2.2 the poset (S, ⊑) of non-empty open intervals of rational numbers
ordered by inclusion. Given intervals (a, b), (c, d), let (a, b)Rf (c, d) if a < c. Then
we claim that Rf satisfies the three conditions in Lemma 5.3.7:
• up-R: suppose (a′ , b′ ) ⊆ (a, b) and (a′ , b′ )Rf (c′ , d′ ). From (a′ , b′ ) ⊆ (a, b),
we have a ≤ a′ , and from (a′ , b′ )Rf (c′ , d′ ), we have a′ < c′ . Thus, a < c′ ,
which yields (a, b)Rf (c′ , d′ ).
• R-down: suppose (a, b)Rf (c, d) and (c′ , d′ ) ⊆ (c, d). From (a, b)Rf (c, d), we
have a < c, and from (c′ , d′ ) ⊆ (c, d), we have c ≤ c′ . Thus, a < c′ , which
yields (a, b)Rf (c′ , d′ ).
• R-refinability: suppose (a, b)Rf (c, d), so a < c. If b ≤ c, let (a′ , b′ ) =
(a, b). If c < b, let (a′ , b′ ) = (a, c). Observe that (a′ , b′ ) ⊆ (a, b) and for all
(a′′ , b′′ ) ⊆ (a′ , b′ ), a′′ < c, so (a′′ , b′′ )Rf (c, d).
Thus, by Lemma 5.3.7, (S, ⊑, RO(S, ⊑), Rf ) is a full relational possibility frame.
In line with the temporal interpretation of (S, ⊑) mentioned in Example 3.1.2, we
may regard Rf as the future accessibility relation for future modalities ✷f and
✸f (similarly, we can introduce a past accessibility relation by (a, b)Rp (c, d) if
d < b). Note that despite Rf being irreflexive, ✷f p → p (resp. p → ✸f p) is valid:
supposing ✷f p is true at (a, b), observe that ∀(a′ , b′ ) ⊆ (a, b) ∃(a′′ , b′′ ) ⊆ (a′ , b′ )
such that a < a′′ and hence (a, b)Rf (a′′ , b′′ ), which with ✷f p being true at (a, b)
implies that p is true at (a′′ , b′′ ), which by refinability implies that p is true at
Possibility Semantics
(a, b) (cf. [147, p. 459]). For the close relation between this possibility semantics
for temporal logic and interval semantics for temporal logic, see § 6.1.
In Example 5.3.10 we showed that the relation R actually satisfies the following
strengthening25 of R-refinability:
• R-refinability++ : if xRy, then ∃x′ ⊑ x ∀x′′ ⊑ x′ : x′′ Ry.
This was Humberstone’s original condition for modal semantics instead of Rrefinability, but for reasons explained in [93, 96], it proved too strong for the
general theory of possibility semantics for modal logic.
Below are intuitive, general explanations of each of the three conditions in
Lemma 5.3.7, working with the standard idea of “accessibility” that xRi y iff for
every proposition Z, if ✷i Z is true at x, then Z is true at y:
• up-R. Assume x′ ⊑ x and x′ Ri y ′ . For xRi y ′ , we must argue that whenever
✷Z is true at x, Z should be true at y ′ . Suppose ✷Z is true at x. Then
since x′ ⊑ x, by persistence, ✷Z should be true at x′ . Then since x′ Ri y ′ , Z
should be true at y ′ .
• R-down. Assume xRi y and y ′ ⊑ y. For xRi y ′ , suppose ✷Z is true at x.
Then since xRi y, Z should be true at y, and then since y ′ ⊑ y, Z should be
true at y ′ by persistence.
• R-refinability. Think of obtaining x′ by extending the description of x
with ✸i ↓y, using the fact that ↓y is regular open if (S, ⊑) is separative
(recall Definition 3.2.4 and Proposition 3.2.5).
It will be convenient to have a term for frames satisfying the three properties
above.
Definition 5.3.11. A paradigm relational possibility frame is a relational possibility frame satisfying up-R, R-down, and R-refinability.
The possibility frames we will define from algebras also satisfy an additional
property that can be useful:
• R-dense – xRi y if ∀y ′ ⊑ y ∃y ′′ ⊑ y ′ : xRi y ′′ (see Figure 8).
Note that together R-down and R-dense are equivalent to the condition that
for each x ∈ S, we have Ri (x) ∈ RO(S, ⊑).
25
Another condition dubbed R-refinability+ is considered in [96].
Wesley H. Holliday
x
y
⇒
x
y
∀
y′
∃
y ′′
Figure 8: the R-dense condition. The quantifiers in the diagram correspond to
the quantifiers in the definition of R-dense in the text.
Definition 5.3.12. A strong relational possibility frame is a relational possibility
frame satisfying up-R, R-down, R-refinability, and R-dense.
Example 5.3.13. It is easy to check that the temporal frame in Examples 5.3.9
satisfies R-dense. However, the temporal frame in Example 5.3.10 does not, for a
reason raised in our earlier discussion: although ∀(a′ , b′ ) ⊆ (a, b) ∃(a′′ , b′′ ) ⊆ (a′ , b′ )
such that a < a′′ and hence (a, b)Rf (a′′ , b′′ ), it is not the case that (a, b)Rf (a, b),
because a 6< a. In fact, we obtain a frame that realizes the same BAO if we
define (a, b)Rf′ (c, d) by a ≤ c, instead of a < c, and then R-dense is satisfied. We
adopted the definition with a < c to facilitate comparison with [147] in § 6.1.
In Appendix A, we give a single condition equivalent to the conjunction of
the four in Definition 5.3.12. We can relate the notion of a strong frame to the
standard notion of tightness of general frames (see, e.g., [37, p. 251]) as follows.
Definition 5.3.14. A relational possibility frame F = (S, ⊑, P, {Ri }i∈I ) is Rtight if for all x, y ∈ S, if for all Z ∈ P , x ∈ ✷i Z implies y ∈ Z, then xRi y.
Lemma 5.3.15 ([96]). For any relational possibility frame F:
1. if F is R-tight, then F satisfies up-R, R-down, and R-dense;
2. if F is R-tight and satisfies R-refinability, then F is strong;
3. if F is full, then F is R-tight iff F is strong.
Lemma 5.3.16 ([96]). For any possibility frame F = (S, ⊑, P, {Ri }i∈I ), define
F = (S, ⊑, P, {Ri✷ }i∈I ) by xRi✷ y iff for all Z ∈ P , x ∈ ✷i Z implies y ∈ Z. Then:
1. F b = F ✷b , and F ✷ is R-tight;
Possibility Semantics
2. if F is full, then F ✷ is strong.
Thus, may assume without loss of generality that any full relational possibility
frame we are working with is in fact strong. Indeed, we may assume this without
loss of generality for any relational possibility frame (not just a full one), but we
postpone the reason to § 5.3.8.
One of the appeals of Kripke frame semantics for normal modal logics is the
correspondence theory between modal axioms and first-order properties of the Ri
relations [15, 12]. In the setting of full possibility frames, we also have an appealing
correspondence between modal axioms and first-order properties of Ri and ⊑.
Compare the following two theorems, where for a sequence α = (α1 , . . . , αn ) of
modal indices from I, ✸α ϕ := ✸α1 . . . ✸αn ϕ, ✷α ϕ := ✷α1 . . . ✷αn ϕ, and xRα y iff
there are x0 , . . . , xn with x0 = x, xn = y, and x0 Rα1 x1 , x1 Rα2 x2 , . . . , xn−1 Rαn xn
(if α is the empty sequence, ✸α ϕ = ✷α ϕ = ϕ and xRα y iff x = y).
Theorem 5.3.17 ([120]). Let F be a Kripke frame. Then for any sequences α,
β, δ, and γ of indices from I, ✸α ✷β p → ✷δ ✸γ p is valid on F iff F satisfies:
∀x∀y∀z((xRδ y ∧ xRα z) → ∃u(yRγ u ∧ zRβ u)).
Theorem 5.3.18 ([96]). Let F be a full paradigm relational possibility frame.
Then for any sequences α, β, δ, and γ of indices from I, ✸α ✷β p → ✷δ ✸γ p is valid
on F iff F satisfies
∀x∀y xRδ y → ∃x′ ⊑ x ∀z(x′ Rα z → ∃u(yRγ u ∧ zRβ u)) .
For the case where α is empty, ✷β p → ✷δ ✸γ p is valid on F iff F satisfies
∀x∀y(xRδ y → ∃u(yRγ u ∧ xRβ u)).
Remark 5.3.19. Over strong full possibility frames, the first-order conditions in
Theorem 5.3.18 may simplify: for example, ✷i p → p corresponds just to reflexivity
of Ri and ✷i p → ✷i p✷i p just to transitivity. Diamond formulas also simplify but
still involve ⊑. For example, ✸i p → ✷i✸i p corresponds over strong frames to
∀x∀y xRi y → ∃x′ ⊑ x ∀z(x′ Ri z → yRi z) .
More generally, Yamamoto [171] has proved the analogue of the Sahlqvist
correspondence theorem (see [30, § 3.6]) for full possibility frames.
Theorem 5.3.20 ([171]). For every Sahlqvist formula ϕ of L(I), there is a formula
cϕ in the first-order language with relation symbols for Ri and ⊑ such that a full
possibility frame F = (S, ⊑, RO(S, ⊑), {Ri }i∈I ) validates ϕ iff (S, ⊑, {Ri }i∈I ) as
a first-order structure satisfies cϕ . Moreover, cϕ is effectively computable from ϕ.
This result was generalized to the larger class of inductive formulas in [172, 173].
Wesley H. Holliday
5.3.2
Relational possibility models and Kripke models
As noted in Remark 5.3.6, our focus in this chapter is on possibility frames, frame
validity of formulas, and algebras arising from frames. At this level, the limitation
of Kripke frames is that they can realize only atomic algebras. However, here we
will briefly discuss the connection between possibility models and Kripke models.
One direction is obvious: every Kripke model can be viewed as a possibility model
where ⊑ is the identity relation. In the other direction, given any possibility model
M and formula ϕ, we can construct a Kripke model based on the ϕ-decisive
possibilities in M, which decide the truth value of every subformula of ϕ.
Definition 5.3.21. Given a possibility model M = (S, ⊑, {Ri }i∈I , π) and formula
ϕ,26 we define the Kripke model Mϕ = (Sϕ , {Ri,ϕ }i∈I , πϕ ) as follows:
1. Sϕ = {x ∈ S | for all subformulas ψ of ϕ, M, x
ψ or M, x
¬ψ}.
2. xRi,ϕ y iff ∃z ∈ Ri (x): y ⊑ z;
3. πϕ (p) = π(p) ∩ Sϕ .
Lemma 5.3.22. Let M = (S, ⊑, {Ri }i∈I , π) be a possibility model and ϕ a formula. Then:
1. for every x ∈ S, there is an x′ ∈ Sϕ such that x′ ⊑ x;
2. for every x ∈ Sϕ and subformula ψ of ϕ, M, x
ψ iff Mϕ , x
ψ.
Proof. For part 1, note that for any y ⊑ x, if M, y 1 ψ, then by refinability there
is a z ⊑ y such that M, z ¬ψ; and if M, y ψ, then by persistence, for any
z ⊑ y, M, z ψ. Thus, starting with x, we can apply refinability and persistence
for each subformula ψ of ϕ in turn until reaching the desired x′ ∈ Sϕ with x′ ⊑ x.
Part 2 is by induction. The base case and ∧ case are obvious. For any
subformula ¬ψ of ϕ, we have M, x ¬ψ iff M, x 1 ψ (since x ∈ Sϕ ) iff Mϕ , x 1 ψ
(by the inductive hypothesis) iff Mϕ , x ¬ψ (since Mϕ is a Kripke model). For
any subformula ✷i ψ of ϕ, if Mϕ , x 1 ✷i ψ, then there is a y ∈ Sϕ such that xRi,ϕ y
and Mϕ , y 1 ψ, which implies M, y 1 ψ by the inductive hypothesis. Since
xRi,ϕ y, there is a z ∈ Ri (x) with y ⊑ z. Then by persistence, M, y 1 ψ implies
M, z 1 ψ, so M, x 1 ✷i ψ. Conversely, if M, x 1 ✷i ψ, so there is a y ∈ Ri (x)
such that M, y 1 ψ, then by refinability there is a y ′ ⊑ y such that M, y ′ ¬ψ.
Then by part 1, there is a y ′′ ∈ Sϕ with y ′′ ⊑ y ′ , which implies M, y ′′ 1 ψ and
y ′′ ⊑ y. It follows that xRi,ϕ y ′′ and hence Mϕ , x 1 ✷i ψ.
26
We have omitted the specification of the set P of admissible propositions of the underlying
frame of M, as only the valuation π matters for the model.
Possibility Semantics
Note, however, that the underlying Kripke frame of Mϕ may fail to validate
the same formulas as the underlying full possibility frame of M. Nonetheless,
Lemma 5.3.22 and the soundness of the minimal normal modal logic K with respect
to Kripke models yields soundness of K with respect to possibility models without
any additional checking of Boolean or modal axioms.
5.3.3
From V-BAOs to full and principal frames
We now return to our discussion of the relation between possibility frames and
BAOs. In this subsection and the next, we will make precise the following claim:
from an algebraic point of view, the essence of relational semantics for modal
logic in its most basic form (i.e., before the step to general frames or topological
frames that can realize all modal algebras) is that the ✷i modalities distribute
over arbitrary conjunctions—or dually, the ✸i modalities distribute over arbitrary disjunctions. The atomicity of the dual algebras of Kripke frames can be
disassociated from the essence of the basic relational semantics.
Formally, following [121], we call a Boolean algebra equipped with unary ✷i
operations a V-BAO if ✷i distributes over any existing meets:
if
V
a exists, then ✷i (
V
a) =
a∈A
a∈A
V
✷i a.
a∈A
Then ✷i is said to be completely multiplicative. This is equivalent to the ✸i
operation related by ✸i a = ¬✷i ¬a being completely additive:
if
W
a exists, then ✸i (
W
a) =
a∈A
a∈A
W
✸i a.
a∈A
W
The ‘V’ is chosen to suggest the big join , thinking in terms of ✸i as primitive
(in contrast to our approach of taking ✷i as primitive).
We can turn V-BAOs into relational possibility frames as follows.
Theorem 5.3.23 ([96]). For any V-BAO B = (B, {✷i }i∈I ), define:
Bu = (B+ , ≤+ , RO(B+ , ≤+ ), {Ri }i∈I ), the full frame of B, and
Bp = (B+ , ≤+ , {↓+ x | x ∈ B+ } ∪ {∅}, {Ri }i∈I ), the principal frame of B,
where (B+ , ≤+ ) is as in Theorem 3.2.3.2 and for x, y ∈ B+ and i ∈ I,
xRi y iff ∀y ′ ∈ B+ , if y ′ ≤+ y then x ∧ ✸i y ′ 6= 0 in B.
Then:
1. Bu and Bp are strong relational possibility frames, with Bu being full;
(7)
Wesley H. Holliday
2. if B is a complete Boolean algebra, then Bu = Bp ;
3. (Bu )b is the lower MacNeille completion (or Monk completion) of B;
4. (Bp )b is isomorphic to B.
Remark 5.3.24. The proof of part 4 takes advantage of the surprising fact—to
which the study of possibility semantics led—that complete multiplicativity of ✷i
(or complete additivity of ✸i ), an ostensibly second-order condition on a BAO, is
in fact equivalent to the following first-order condition on a BAO, defined using
the Ri relation from (7):
if x ∧ ✸i y 6= 0, then ∃y ′ ∈ B+ : y ′ ≤+ y and xRi y ′ .
This is proved in [94, 101] for BAOs (for a generalization to posets, see [2]).
By Theorem 5.3.23, any complete V-BAO can be represented as the algebra
associated with a full possibility frame, while any V-BAO can be represented as
the algebra associated with a possibility frame whose admissible sets are precisely
the principle downsets together with ∅. We will use these facts in what follows.
They imply that any logic complete with respect to a class of complete V-BAOs
(resp. V-BAOs) is complete with respect to a class of the corresponding frames.
A T -BAO [121] is a BAO B = (B, {✷i }i∈I ) in which for each i ∈ I, there is
an fi : B → B that is a residual of ✷i ,27 i.e., for all a, b ∈ B:
a ≤ ✷i b iff fi (a) ≤ b.
(8)
A useful fact is that a complete BAO is a V-BAO iff it is a T -BAO. The right to
left direction is obvious. For the left to right, simply take
fi (a) =
^
{c ∈ B | a ≤ ✷i c}.
Now we have an analogue of Proposition 5.2.10 for relational possibility frames.
Theorem 5.3.25. Suppose B is a Boolean algebra isomorphic to F b for some
possibility frame F = (S, ⊑, P ) in which every principal downset belongs to P .28
Then for any T -BAO B = (B, {✷i }i∈I ), there is a strong relational possibility
frame BF = (S, ⊑, P, {Ri }i∈I ) such that (BF )b is isomorphic to B.
27
T stands for ‘tense’, as in tense logics the past diamond operator is a residual of the future
box operator.
28
E.g., P = RO(S, ⊑) for a separative poset (S, ⊑).
Possibility Semantics
Proof. We use an isomorphism σ from F b to B to define accessibility relations Ri
on S. Let fi be the residual of ✷i in the T -BAO based on B. Then for x ∈ S, let
Ri (x) = σ −1 (fi (σ(↓x))).
With this definition, we claim that for any Z ∈ P , σ(✷i Z) = ✷i σ(Z). Indeed:
σ(✷i Z) = σ({x ∈ S | Ri (x) ⊆ Z}
= σ({x ∈ S | fi (σ(↓x)) ≤ σ(Z)}
= σ({x ∈ S | σ(↓x) ≤ ✷i σ(Z)} by (8)
= σ({x ∈ S | ↓x ⊆ σ −1 (✷i σ(Z))}
= σ(σ −1 (✷i σ(Z)))
= ✷i σ(Z).
Hence (BF )b is isomorphic to B. Let fi : P → P be the residual of ✷i in (BF )b , so
Ri (x) = fi (↓x).
We now show that BF is strong. First, up-R is equivalent to the condition that
x′ ⊑ x implies Ri (x′ ) ⊆ Ri (x). Now if x′ ⊑ x, so ↓x′ ⊆ ↓x, then fi (↓x′ ) ⊆ fi (↓x), so
Ri (x′ ) ⊆ Ri (x). R-down and R-dense are immediate from the fact that fi (↓x) ∈
P ⊆ RO(S, ⊑). Finally, for R-refinability, suppose xRi y. Then x 6∈ ✷i ¬↓y, so
there is an x′ ⊑ x such that x′ ∈ ✸i ↓y. Consider any x′′ ⊑ x′ . Then ↓x′′ 6⊆ ✷i ¬↓y,
so by residuation, fi (x′′ ) 6⊆ ¬↓y, i.e., Ri (x′′ ) 6⊆ ¬↓y. Hence there is some z ∈
Ri (x′′ ) such that z 6∈ ¬↓y, which implies there is a y ′ ⊑ z such that y ′ ⊑ y. Then
since Ri (x′′ ) is a downset, we have x′′ Ri y ′ , which establishes R-refinability.
Theorem 5.3.25 has a tree completeness corollary directly analogous to Corollary 5.2.27, but now for relational possibility frames based on 2<ω .
5.3.4
Dual equivalence with complete V-BAOs
The representation theorem for complete V-BAOs in Theorem 5.3.23 can be
turned into a full categorical duality. On the algebra side, we have the following
category.
Proposition 5.3.26. Complete V-BAOs with complete BAO-homomorphisms,
i.e., Boolean homomorphisms preserving arbitrary meets and each ✷i operator,
form a category, CV-BAO.
On the possibility side, we first characterize the possibility frames that arise
from complete V-BAOs via the (·)u map.
Wesley H. Holliday
Lemma 5.3.27 ([96]). Let F = (S, ⊑, RO(S, ⊑), {Ri }i∈I ) be a full relational
possibility frame. Then F is isomorphic to (F b )u iff the underlying poset of F is
obtained from a complete Boolean lattice by deleting its bottom element and for
all i ∈ I and x ∈ S, Ri (x) is a principal downset in (S, ⊑).
Definition 5.3.28. A rich possibility frame is a full possibility frame satisfying
the right-hand side of the equivalence in Lemma 5.3.27.
To define categories of full and rich possibility frames, we need to introduce
morphisms. We use the notation x ≬ y for ↓x ∩ ↓y 6= ∅.
Definition 5.3.29. Given full possibility frames F and F ′ , a strict possibility
morphism from F to F ′ is an h : S → S ′ such that for all x, y ∈ S and y ′ ∈ S ′ :
1. ⊑-forth – if y ⊑ x, then h(y) ⊑′ h(x);
2. ⊑-back – if y ′ ⊑′ h(x), then ∃y: y ⊑ x and h(y) ⊑′ y ′ ;
3. R-forth – if xRy, then h(x)R′ h(y);
4. R-back – if h(x)R′ y ′ and z ′ ⊑′ y ′ , then ∃y: xRy and h(y) ≬′ z ′ .
A p-morphism is defined in the same way as a strict possibility morphism, but
with strengthened versions of the two back conditions:
2′ . p-⊑-back – y ′ ⊑′ h(x), then ∃y: y ⊑ x and h(y) = y ′ ;
4′ . p-R-back – if h(x)R′ y ′ , then ∃y: xRy and h(y) = y ′ .
Proposition 5.3.30 ([96]).
1. Full possibility frames with strict possibility morphisms form a category,
FullPoss;
2. Rich possibility frames with p-morphisms form a category, RichPoss.
Now we can make precise the categorical relationship between full possibility
frames and complete V-BAOs.
Theorem 5.3.31 ([96]).
1. RichPoss is a reflective subcategory of FullPoss;
2. RichPoss is dually equivalent to CV-BAO.
Recall, by contrast, that the category of Kripke frames with p-morphisms is
dually equivalent to the category of complete and atomic V-BAOs (CAV-BAOs)
with complete BAO-homomorphisms [166].
Possibility Semantics
5.3.5
Quasi-normal possibility frames
In this section, we briefly comment on possibility semantics for quasi-normal
modal logics. These logics have important applications in provability logic [32],
which is the subject of §§ 5.3.6-5.3.7.
A quasi-normal modal logic is a set L of L(I) formulas that contains all theorems of the smallest normal modal logic, K, and is closed under modus ponens
and uniform substitution—but not necessarily under the necessitation rule stating
that if ϕ ∈ L, then ✷i ϕ ∈ L. Algebraic semantic for quasi-normal modal logics
can be given using matrices (B, F ) where B = (B, {✷i }i∈I ) is a BAO and F is a
distinguished filter in B. Instead of defining a formula ϕ to be valid if for any
valuation θ on B, θ̃(ϕ) = 1B , we define ϕ to be valid if for any valuation θ on B,
θ̃(ϕ) ∈ F . Then it is possible for ϕ to be valid on B while ✷i ϕ is not valid. Yet
modus ponens and uniform substitution still preserve validity.
One obvious way to give possibility semantics for quasi-normal modal logics
is to equip possibility frames with a distinguished filter of propositions.
Definition 5.3.32. A relational possibility frame with distinguished filter is a pair
(F, F ) where F = (S, ⊑, P, {Ri }i∈I ) is a relational possibility frame and F is a
filter in the algebra P of propositions.
Remark 5.3.33. When ⊑ is the identity relation, P = ℘(S), and F is a principal
filter, we obtain the standard notion of a Kripke frame with distinguished worlds
as a special case of Definition 5.3.32.
A perhaps more appealing approach, given our move from worlds to possibilities, is to distinguish a directed set S0 of possibilities.
Definition 5.3.34. A quasi-normal relational possibility frame is a tuple Q =
(S, ⊑, P, {Ri }i∈I , S0 ) where FQ = (S, ⊑, P, {Ri }i∈I ) is a relational possibility
frame and S0 ⊆ S is such that if x, y ∈ S0 , then there is a z ∈ S0 such that
z ⊑ x and z ⊑ y. A formula ϕ is valid on Q if for every model M based on FQ ,
we have M, x ϕ for some x ∈ S0 .
If S0 contains a maximum element with respect to ⊑, we call Q a relational
possibility frame with a distinguished possibility.
Any such frame gives us a matrix Qm = ((FQ )b , F ) with (FQ )b as in Proposition 5.3.5 and the filter F defined by:
F = {Z ∈ P | S0 ∩ Z 6= ∅}
(which is principal if Q has a distinguished possibility). This yields the following,
which can also be easily confirmed directly.
Wesley H. Holliday
Proposition 5.3.35. The set of formulas valid on any class of quasi-normal
relational possibility frames is a quasi-normal modal logic.
Conversely, any matrix (B, F ) where B is a complete V-BAO and F a proper
filter gives us a quasi-normal relational possibility frame (Bu , S0 ) with Bu as in
Theorem 5.3.23 and S0 = F , such that (Bu , S0 )m is isomorphic to (B, F ).
In addition, we note that any rooted relational possibility frame, i.e., in which
the poset (S, ⊑) has a maximum element m, may be regarded as a quasi-normal
relational possibility frame in which S0 = {m}. There is no loss of generality
in possibility semantics for L(I) in working with rooted frames (note that Bu is
rooted, as is Bg in § 5.3.8), so in this sense quasi-normal relational possibility
frames are a generalization of relational possibility frames.
5.3.6
Full frames and world incompleteness in provability logic
In this section, we illustrate the greater generality of full relational possibility
frames over Kripke frames using examples inspired by provability logic. Recall
that in possibility semantics, Kripke frames may be regarded as full relational
world frames (see Remark 5.3.6). Already for the basic unimodal logic, there are
many Kripke incomplete but full-possibility-frame complete logics. The following
result adapts results of [122] and [113] to the setting of possibility semantics.
Theorem 5.3.36 ([96]). There are continuum-many normal unimodal logics L
such that L is the logic of a class of full relational possibility frames but L is not
the logic of any class of full relational world frames.
If we go beyond the basic unimodal language, we can obtain very natural
examples of Kripke incomplete but full-possibility-frame complete logics. Here
we consider the language of propositional term modal logic [67, 141, 142, 143],
which allows quantification over modalities, as well as what can be regarded as
a fragment of that language—the language of “someone believes” from doxastic
logic [86, 82]. These languages have natural applications in multi-agent doxastic
or epistemic logic, as well as (we will propose) polymodal provability logic. In
addition to the devices for quantifying over modalities, we consider extensions
of the languages with 0-ary modalities G(i) expressing that agent i belongs to
the group of agents over whom we are quantifying, and 0-ary O(i) modalities
expressing the “opinionatedness” (in doxastic logic) or “omniscience” (in epistemic
logic) or “negation-completeness” (in provability logic) of i. Fixing a countably
Possibility Semantics
infinite set Var of variables, the languages are:
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷v ϕ | ∀vϕ | G(v) | O(v)
T MLGO
T MLG
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷v ϕ | ∀vϕ | G(v)
T MLO
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷v ϕ | ∀vϕ | O(v)
T ML
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷v ϕ | ∀vϕ
LSG (I)
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷i ϕ | Sϕ | G(i)
LS (I)
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | ✷i ϕ | Sϕ
where p ∈ Prop, v ∈ Var, and i ∈ I. We define ∃vϕ as ¬∀v¬ϕ.
As the semantics below will show, the languages with S may be regarded as
fragments of the term modal languages by translating Sϕ as ∃v✷v ϕ.
We can interpret all of these languages using the following frames.
Definition 5.3.37. A varying agent-domain full relational possibility frame is a
tuple (F, G) where F = (S, ⊑, RO(S, ⊑), {Ri }i∈I ) is a full relational possibility
frame for L(I) and G : S → ℘(I) \ {∅} satisfies the following:
1. persistence for G: if x′ ⊑ x, then G(x′ ) ⊇ G(x);
2. refinability for G: if i ∈ I \ G(x), then ∃x′ ⊑ x ∀x′′ ⊑ x′ i 6∈ G(x′′ ).
We say F has universal agent-domain if G(x) = I for all x ∈ S.
Varying agent-domain domain full neighborhood possibility frames (see § 5.2)
are defined analogously. Quasi-normal versions of these frames are defined as in
Definitions 5.3.32-5.3.34 in the obvious way.
We first explain the simpler semantics for LSG (I).
Definition 5.3.38. Given a possibility model M based on a frame F as in Definition 5.3.37 and ϕ ∈ LSG (I), we define M, x ϕ with the usual clauses for ¬,
∧, and ✷i , plus the following:
1. M, x
Sϕ iff ∀x′ ⊑ x ∃x′′ ⊑ x′ ∃i ∈ G(x′′ ): M, x′′
2. M, x
G(i) iff i ∈ G(x).
✷i ϕ;
A formula ϕ is valid on F if for every model M based on F and every possibility
x in F, we have M, x ϕ.
The following is immediate from the semantic clause for S.
Lemma 5.3.39. If M is a world model, i.e., ⊑ is the identity relation, then:
M, w
Sϕ iff ∃i ∈ G(w): M, w
✷i ϕ.
Wesley H. Holliday
Given the semantic clauses of Definition 5.3.38, we have the validity of the
(free logical) existential generalization principle:
(✷i ϕ ∧ G(i)) → Sϕ.
Now we turn to the term modal languages.
Definition 5.3.40. Given a varying agent-domain full relational or neighborhood possibility frame F for a language L(I), a model M based on F, variable assignment g : Var → I, and ϕ ∈ T MLGO , we define M, x g ϕ and
JϕKM
g = {y ∈ S | M, y g ϕ} as follows:
• M, x
g
✷v ϕ iff x ∈ ✷g(v) JϕKM
g ;
• M, x
g
∀vϕ iff for all i ∈ G(x), M, x
• M, x
g
G(v) iff g(v) ∈ G(x);
• M, x
g
O(v) iff for all Z ∈ RO(S, ⊑), x ∈ ✷g(v) Z ∨ ✷g(v) ¬Z.
g[v:=i]
ϕ;
A formula ϕ is valid with respect to F, g if for every model M based on F and
every possibility x in F, M, x g ϕ. A formula is valid on F if for every variable
assignment g, ϕ is valid with respect to F, g.
Note the validity according to this semantics of the principle
O(v) → (✷v p ∨ ✷v ¬p).
The inadequacy of Kripke frames for the study of these languages is shown
by a series of related problems to follow. In fact, they afflict not only Kripke
frames but more generally any full neighborhood world frame (recall Remark
5.2.33) that is monotonic, i.e., each Ni (w) is closed under supersets, which is the
condition corresponding to the validity of ✷i (p ∧ q) → ✷i p. The essential problem
is that world frames commit us to the idea that if every truth is believed by some
agent/theory or other, formalized by the frame validity of
p → ∃v✷v p,
then there is a single agent/theory who is fully opinionated:
∃vO(v).
To see that world frames have this commitment, consider for a given world w the
“world proposition” {w}; then the agent who believes {w} is fully opinionated.
This can easily be turned into a proof of Proposition 5.3.42 below.
Possibility Semantics
Remark 5.3.41. First, we note a connection to “Fitch’s paradox” of knowability
[64, 34]: under weak assumptions, the “verifiability principle” that every truth
could in principle be known (at some time) by some agent or other, p → ∃v✷v p
(where is some kind of possibility modal), entails the stricter verificationist
principle that every truth is known (at some time) by some agent or other, p →
∃v✷v p. Now we add that world frames commit the strict verificationist to the
implausible principle that there is (at some time) a single omniscient agent.
Proposition 5.3.42. Any varying agent-domain full monotonic neighborhood
world frame that validates p → ∃v✷v p validates ∃vO(v).
Yet it is straightforward to construct a full relational possibility frame without
the unwanted consequence.
Proposition 5.3.43. There is a universal agent-domain full relational possibility
frame that validates p → ∃v✷v p but not ∃vO(v).
Proof. Where I is countably infinite, we build a possibility frame for L(I) based
on the full infinite binary tree 2<ω regarded as a poset (2<ω , ⊑) as in Examples
3.1.1 and 3.2.1. Fix a bijection f from I to 2<ω , and for x ∈ 2<ω , let Ri (x) = ↓f (i).
Then it is easy to see that Ri satisfies up-R, R-down, and R-refinability, so by
Lemma 5.3.7, (2<ω , ⊑) equipped with RO(2<ω , ⊑) and {Ri }i∈I is a full relational
possibility frame, and setting G(x) = I gives us a universal agent-domain frame
F. Let M be any model based on F and g0 : Var → I. To see that p → ∃v✷v p
is globally true in M, observe that if M, x g0 p, then M, x g ✷v p where
g(v) = f −1 (x), so M, x g0 ∃v✷v p. Finally, since there is no x ∈ X and i ∈ I
such that for all Z ∈ RO(2<ω , ⊑), x ∈ ✷i Z ∨ ✷i ¬Z, the formula ∃vO(v) is not
true at any possibility in the model.
Let us now consider a related example in the language LS (I). For this, we
draw an explicit connection to provability logic [4]. The key idea is to use the
formula corresponding to Gödel’s Second Incompleteness Theorem:
✷i ¬✷i ⊥ → ✷i ⊥.
The Second Incompleteness Theorem entails, for the theories it covers, that if a
single theory can prove every truth, then it is inconsistent, so there is an inconsistent theory: S⊥. World frames commit us to the idea that if every truth is
provable in some theory or other in a class, then there is a single theory that
proves every truth—hence the class contains an inconsistent theory by the Second Incompleteness Theorem. But we know from Gödel that this is not so for
every class of theories: it can be that every truth is provable in some theory or
Wesley H. Holliday
other in a class, yet there is no inconsistent theory in the class. This reasoning
inspires the following world-incompleteness result in Theorem 5.3.44. Recall that
the Second Incompleteness formula is a special case (substituting ⊥ for p) of the
formula corresponding to Löb’s Theorem in provability logic:
✷i (✷i p → p) → ✷i p.
Theorem 5.3.44.
1. Any varying agent-domain full monotonic neighborhood world frame that
validates p → Sp and ✷i ¬✷i ⊥ → ✷i ⊥ for each i ∈ I also validates S⊥.
2. There are varying agent-domain full relational possibility frames validating
p → Sp and ✷i (✷i p → p) → ✷i p for each i ∈ I but not S⊥.
Proof. For part 1, let F be a frame satisfying the hypothesis. To show that F
validates S⊥, consider any world w in F. Let M be a model based on F such
that π(p) = {w}. Then since F validates p → Sp, we have M, w Sp, so there
is some i ∈ G(w) such that M, w
✷i p and hence {w} ∈ Ni (w). Then by
monotonicity, Ni (w) contains every A ⊆ W with w ∈ A. Thus, if M, w ¬✷i ⊥,
then M, w
✷i ¬✷i ⊥. But F validates ✷i ¬✷i ⊥ → ✷i ⊥, so we conclude that
M, w ✷i ⊥. Hence M, w S⊥. Since S⊥ contains no propositional variables,
it follows that F, w validates S⊥. Since w was arbitrary, F validates S⊥.
For part 2, let T be any tree in which every node has at least 2, but only finitely
many, children. For x, y ∈ T , we set x ⊑ y if x = y or x is a descendent of y. Fix
a bijection f from I to T . Then for any i ∈ I and x ∈ T , where c1 , . . . , cn are the
children of f (i), define Ri (x) as follows (recalling that ↓cj = {y ∈ S | y ⊑ cj }):
Ri (x) =
S
↓cj
1<j≤n
S
↓cj
k<j≤n
S
↓cj
if f (i) ⊑ x
if ∃k : x ⊑ ck
otherwise,
ki <j≤n
where ki can be chosen to be any integer from 1 to n. See Figure 9 below.
We claim that Ri satisfies the following properties:
1. up-R – if x′ ⊑ x and x′ Ri y ′ , then xRi y ′ ;
2. R-down – if xRi y and y ′ ⊑ y, then xRi y ′ ;
3. R-refinability – if xRi y, then ∃x′ ⊑ x ∀x′′ ⊑ x′ ∃y ′ ⊑ y: x′′ Ri y ′ .
Possibility Semantics
For up-R, this is equivalent to the condition that x′ ⊑ x implies Ri (x′ ) ⊆
Ri (x), which is clear from the definition of Ri . That R-down holds is immediate
from the definition of Ri (x) as a downset. Finally, for R-refinability, suppose
xRi y. Then either (i) f (i) ⊑ x, (ii) x ⊑ ck for some child ck of f (i), or (iii) neither
(i) nor (ii) holds. In addition, y ⊑ cj for some child cj of f (i) with j > 1 in cases
(i) and (iii) and j > k in case (ii). In case (i), let x′ := c1 , while in cases (ii) and
(iii), let x′ := x. Then note that that for all x′′ ⊑ x′ , we have x′′ Ri y.
Thus, by Lemma 5.3.7, (T, ⊑) equipped with RO(T, ⊑) and {Ri }i∈I is a full
relational possibility frame. Now define G : T → ℘(I) \ {∅} as follows:
G(x) = {i ∈ I | ∀x′ ⊑ x Ri (x′ ) 6= ∅}.
Then we claim that persistence and refinability hold for G. Persistence is immediate from the definition. For refinability, suppose i ∈ I \ G(x), so there is an x′ ⊑ x
such that Ri (x′ ) = ∅. Then by up-R, for all x′′ ⊑ x′ we have Ri (x′′ ) = ∅ and
hence i 6∈ G(x′′ ). Thus, we have a varying agent-domain full relational possibility
frame. Let M be any model based on the frame.
First, we show that ✷i (✷i p → p) → ✷i p is true at every x in M. Suppose
M, x ✷i (✷i p → p). If Ri (x) = ∅, then immediately M, x ✷i p. Otherwise
xRi cn where cn is the last of the children of f (i) in our enumeration of f (i)’s
children, so from M, x ✷i (✷i p → p) we have M, cn ✷i p → p. Since Ri (cn ) =
∅, we have M, cn ✷i p and hence M, cn p. Then since Ri (cn−1 ) = ↓cn , we have
M, cn−1 ✷i p. Hence if x ⊑ cn−1 , we have M, x ✷i p, so we are done. If instead
xRi cn−1 , then from M, x ✷(✷i p → p) we have M, cn−1 ✷i p → p, which with
M, cn−1
✷i p from above yields M, cn−1
p and hence M, cn−2
✷i p, so
M, x
✷i p if x ⊑ cn−2 . Repeating this reasoning, if x is under some child of
f (i), we obtain M, x ✷i p, and otherwise we obtain M, ck p for every child ck
of f (i) and hence M, x ✷i p for every x in M.
Next, we show that p → Sp is globally true. Suppose M, x p and x′ ⊑ x.
Let x′′ be the first child of x′ , i.e., c1 , in our enumeration of the children of x′ .
Then observe that f −1 (x′ ) ∈ G(x′′ ) and M, x′′ ✷f −1 (x′ ) p. Hence M, x Sp.
Finally, it is immediate from our definition of G that if M, x
✷i ⊥, then
i 6∈ G(x). Hence there is no possibility that makes S⊥ true.
Corollary 5.3.45. There is a varying-agent domain full relational possibility
frame whose LS (I)-logic is not that of any varying agent-domain full monotonic
neighborhood world frame.
For actual arithmetic interpretations of LS (I), we may want p → Sp to be
valid while ✷i (p → Sp) is not, as in the following example.
.
..
..
.
..
.
..
.
..
.
.
.
..
..
.
..
c3
..
.
.
.
c2
..
..
c1
..
f (i)
.
Wesley H. Holliday
Figure 9: illustration of the construction in the proof of Theorem 5.3.44.2 for
n = ki = 3. Solid lines are for ⊑ and dashed lines are for Ri . Every possibility
with an accessibility arrow to cj also has accessibility arrows to all descendants
of cj , but these arrows are not shown in the diagram.
Example 5.3.46. Consider the language of arithmetic and the theories Q (Robinson’s Q), EA (Elementary Arithmetic), and PA (Peano Arithmetic). Let T be an
arithmetically definable set of elementarily presented consistent extensions of Q,
containing at least EA and any consistent extension of EA with a single sentence.
Let T be an arithmetic predicate expressing that a number is the Gödel number
of a theory in T or the Gödel number of a consistent extension of EA with a single
sentence. The latter disjunct is redundant from the point of view of truth, but it
ensures that PA proves “if x is the Gödel number of a consistent extension of EA
with a single sentence, then T (x),” which we will use below.
Say that a T-realization of a formula ϕ of LS (I) is a sentence of arithmetic
obtained by: uniformly replacing the propositional variables in ϕ by sentences of
arithmetic; for each i ∈ I, uniforming replacing ✷i by the provability predicate of
some theory in T, given by its elementary presentation;29 and replacing S by the
arithmetic predicate using T that codes “there is some theory in T that proves. . . ”.
Say that ϕ is arithmetically T-valid if all of its T-realizations are provable in PA.
(By analogy with modal validity, given a class T of such sets T, we could define
ϕ to be arithmetically T-valid if ϕ is arithmetically T-valid for every T ∈ T.)
Then by Löb’s theorem, ✷i (✷i p → p) → ✷i p is arithmetically T-valid for each
i ∈ I, as is the Second Incompleteness Theorem formula, ✷i ¬✷i ⊥ → ✷i ⊥. In
addition, we claim that p → Sp is arithmetically T-valid. First, PA proves all
29
Note that different realizations may interpret ✷i as the provability predicate of different
theories in T, just as they may interpret p as different arithmetic sentences.
Possibility Semantics
local reflection principles for EA [115, 5]:
PA ⊢ ProvEA (pχq) → χ
and hence
PA ⊢ φ → ¬ProvEA (pφ → 0 = 1q).
In addition, we have the following formalized deduction theorem [58]:
PA ⊢ ProvEA+φ (pψ q) ↔ ProvEA (pφ → ψ q).
From the previous two facts, we obtain:
PA ⊢ φ → ¬ProvEA+φ (p0 = 1q)
and hence
PA ⊢ φ → ∃x(T (x) ∧ Provx (pφq)),
as EA + φ is a witness for the existential statement (recall our specification of the
predicate T above). Next, observe that S⊥ is not arithmetically T-valid, because
all theories in T are consistent and PA is sound. Finally, note that ✷i (p → Sp)
may fail to be arithmetically T-valid, if we realize i as EA and EA is unable to
prove the consistency of other theories in T and hence their membership in T.
To validate p → Sp but not ✷j (p → Sp), we can use a possibility frame with
a distinguished possibility—or equivalently, a distinguished principal filter—as
in § 5.3.5. Recall that polymodal GL (Gödel-Löb logic) is the smallest normal
polymodal logic containing the Löb axiom for each modality ✷i .
Theorem 5.3.47.
1. Any varying agent-domain full monotonic neighborhood world frame with
a distinguished principal filter (“with distinguished worlds”) that validates
p → Sp and ✷i ¬✷i ⊥ → ✷i ⊥ for each i ∈ I also validates S⊥.
2. For j ∈ I, there are varying agent-domain full relational possibility frames
with a distinguished possibility validating p → Sp and all theorems of polymodal GL but not S⊥ or ✷j (p → Sp).
Proof. For part 1, pick any world w that belongs to the proposition that generates
the principal filter. Then re-run the proof of Theorem 5.3.44.1.
For part 2, partition I into infinite sets I1 and I2 with j ∈ I2 , and take a
possibility frame F for L(I1 ) as in the proof of Theorem 5.3.44.2. We will build a
frame based on the disjoint union of F and any world frame G for L(I2 ) validating
Wesley H. Holliday
GL such that for some w ∈ G, Ri (w) 6= ∅ for all i ∈ I2 . Keep all accessibility links
between possibilities in F and between possibilities in G. Then for every i ∈ I2
and x ∈ F, let Ri (x) be the set of all worlds in G. For every i ∈ I1 and x ∈ G,
let Ri (x) = ∅. Then we have a possibility frame validating GL. Moreover, at the
root r of F, p → Sp is valid, but ✷j (p → Sp) is not, due to the fact that rRj w.
Thus, we may select r as the distinguished possibility to obtain the result.
Remark 5.3.48. In response to Theorem 5.3.47.1, one may try to salvage the use
of full world frame semantics for LS (I) by employing distinguished non-principal
filters (e.g., the cofinite filter, in which case the semantic value of p → Sp will
trivially belong to the filter whenever the semantic value of p is a finite set).
However, Theorem 5.3.47.2 suggests that switching to possibility frames for LS (I)
may be a more natural approach.
The use of varying agent-domains was essential for Theorem 5.3.44.2, as the
following shows.
Proposition 5.3.49. Any universal agent-domain quasi-normal relational possibility frame validating p → Sp and ✷i ¬✷i ⊥ → ✷i ⊥ for each i ∈ I also validates
S⊥.
Proof. For a quasi-normal frame (Definition 5.3.34), say that ϕ is super-valid if ϕ
is valid on the underlying frame without the set S0 . Then note the following:
1. ¬S⊥ → S¬S⊥ is valid, substituting ¬S⊥ for p in p → Sp;
2. ✷i ⊥ → S⊥ is super-valid in universal agent-domain frames
⇒ ¬S⊥ → ¬✷i ⊥ is super-valid
⇒ ✷i (¬S⊥ → ¬✷i ⊥) is super-valid30
⇒ ✷i ¬S⊥ → ✷i ¬✷i ⊥ is super-valid
⇒ ✷i ¬S⊥ → ✷i ⊥ is valid since ✷i ¬✷i ⊥ → ✷i ⊥ is valid
⇒ S¬S⊥ → S⊥ is valid
⇒ ¬S⊥ → S⊥ is valid given 1
⇒ S⊥ is valid,
which completes the proof.
30
Although ϕ being valid does not entail that ✷i ϕ is valid in a quasi-normal frame, ϕ being
super-valid does.
Possibility Semantics
However, varying agent-domains are not required in general for semantically
separating principles that cannot be separated by world frames. We give an
example in T ML. Suppose that for every truth p and theory u, there is some
theory v or other such that u correctly believes that v believes p:
p → ∃v(✷v p ∧ ✷u ✷v p).
(9)
In a world frame, take p to be a “world proposition” {w}. Then at w, the theory
u must believe that the witness to the existential quantifier is a negation-complete
theory. But by the Second Incompleteness Theorem, a negation-complete theory
must believe itself to be inconsistent: ✷v ✷v ⊥. For by negation-completeness, we
have ✷v ✷v ⊥ ∨ ✷v ¬✷v ⊥, and the right disjunct entails ✷v ⊥ and hence ✷v ✷v ⊥.
Since u believes all of this, u will believe that v believes itself to be inconsistent.
So as a world frame consequence of (9), we will have:
∃v✷u ✷v ✷v ⊥.
(10)
Yet (10) is not a possibility frame consequence of (9).
Theorem 5.3.50.
1. Any varying agent-domain full monotonic neighborhood world frame (with
distinguished principal filter) that validates p → ∃v(✷v p ∧ ✷u ✷v p) and
✷v ¬✷v ⊥ → ✷v ⊥ for each v ∈ Var also validates ∃v✷u ✷v ✷v ⊥.
2. There are universal agent-domain full relational possibility frames validating
p → ∃v(✷v p ∧ ✷u ✷v p) and ✷v (✷v p → p) → ✷v p for each v ∈ Var but not
∃v✷u ✷v ✷v ⊥.
Proof. For part 1, let F be a frame satisfying the hypothesis. To show that F
validates ∃v✷u ✷v ✷v ⊥, consider any world w in F (that belongs to the proposition
that generates the distinguished principal filter). Let M be a model based on
F such that π(p) = {w}. Then since F validates p → ∃v(✷v p ∧ ✷u ✷v p), we
have M, w g0 ∃v(✷v p ∧ ✷u ✷v p), so there is an i ∈ G(w) such that for g =
g0 [v := i], we have M, w g ✷v p ∧ ✷u ✷v p, which implies (a) {w} ∈ Ng(v) (w) and
(b) J✷v pKM
g ∈ Ng(u) (w). By (a) and monotonicity, Ng(v) (w) contains every A ⊆ W
with w ∈ A. Thus, if M, w g ¬✷v ⊥, then M, w g ✷v ¬✷v ⊥. But F validates
M
✷v ¬✷v ⊥ → ✷v ⊥, so we conclude that M, w g ✷v ⊥. Hence JpKM
g ⊆ J✷v ⊥Kg ,
M
M
so by monotonicity, J✷v pKg ⊆ J✷v ✷v ⊥Kg . Then by (b) and monotonicity again,
J✷v ✷v ⊥KM
g ∈ Ng(u) (w). Hence M, w g ✷u ✷v ✷v ⊥, so M, w g0 ∃v ✷u ✷v ✷v ⊥.
For part 2, we use a possibility frame as in the proof of Theorem 5.3.44.2 such
that the root node r has four children and every other node has at least three
Wesley H. Holliday
children (recall that the proof worked on the assumption that every node has a
finite number of children greater than one) and (b) ki = 1 for all i ∈ I. But now
we let G(x) = I for all x ∈ T , so we have a universal agent-domain frame.
The argument that ✷v (✷v p → p) → ✷v p is valid on the frame is the same
as in the proof of Theorem 5.3.44.2. The proof that p → ∃v(✷v p ∧ ✷u ✷v p) is
valid is also similar to the argument that p → Sp is valid in the proof of Theorem
5.3.44.2. Simply add to that argument that not only do we have M, x′′ g ✷v p
for the assignment g = g0 [v := f −1 (x′ )], but in fact M, y g ✷v p for any y ∈ S,
which implies M, x′′ g ✷v p ∧ ✷u ✷v p. Hence M, x g0 ∃v(✷v p ∧ ✷u ✷v p).
Finally, we claim that ∃v✷u ✷v ✷v ⊥ is not valid on F. Suppose f −1 (r) = i. Let
g0 be an assignment such that g0 (u) = i. We claim that for any model M based on
F, M, r 1g0 ∃v✷u ✷v ✷v ⊥. Where c1 is the first child of r in our enumeration of r’s
children, it suffices to show M, c1 g0 ∀v¬✷u ✷v ✷v ⊥. For j ∈ I, let g = g0 [v := j].
We will show M, c1 g ¬✷u ✷v ✷v ⊥, i.e., for all x ⊑ c1 , M, x 1g ✷u ✷v ✷v ⊥. For
all x ⊑ c1 , we have xRi c2 . Thus, it suffices to show that M, c2 1g ✷v ✷v ⊥. If
i = j, then M, c2 1g ✷v ✷v ⊥ follows from the facts that c2 Ri c3 and c3 Ri c4 . So
suppose i 6= j. Then f (j) 6= f (i), i.e., f (j) 6= r, so c3 is not under any child of
f (j). It follows, given (b) above, that c3 Rj c′2 where c′2 is the second child of f (j)
in the enumeration of f (j)’s children. Then since c′2 Rj c′3 , where c′3 is the third
child in the enumeration of f (j)’s children, we have M, c2 1g ✷v ✷v ⊥.
Corollary 5.3.51. There is a universal agent-domain full relational possibility
frame whose propositional term modal logic is not the logic of any class of varying
agent-domain full monotonic neighborhood world frames.
A systematic study of possibility semantics for the five languages introduced
above—and of the relation of possibility semantics to arithmetic interpretations
of these languages for provability logic—remains to be carried out.
Note that in this chapter, the natural examples of Kripke incomplete but fullpossibility-frame complete normal modal logics all involve devices for quantifying
over modalities or propositions. This leads to the following question.
Question 5.3.52. Is there a “natural” example of a Kripke incomplete but fullpossibility-frame complete normal propositional modal logic without devices for
quantifying over modalities or propositions?
5.3.7
Principal frames and V-incompleteness in provability logic
Since relational possibility frames are more general than Kripke frames, they can
be used not only to overcome some Kripke incompleteness results but also to
deepen other Kripke incompleteness results, i.e., to prove that some logics known
Possibility Semantics
to be Kripke incomplete are in fact incomplete with respect to even more general
semantics. An open problem concerning modal incompleteness from [122, 168] was
whether there is any modal logic that is not the logic of any class of V-BAOs—
whether there is a “V-incomplete” logic. Ideas from possibility semantics led to
a positive answer to this question in [101]. Here we give a possibility semantic
proof of two of the main theorems of [101], reproduced in Corollary 5.3.55 below.
Let vB be the smallest normal modal logic containing the axiom
• ✷¬✷⊥ → ✷(✷(✷p → p) → p).
Van Benthem [8] introduced this logic, inspired by provability logic (the formula
under the ✷ in the consequent, ✷(✷p → p) → p), is a theorem of the provability
logic GLS), and proved that it is Kripke frame incomplete: ✷¬✷⊥ → ✷⊥ is a
Kripke frame consequence of vB but is not a theorem of vB.
Another Kripke incomplete logic—this time not just inspired by provability
logic but a key system in the field—is the bimodal provability logic GLB, which
is the smallest normal bimodal logic containing the following axioms:
• ✷i (✷i → p) → ✷i p for i ∈ {0, 1};
• ✷0 p → ✷1 p;
• ✸0 p → ✷1 ✸0 p.
An arithmetic interpretation of GLB interprets ✷0 as provability in PA and ✷1 as
provability in PA with one application of the ω-rule (or in PA together with all
Π01 arithmetic truths). Japaridze [108] proved the arithmetic completeness of GLB
under this interpretation, as well as the Kripke incompleteness of GLB: ✷1 ⊥ is a
Kripke frame consequence of GLB but is not a theorem of GLB.
The logics vB and GLB are related by the fact that the following is a theorem
of GLB [101]:
• ✷1 (✷0 (✷0 p → p) → p).
Now we will show that ✷¬✷⊥ → ✷⊥ is a consequence of vB and ✷1 ⊥ a consequence of GLB over a class of possibility frames that can represent all V-BAOs.
Lemma 5.3.53. Let F be a paradigm possibility frame in which every principal
downset is an admissible set, and let ϕ be a formula not containing p. Suppose
that ϕ → ✷1 (✷0 (✷0 ¬p → ¬p) → ¬p) (where 0 and 1 are two modal indices that
may be equal) is globally true in every model based on F in which π(p) is a
principal downset. Then ϕ → ✷1 ⊥ is valid on F.
Wesley H. Holliday
Proof. Suppose N is a model based on F in which N , x
ϕ. We claim that
N,x
✷1 ⊥. Suppose for contradiction that N , x 1 ✷1 ⊥, so there is some y
such that xR1 y. Let M be the model on F that differs from N at most in that
π(p) = ↓y. Then since ϕ does not contain p, we have M, x ϕ. We claim that
M, y 1 ✸0 (p ∧ ✷0 ¬p).
(11)
✸0 (p ∧ ✷0 ¬p),
(12)
For if
M, y
then ↓y = JpKM ⊆ J✸0 (p ∧ ✷0 ¬p)KM , in which case (12) and the monotonicity of
✸0 together imply
M, y ✸0 (✸0 (p ∧ ✷0 ¬p) ∧ ✷0 ¬p),
(13)
which is impossible, since
✸0 (✸0 (p ∧ ✷0 ¬p) ∧ ✷0 ¬p)
⇒
✸0 (✸0 p ∧ ✷¬p)
⇒
✸0 ⊥
⇒
⊥.
Given (11) and M, y p, by refinability and persistence there is a y ′ ⊑ y such that
M, y ′ ¬✸0 (p ∧ ✷0 ¬p) ∧ p, which is equivalent to M, y ′ ✷0 (✷0 ¬p → ¬p) ∧ p.
Now by R-down and R-refinability, given xR1 y and y ′ ⊑ y, there is an x′ ⊑ x
such that for all x′′ ⊑ x′ , there is a y ′′ ⊑ y ′ such that x′′ R1 y ′′ . Since y ′′ ⊑ y ′ and
M, y ′ ✷0 (✷0 ¬p → ¬p)∧p, we have M, y ′′ ✷0 (✷0 ¬p → ¬p)∧p. Thus, for every
x′′ ⊑ x′ , there is a y ′′ ⊑ y ′ such that x′′ R1 y ′′ and M, y ′′ ✷0 (✷0 ¬p → ¬p) ∧ p,
which implies that M, x′
✸1 (✷0 (✷0 ¬p → ¬p) ∧ p). Then since x′ ⊑ x and
M, x ϕ, we have M, x′
ϕ ∧ ✸1 (✷0 (✷0 ¬p → ¬p) ∧ p). But this contradicts
the fact that M, x′ ϕ → ✷1 (✷0 (✷0 ¬p → ¬p) → ¬p).
Theorem 5.3.54.
1. vB is not the logic of any class of paradigm relational possibility frames in
which every principal downset is an admissible set.
2. GLB is not the logic of any class of paradigm relational possibility frames in
which every principal downset is an admissible set.
Proof. For part 1, by Lemma 5.3.53, any such possibility frame validating vB also
validates the non-theorem ✷¬✷⊥ → ✷⊥. For part, by Lemma 5.3.53, any such
possibility frame validating GLB also validates the non-theorem ✷1 ⊥.
Possibility Semantics
Corollary 5.3.55.
1. vB is not the logic of any class of V-BAOs.
2. GLB is not the logic of any class of V-BAOs.
Proof. By Theorem 5.3.23, every V-BAO has the same logic as a paradigm relational possibility frame in which every principal downset is admissible. Thus, if L
is the logic of a class of V-BAOs, then it is the logic of a class of possibility frames
in which every principal downset is admissible. Now apply Theorem 5.3.54.
We have now seen several fruitful connections between possibility semantics
and provability logic in Remark 5.2.20, § 5.3.6, and the results above.
5.3.8
General frames
As in § 5.2.3, where we saw that general neighborhood possibility frames can
represent arbitrary BAEs, let us briefly consider how general relational possibility
frames can represent arbitrary BAOs—choice free.
Proposition 5.3.56 ([96]). Given any BAO B = (B, {✷i }i∈I ), define
Bf = (Bf , {Ri }i∈I ) and Bg = (Bg , {Ri }i∈I )
where Bf and Bg are as in Definition 3.3.7, and for F, F ′ ∈ PropFilt(B),
F Ri F ′ iff for all a ∈ B, ✷i a ∈ F implies a ∈ F ′ .
Then:
1. Bf and Bg are strong relational possibility frames;
2. (Bf )b is (up to isomorphism) the canonical extension of B;
3. (Bg )b is isomorphic to B.
Proposition 5.3.57 ([96]). Let F = (S, ⊑, P, {Ri }i∈I ) be a general relational
possibility frame. Then F is isomorphic to (F b )g if and only if (S, ⊑, P ) is a
filter-descriptive possibility frame and F is R-tight (recall Definition 5.3.14). We
also call such an F filter-descriptive.
Morphisms between filter-descriptive relational possibility frames are
p-morphisms as in Definition 5.3.29 such that the inverse image of any admissible
proposition is also an admissible proposition.
Wesley H. Holliday
Definition 5.3.58. Given relational possibility frames F = (S, ⊑, P, {Ri }i∈I )
and F ′ = (S ′ , ⊑′ , P ′ , {Ri′ }i∈I ), a p-morphism from F to F ′ is a map h : S → S ′
satisfying the back and forth conditions 1, 2′ , 3, and 4′ from Definition 5.3.29, as
well as the condition that for all Z ′ ∈ P ′ , h−1 [Z ′ ] ∈ P .
We can now relate filter-descriptive relational possibility frames and BAOs
categorically as follows.
Proposition 5.3.59.
1. Filter-descriptive relational possibility frames together with p-morhisms
form a category, FiltRelPoss [96];
2. BAOs together with BAO-homomorphisms form a category, BAO.
Theorem 5.3.60 ([96]). FiltRelPoss is dually equivalent to BAO.
Remark 5.3.61. Theorem 5.3.60 is a choice-free possibility semantic analogue of
Goldblatt’s [77] result that the category of descriptive world frames is dually
equivalent to BAO. In [96] it is also shown that FiltRelPoss is a reflective
subcategory of the category Poss of all relational possibility frames together
with maps called possibility morphisms. This is an analogue of Goldblatt’s [78]
result that the category of descriptive world frames is a reflective subcategory of
the category of all general world frames with modal maps.
Building on § 3.4.1, general relational possibility frames can be related to
topological relational possibility frames by using the admissible propositions to
generate a topology. Filter-descriptive relational possibility frames then correspond to “modal UV-spaces” [26].
5.4
Functional frames
The move from worlds to possibilities makes possible a move from accessibility
relations to accessibility functions. Suppose F = (S, ⊑, P, {Ri }i∈I ) is a possibility
frame for L(I) such that for each i ∈ I and x ∈ S, there is an fi (x) ∈ S such that
Ri (x) = ↓fi (x),
recalling that ↓y = {y ′ ∈ S | y ′ ⊑ y}. Then we may replace the relations Ri by
functions fi , obtaining a functional frame F = (S, ⊑, P, {fi }i∈I ), so that the truth
clause for ✷i becomes:
M, x
✷i ϕ iff M, fi (x)
ϕ.
Possibility Semantics
This idea was developed in [93] (cf. [102]). For a reading of ✷i as a belief/knowledge modality, fi (x) represents “the world as agent i in x believes/knows it to
be,” which may of course be partial. As observed in [96], full functional possibility
frames correspond to complete T -BAOs as in § 5.3.3, which we saw were equivalent to complete V-BAOs. Hence we may always switch from a full relational
possibility frames to a full functional possibility frame realizing the same BAO.
For functional frames, we have the following elegant analogues of the conditions
on the interaction of Ri and ⊑ from § 5.3:
• f -persistence: if x′ ⊑ x, then fi (x′ ) ⊑ fi (x);
• f -refinability: if y ⊑ fi (x), then ∃x′ ⊑ x ∀x′ ⊑ x fi (x′′ ) ≬ y.
Example 5.4.1. The full possibility frame constructed in the proof of Proposition
5.3.43 may be viewed as a functional possibility frame.
5.5
First-order modal logic
In this section, we combine possibility semantics for first-order logic (§ 4) and
possibility semantics for modal logic (§ 5) to obtain possibility semantics for firstorder modal logic. This was done first by Harrison-Trainor [90], though our setup
below will be slightly different. The language is defined just as in first-order logic
but with the extra inductive clause that if ϕ is a formula, so is ✷i ϕ for any i ∈ I.
In addition to adding accessibility relations (or neighborhood functions) to
the first-order possibility models of Definition 4.1.1 to interpret the ✷i , we add a
varying domain function for the usual reasons that varying domains are allowed
in first-order modal logic (see [66, § 4.7]). As a result, the base logic is a free logic
instead of classical first-order logic, as ∀xϕ → ϕxt is no longer valid and must be
replaced by
(∀xϕ ∧ ∃x t = x) → ϕxt .
For simplicity, here we only consider full frames in which all regular open
downsets are admissible propositions.
Definition 5.5.1. A (full) first-order relational possibility frame is a tuple
F = (S, ⊑, D, ≍, d, {Ri }i∈I ) where:
1. (S, ⊑) is a poset;
2. D is a nonempty set;
3. ≍ is a function assigning to each s ∈ S an equivalence relation ≍s on D as
in Definition 4.1.1;
Wesley H. Holliday
4. d is a function assigning to each s ∈ S a subset d(s) ⊆ D satisfying:
• persistence for d: if a ∈ d(s), s′ ⊑ s, and a ≍s′ b, then b ∈ d(s′ );
• refinability for d: if a ∈ D \ d(s), then ∃s′ ⊑ s ∀s′′ ⊑ s′ a 6∈ d(s′′ ).
5. (S, ⊑, RO(S, ⊑), {Ri }i∈I ) is a full relational possibility frame as in Definition 5.3.4.
A first-order relational world frame is a first-order relational possibility frame in
which ⊑ is the identity relation.
A first-order relational possibility model is a tuple
M = (S, ⊑, D, ≍, d, {Ri }i∈I , V )
where (S, ⊑, D, ≍, d, {Ri }i∈I ) is a first-order relational possibility frame and
(S, ⊑, D, ≍, V ) is a first-order possibility model as in Definition 4.1.1.
Definition 5.5.2. We define the satisfaction relation M, s g ϕ as in Definition
4.1.6 but with a modified quantifier clause and the additional modal clause:
• M, s
g
∀xϕ iff for all a ∈ d(s), M, s
• M, s
g
✷i ϕ iff for all s′ ∈ Ri (s), M, s′
g[x:=a]
g
ϕ.
ϕ.
In addition to satisfaction, we have the usual notion of frame validity from
modal logic.
Definition 5.5.3. Given a first-order relational possibility frame F and formula
ϕ, we say that ϕ is valid on F if for every model M based on F, every possibility
s in F, and every variable assignment g, we have M, s g ϕ.
In the first-order setting, we have some of the simplest world-incompleteness
results. Consider a first-order modal language with a single modality ✷ and a
single unary predicate Q. Let E(x) (“x exists”) abbreviate the formula ∃x′ x′ = x,
where x′ is a variable distinct from x. Now consider the following principle:
Q(c) → ∃x✷(E(x) ↔ Q(c)).
(Fact)
Intuitively, (Fact) says that if c has property Q, then there exists an x—namely
the fact that c has property Q—such that necessarily, x exists iff c has property
Q. This principle was considered for first-order modal logic by Fine [62, p. 88]. It
can be traced back to Bertrand Russell’s logical atomism, as explained in [112]:
Possibility Semantics
The simplest propositions in the language of Principia Mathematica
are what Russell there called “elementary propositions”, which take
forms such as “a has quality q”, “a has relation [in intension] R to b”,
or “a and b and c stand in relation S” (PM, 43-44). Such propositions consist of a simple predicate, representing either a quality or a
relation, and a number of proper names. According to Russell, such
a proposition is true when there is a corresponding fact or complex,
composed of the entities named by the predicate and proper names
related to each other in the appropriate way. E.g., the proposition “a
has relation R to b” is true if there exists a corresponding complex in
which the entity a is related by the relation R to the entity b. If there
is no corresponding complex, then the proposition is false.
Any world frame validating (Fact) validates the additional principle, also
considered by Fine [62, p. 88], that there is a single fact—the “world fact”—whose
existence entails the existence of everything else:
∃x∀y✷(E(x) → E(y)).
(World)
Yet (World) is not a theorem of the minimal free first-order modal logic containing as axioms (Fact) and the axioms of S5 for ✷. Both of these claims are
shown by the following.
Proposition 5.5.4.
1. Any first-order relational world frame that validates (Fact) also validates
(World).
2. There are first-order relational possibility frames in which R is the universal
relation that validate (Fact) but not (World).
Proof. For part 1, suppose (Fact) is valid on a world frame F. Define a model
M based on F where Q(c) is true only at a single world w. Then since (Fact) is
valid, it follows that there is an a ∈ d(w) such that for all v ∈ R(w), if v 6= w, then
a 6∈ d(v). Thus, E(x) is true only at w relative to the relevant variable assignment
mapping x to a. It follows that a is a witness to the truth of (World) at w.
For part 2, we define a first-order relational possibility frame based on the full
infinite binary tree 2<ω regarded as a poset (2<ω , ⊑) as in Examples 3.1.1 and
3.2.1. Let D, the domain of objects, be RO(S, ⊑), the set of propositions. For
each s ∈ 2<ω , let ≍s be the identity relation, and let d(s) be the set of all sets
from RO(S, ⊑) to which s belongs. Then note that persistence and refinability
for d both hold. Finally, let R be the universal relation on 2<ω . Then we have
Wesley H. Holliday
defined a first-order relational possibility frame. Note that for any proposition
Z ∈ RO(S, ⊑), there is an object in D, namely Z itself, such that for all s ∈ 2<ω ,
Z ∈ d(s) iff s ∈ Z.
It follows that (Fact) is valid on the frame. But since RO(S, ⊑) is atomless, it
is easy to see that (World) is not valid on the frame.
Remark 5.5.5. In part 1, we can more generally consider first-order monotonic
neighborhood world frames [3] and obtain the same result that any such frame
that validates (Fact) validates (World), which yields the following.
Corollary 5.5.6. There is a first-order modal logic that is the logic of a firstorder relational possibility frame but not of any class of first-order monotonic
neighborhood world frames.
The theory of possibility semantics for first-order modal logic is still in its
infancy. Any logic complete with respect to first-order world frames is immediately complete with respect to first-order possibility frames, but it remains to
systematically study first-order modal logics that are possibility complete but
world incomplete.
6
Further connections and directions
In this section, we briefly cover some connections between possibility semantics
and other frameworks (for more connections, see [96, § 8.1]), namely interval semantics for temporal logic (§ 6.1) and Kripke semantics for bimodal logics (§ 6.2).
We then turn to two non-classical generalizations of possibility semantics, namely
possibility semantics for intuitionistic logic (§ 6.3) and inquisitive logic (§ 6.4).
As noted in § 1, another impossibility non-classical generalization is possibility
semantics for orthologic, for which we refer the reader to [102].
6.1
Interval semantics
Humberstone’s [104] original development of possibility semantics for modal logic
was inspired by interval semantics for temporal logic (see [103], [106, Ex. 4.1.12]).
In such semantics, the truth of a formula of temporal logic is defined not with
respect to an instant of time, as in the usual semantics (see, e.g., [81]), but
with respect to an interval of time. For influential early developments of interval
semantics, see [103, 110, 147, 9]. We will briefly compare one version of interval
semantics, due to Röper [147], to possibility semantics for temporal logic.
Possibility Semantics
Given a relational possibility frame as in § 5.3, call each x ∈ S a stretch of
time, and take x ⊑ y to mean that x is a substretch of y. Röper [147] argued that
propositions about “states and processes” should correspond to regular downsets
in (S, ⊑). The main difference between Röper’s approach and one using relational
possibility frames concerns the accessibility relation used to interpret temporal
modalities. In possibility semantics for temporal logic, we adopt the following
reading of xRy:
xRy iff the stretch x begins before the stretch y does.
We claim that under this interpretation of R, the conditions up-R, R-down,
R-refinability, and R-dense from § 5.3 all make good sense, not only for the
specific temporal poset in Example 5.3.10 but more generally for any poset of
temporal stretches.
We then define “ϕ is true throughout the stretch x” with our usual clauses,
including those for ✷ and ✸ (in frames satisfying R-down), now read as “it will
always be that. . . ” and “it will be sometime be that. . . ”:
• M, x
✷ϕ iff ∀y ∈ S, if xRy then M, y
• M, x
✸ϕ iff ∀x′ ⊑ x ∃y ∈ S: x′ Ry and M, y
ϕ.
ϕ.
Instead of working with the relation xRy interpreted to mean “x begins before
y,” Röper, like many authors in the literature on interval semantics, works with
a relation > interpreted as follows:
x > y iff “x wholly precedes y.”
Now the following principle, which yields persistence for modal formulas in possibility semantics, holds for Q = R but not for Q = >:
if x′ ⊑ x and x′ Qy ′ , then xQy ′ .
(For R, this is just up-R.) Thus, Röper is forced to build downward persistence
into his semantic clauses for ✷ and ✸ that use >:
• M, x
✷ϕ iff ∀x′ ⊑ x ∀y ∈ S, if x′ > y then M, y
• M, x
✸ϕ iff ∀x′ ⊑ x ∃x′′ ⊑ x′ ∃y ∈ S: x′′ > y and M, y
ϕ.
ϕ.
Thus, by working with > instead of R, Röper needs one more quantifier in each
clause. However, the two approaches can be related by assuming the following:
x > y iff ∀x′ ⊑ x x′ Ry,
Wesley H. Holliday
i.e., x wholly precedes y iff every substretch of x begins before y does. Indeed, in
Röper’s canonical model construction in [147], he defines x > y in precisely this
way from the canonical relation that we would take to be R.
There is much more than this brief sketch to say about the relation between
possibility semantics and interval semantics. We refer the interested reader to van
Benthem’s textbook [11] as a starting point for systematic comparison.
6.2
A bimodal perspective
Given a full relational possibility frame (S, ⊑, RO(S, ⊑), R) for a unimodal language, we may regard the triple (S, ⊑, R) as a bimodal Kripke frame, or in possibilistic terms, as a bimodal full world frame (S, =, ℘(S), {⊑, R}) with ⊑ now
thought of as an accessibility relation for a modality ✷⊑ alongside the modality
✷R , and the poset (S, =) now being discrete. Corresponding to this semantic
transformation is the following syntactic translation:
• p(p) = ✷⊑ ✸⊑ p;
• p(¬ϕ) = ✷⊑ ¬p(ϕ);
• p(ϕ ∧ ψ) = p(ϕ) ∧ p(ψ);
• p(✷ϕ) = ✷R p(ϕ).
This bimodal perspective on possibility semantics is the topic of [19], where it
is shown to lead to interesting translations between some well-known unimodal
logics and bimodal logics that can be seen as dynamic topological logics.
6.3
Intuitionistic case
In this section, we briefly sketch the generalization of possibility semantics for
intuitionistic logic. A much more thorough investigation of these ideas appears in
[24], which studies a hierarchy of semantics for intuitionistic logic.
Recall that algebraic semantics for intuitionistic logic is given using Heyting
algebras. A Heyting algebra H may be defined as a bounded distributive lattice
with a binary operation → that is a residual of ∧, i.e., such that for all a, b, x ∈ H,
a ∧ x ≤ b iff x ≤ a → b.
A complete Heyting algebra is a Heyting algebra that is complete as a lattice.
Complete Heyting algebras, also called locales or frames in pointfree topology,31
31
The use of ‘frame’ in both modal logic and pointfree topology for different kinds of objects
is unfortunate but now well established.
Possibility Semantics
can be equivalently defined as complete lattices L satisfying the join-infinite distributive law [29, p. 128] stating that for all a ∈ L and B ⊆ L,
a∧
_
B=
_
{a ∧ b | b ∈ B}.
Given any poset (S, ⊑), the collection Down(S, ⊑) of all downward closed sets,
ordered by inclusion, forms a locale. More generally, given any topological space
X, the collection Ω(X) of all open sets of X, ordered by inclusion, forms a locale.
These two facts are the basis for the Kripke frame semantics [52, 117, 83, 118] and
topological semantics [159, 165] for intuitionistic logic, respectively. However, only
special locales arise in these ways, as shown by the following well-known results
(see, e.g., [43, Prop. 1.1] for part 1 and [144, Prop. 5.3] for part 2).
Proposition 6.3.1. Let L be a lattice.
1. L is isomorphic to Down(S, ⊑) for a poset (S, ⊑) iff L is a locale in which
every element is a join of completely join-prime elements (where an element
W
p 6= 0 is completely join-prime if p ≤ A implies p ≤ a for some x ∈ A).
2. L is isomorphic to Ω(X) for some space X iff L is a locale in which every
element is a meet of meet-prime elements (where an element p 6= 1 is meetprime if a ∧ b ≤ p implies that a ≤ p or b ≤ p).
Just as classical possibility semantics seeks to represent all complete Boolean
algebras, not just atomic ones, intuitionistic possibility semantics seeks to represent all locales, not just the special ones in Proposition 6.3.1. This is achieved
by equipping a poset with an additional datum. To motivate the key definition
(Definition 6.3.5), we first recall relevant notions and results (cf. [69]).
Definition 6.3.2. A nucleus on a Heyting algebra H is a unary function
j : H → H such that for all a, b ∈ H:
1. a ≤ ja (increasing);
2. jja ≤ ja (idempotent);
3. j(a ∧ b) = ja ∧ jb (multiplicative).
A nucleus j is dense if j0 = 0.
A nuclear algebra is a pair (H, j) of a Heyting algebra H and a nucleus j on H.
Example 6.3.3. For any Heyting algebra H, the operation j¬¬ defined by
j¬¬ (a) = ¬¬a is a nucleus, the double negation nucleus. In the case of a poset
(S, ⊑), the nucleus j¬¬ on Down(S, ⊑) is given by:
j¬¬ Z = {x ∈ S | ∀x′ ⊑ x ∃x′′ ⊑ x′ : x′′ ∈ Z}.
Wesley H. Holliday
Another nucleus jB on Down(S, ⊑), the Beth nucleus, is defined by:
jB Z = {x ∈ S | every maximal chain in (S, ⊑) containing x intersects Z}.
We return to the connection between these nuclei and possibility semantics below.
For our purposes, the key point about nuclei is that the fixpoints of a nucleus
on a Heyting algebra again form a Heyting algebra, as in the following well knowntheorem (see, e.g., [126] or [51, p. 71]), where part 3 gives an algebraic version of
Glivenko’s Theorem (see, e.g., [146, p. 134]).
Theorem 6.3.4. For any nuclear algebra (H, j), consider the collection
Hj = {a ∈ H | ja = a}
of fixpoints of j.
1. Hj is a Heyting algebra, called the algebra of fixpoints in (H, j), under the
following operations for a, b ∈ Hj :
0j
= j0;
a ∧j b = a ∧ b;
a ∨j b = j(a ∨ b);
a →j b = a → b.
2. If H is a locale, then so is Hj , where
A ⊆ Hj .
V
j
A=
V
A and
W
j
W
A = j( A) for
3. If j is the nucleus of double negation, then Hj is a Boolean algebra.
It is easy to see in light of part 3 that classical possibility semantics is based
on taking the double negation fixpoints in the Heyting algebra Down(S, ⊑) of
downsets of a poset. By switching the nucleus from double negation to some
other non-Boolean nucleus, we obtain an intuitionistic generalization.
Definition 6.3.5. A nuclear frame is a triple (S, ⊑, j) such that (S, ⊑) is a poset
and j is a nucleus on Down(S, ⊑). The nuclear frame is dense if j is dense.
Then any nuclear frame gives us a locale Down(S, ⊑)j . As explained in [50, 51,
24], Beth semantics for intuitionistic logic [20], which predated Kripke semantics,
can be seen as a special case of this construction. Crucially, as shown by Dragalin
[50, 51, p. 75], every locale can be represented as Down(S, ⊑)j for some nuclear
frame, in contrast to Proposition 6.3.1. We include a proof, presented as in [23],
to show its connection with the analogous Theorem 3.2.3.2 in the Boolean case.
Possibility Semantics
Theorem 6.3.6 ([50]). A lattice L is a locale iff L is isomorphic to Down(S, ⊑)j
for some dense nuclear frame (S, ⊑, j).
Proof. The right-to-left direction is given by Theorem 6.3.4. For the left-to-right,
as in Theorem 3.2.3.2, let L+ be L\{0} and ≤+ the restriction to L+ of the lattice
order ≤ of L. Define a unary function j on Down(L+ , ≤+ ) by
jX = ↓
_
X,
(14)
W
where X is the join of X in L, and ↓ indicates the downset in (L+ , ≤+ ). It is easy
W
W
W
to see that j is inflationary, idempotent, and that ↓ (X ∩ Y ) ⊆ (↓ X) ∩ (↓ Y )
W
W
W
for X, Y ∈ Down(S, ⊑). To see that ↓ (X ∩ Y ) ⊇ (↓ X) ∩ (↓ Y ), suppose
W
W
that a ∈ L+ is in the right-hand side, so a ≤ X and a ≤ Y , whence a ≤
W
W
( X) ∧ ( Y ). By the join-infinite distributive law for locales,
_
(
_
X) ∧ (
Y)=
_
{x ∧ y | x ∈ X, y ∈ Y },
W
so a ≤ {x ∧ y | x ∈ X, y ∈ Y }. Since X and Y are downsets, we have {x ∧ y |
W
W
x ∈ X, y ∈ Y } ⊆ (X ∩ Y ) ∪ {0}, so {x ∧ y | x ∈ X, y ∈ Y } ≤ ((X ∩ Y ) ∪ {0}) =
W
W
W
(X ∩ Y ). Thus, a ≤ (X ∩ Y ) and hence a ∈ ↓ (X ∩ Y ). Therefore, j is a
W
nucleus. To see that j is dense, observe that j∅ = ↓ ∅ = ↓0 = ∅ since 0 6∈ S.
Finally, we must check that L is isomorphic to Down(L+ , ≤+ )j . Observe that
the elements of Down(L+ , ≤+ )j are exactly the principal downsets in (L+ , ≤+ )
plus ∅. Thus, the map sending each x to ↓x is the desired isomorphism.
The next step is to realize j in some more concrete way. For example, [23]
considers Dragalin frames, originating in [50, 51], which are triples (S, ⊑, D) where
D is a certain function from which a nucleus jD of Down(S, ⊑) can be defined.
It is then shown in [23] that for every nuclear frame (S, ⊑, j), there is a Dragalin
frame (S, ⊑, D) (based on the same poset) such that j = jD . Thus, by Theorem
6.3.6, Dragalin frames can be used to represent all locales.
Another approach, originating in [57], works with triples (S, ⊑, 6) where 6 is
another partial order (or preorder) and an operation j✷⊑ ✸6 is defined by
j✷⊑ ✸6 Z = {x ∈ S | ∀x′ ⊑ x ∃x′′ 6 x′ : x′′ ∈ Z}.
A sufficient (but not necessary) condition for j✷⊑ ✸6 to be a nucleus on Down(S, ⊑)
is that 6 is a subrelation of ⊑. Such frames are called FM-frames after Fairtlough
and Mendler [57]. For a short proof of the following, see [24, Thm. 4.33].
Theorem 6.3.7 ([23, 129]). For every locale L, there is an FM-frame (S, ⊑, 6)
such that L is isomorphic to Down(S, ⊑)j✷⊑ ✸6 .
Wesley H. Holliday
This representation of locales is a special case of a representation of arbitrary
complete lattices using doubly ordered structures [1], which is compared to other
representations of complete lattices in [99].
Let us now connect these algebraic issues back to logic. A superintuitionistic
logic (si-logic) is a set of formulas in the language of propositional logic containing
all theorems of the intuitionistic propositional calculus and closed under modus
ponens and uniform substitution (see [37]). Every class of Heyting algebras determines an si-logic in the obvious way, and every si-logic is the logic of a Heyting
algebra, namely the Lindenbaum-Tarski algebra of the logic. Well-known si-logics
are determined by more concrete Heyting algebras, such as those arising from
posets (Kripke frames) or topological spaces as in Proposition 6.3.1. However,
Shehtman [155, 156, 157] has shown the following.
Theorem 6.3.8 ([155, 156, 157]). There is an si-logic that is not the logic of any
class of Kripke frames.
In fact, there are continuum-many such si-logics [123].
A famous open problem, posed by Kuznetsov [119] in 1975, is whether there
are not only Kripke-incomplete but even topologically-incomplete si-logics.
Question 6.3.9 (Kuznetsov’s Problem for spaces). Is every si-logic the logic of
a class of topological spaces?
We can also consider the analogous question for locales.
Question 6.3.10 (Kuznetsov’s Problem for locales). Is every si-logic the logic of
a class of locales?
Given that FM-frames can represent arbitrary locales, one may hope that they
or related structures could be used as tools to make progress on Questions 6.3.96.3.10. Indeed, Massas [130] has developed a theory of intuitionistic possibility
frames (S, ⊑, 6) leading to a new kind of si-incompleteness result. We recall that
a complete bi-Heyting algebra is a locale L whose order dual L∂ is also a locale
(for recent applications of these algebras in topos quantum theory, see [48, 68]).
The following result was also obtained independently in [72] using Esakia duality.
Theorem 6.3.11 ([72, 130]). There is an si-logic that is not the logic of any class
of complete bi-Heyting algebras.
Indeed, there are continuum-many such si-logics [72].
To develop possibility semantics not only for complete Heyting algebras but
for arbitrary Heyting algebras, we need an appropriate analogue of “choice-free
Stone duality” for Heyting algebras. We leave this as an open problem.
Question 6.3.12. What is the most natural analogue for Heyting algebras of
UV-spaces for Boolean algebras?
Possibility Semantics
6.4
Inquisitive case
>
So far we have used possibility frames to interpret standard languages, e.g., the
language of propositional logic, the language of modal logic, the language of firstorder logic, etc. However, as suggested in [16], once we make the switch from structures based on a set W to structures based on a poset (S, ⊑), we may naturally
introduce new logical operators, beyond those in the traditional repertoire, that
exploit the partial order structure. An example of exactly this move comes from
the program of inquisitive logic [41, 40, 38, 42, 39], which adds a new questionto the language of propositional (or modal or first-order)
forming disjunction
logic. Intuitively, the formula p q represents the question of whether p or q, in
contrast to the formula p ∨ q, which represents the declarative sentence p or q.
Classical inquisitive logic is based on the language L defined by
>
>
>
ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | (ϕ
ϕ).
>
Let L be the fragment without . Then classical inquisitive semantics uses posets
(S, ⊑) (in fact, special posets, as in Remark 6.4.2 below), takes the semantic values
of propositional variables to be elements of RO(S, ⊑), and interprets L exactly
as in possibility semantics. But
is interpreted as follows:
>
ϕ
>
• M, x
ψ iff M, x
ϕ or M, x
ψ.
Compare this with the clause for ∨, which is defined in terms of ¬ and ∧ by the
classical definition (ϕ ∨ ψ) := ¬(¬ϕ ∨ ¬ψ):
• M, x
ϕ ∨ ψ iff ∀x′ ⊑ x′ ∃x′′ ⊑ x′ : M, x′′
ϕ or M, x′′
ψ.
>
While M, x
ϕ ∨ ψ represents that x, now conceived as an information state,
settles that the disjunctive statement is true, M, x ϕ ψ represents that the
information state x answers the question of whether ϕ or ψ. This idea has numerous applications to natural language semantics (see the references above). For
further discussion of inquisitive disjunction and related issues in connection with
possibility semantics, see [107].
Example 6.4.1. Consider a possibility model with the following poset and valuation (so p is forced only at the bottom left node, etc.):
x
y
p
q
r
z
Wesley H. Holliday
>
At the root node x, we have M, x (p ∨ q) ∨ r but M, x 1 (p ∨ q) r. For the
latter claim, observe that M, x 1 r by the valuation and M, x 1 (p ∨ q) because
z ⊑ x and M, z 1 p ∨ q. By contrast, M, y (p ∨ q) r because M, y p ∨ q.
>
Remark 6.4.2. Inquisitive logicians typically assume that their poset (S, ⊑) is of
a very special form: it is the poset of all nonempty subset of a set W , ordered
by inclusion (in the finite case these posets—or their duals—appear as Medvedev
frames in semantics for the si-logic known as Medvedev’s logic [136], as in [95]).
The poset used in Example 6.4.1 is isomorphic to such a poset. One can check
that the description of classical inquisitive semantics above is equivalent to the
usual one in the literature, assuming (S, ⊑) is of the special form.
For the special posets in Remark 6.4.2, an axiomatization of the validities of
is available, yielding classical inquisitive propositional logic (see, e.g., [41]).
Classical inquisitive logic may be understood algebraically in the terms of
§ 6.3 as follows: working with the Heyting algebra Down(S, ⊑) of downsets of a
poset, inquisitive logicians (i) interpret propositional variables as elements of the
fixpoint algebra Down(S, ⊑)j¬¬ of the double negation nucleus j¬¬ , (ii) interpret
∨ (when defined in terms of ¬ and ∧) as j¬¬ applied to the join in Down(S, ⊑), as
in possibility semantics, and (iii) interpret as the ordinary join in Down(S, ⊑).
The same perspective is adopted in [25], where a new topological semantics for
classical inquisitive logic is given using the UV-spaces discussed in § 3.4.2.
Once we have this perspective, there is a natural way to study inquisitive logic
on an intuitionistic base [98]. That is, we now fix the language L∨, as
>
L
>
>
>
ϕ ::= ⊥ | p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | (ϕ → ϕ) | (ϕ
ϕ),
>
and we interpret L∨, in nuclear frames (S, ⊑, j) where j need not be the nucleus
of double negation. As before, we (i) interpret propositional variables as elements
of Down(S, ⊑)j , (ii) interpret ∨ as j applied to the join in Down(S, ⊑), and (iii)
interpret as the ordinary join in Down(S, ⊑). In [98] this approach is instantiated
using the Beth nucleus defined in Example 6.3.3, yielding a Beth semantics (cf. [20,
24]) for inquisitive intuitionistic logic, which is completely axiomatized.
>
6.5
Further language extensions
>
The distinction between the declarative disjunction ∨ and the inquisitive disjuncin § 6.4 is an example of the kind of distinction that can be made in a
tion
semantics based on posets (S, ⊑) instead of sets W .32 There is a large literature
32
While the poset typically used in inquisitive semantics is the poset (℘(W )\{∅}, ⊆) for some
set W (see Remark 6.4.2), we still consider inquisitive semantics a semantics based on posets,
since it uses the structure of the poset (℘(W ) \ {∅}, ⊆), not just the structure of the set W .
Possibility Semantics
on using posets and related structures to interpret operators beyond the standard connectives, modalities, and quantifiers (see, e.g., [11] and [88] in a temporal
context and [17] and [18, § 12] in an informational context). Less has been written about adding semantic clauses for new operators alongside the specifically
possibility-semantic clauses for the standard connectives, modalities, and quantifiers. Thus, we finish this survey with a broad open problem for the future
development of possibility semantics.
Question 6.5.1. What other operators can we add to the languages already interpreted by possibility semantics, exploiting the extra structure in a poset of
possibilities beyond that of a flat set of worlds? For example: Epistemic modals
[102]? Indicative conditionals [102]? Counterfactuals [150]? Determinacy operators [36]? Awareness operators (cf. [87])?
7
Conclusion
Classical possibility semantics offers, in the case of discrete duality, freedom from
the restriction to atomic algebras, and in the case of topological duality, freedom
from the reliance on nonconstructive choice principles. In both cases, possibility
semantics shares some of the spirit of other research programs.
In the first case, just as forcing and Boolean-valued semantics for set theory made possible proofs of independence results in set theory that could not be
achieved by previous methods, possibility semantics for modal logic makes possible proofs of independence results that cannot be proved using atomic algebras
(Propositions 5.1.4, 5.2.21-5.2.23, and 5.3.42-5.3.43, Theorems 5.3.44, 5.3.47, and
5.3.50, and Proposition 5.5.4). Indeed, we have not only independence results
but completeness results for logics that are incomplete with respect to semantics
based on atomic algebras (for axiomatized logics, recall Theorems 5.1.6, 5.1.7,
5.2.18, 5.2.24, 5.2.28, and for semantically defined logics, recall Theorems 5.3.36
and Corollaries 5.3.45, 5.3.51, and 5.5.6). And as we have seen, possibility semantics for first-order logic essentially is forcing and Boolean-valued semantics writ
large—applied not only to the language of set theory but to arbitrary first-order
languages. Possibility semantics for first-order modal logic remains to be systematically explored, as do further applications of possibility semantics in provability
logic (recall Remark 5.2.20 and §§ 5.3.6-5.3.7).
In the second case mentioned above, just as pointfree topology seeks to separate classical results on spaces into a constructive localic argument plus the
application of a nonconstructive choice principle, “choice-free Stone duality” and
“neighborhood possibility systems” may effect a similar separation but now using
a spatial or poset-based—but still constructive—argument.
Wesley H. Holliday
Finally, there lies the frontier beyond classical logics—intuitionistic logics and
logics for extended languages that exploit the extra structure of partial states, not
provided by worlds. It remains to be seen how in these areas possibility semantics
may open up yet more semantic possibilities.
Acknowledgment
For helpful feedback on this chapter, I thank Johan van Benthem, Guram Bezhanishvili, Yifeng Ding, Mikayla Kelley, Guillaume Massas, James Walsh, and the
anonymous referee for Landscapes in Logic. For other helpful discussions of possibility semantics, I thank Nick Bezhanishvili, Matthew Harrison-Trainor, Lloyd
Humberstone, Tadeusz Litak, and Kentarô Yamamoto.
A
Appendix
In this appendix, extending § 5.3.1, we identify the interaction conditions on Ri
and ⊑ that are necessary and sufficient for RO(S, ⊑) to be closed under ✷i , as
required for a full relational possibility frame as in Definition 5.3.4. For x, y ∈ S,
let
x ⊥ y iff ↓x ∩ ↓y = ∅;
x ≬ y iff ↓x ∩ ↓y 6= ∅.
Proposition A.1 ([96]). For any poset (S, ⊑) and binary relation R on S, the
following are equivalent:
1. RO(S, ⊑) is closed under the operation ✷i defined for Z ∈ RO(S, ⊑) by
✷i Z = {x ∈ S | Ri (x) ⊆ Z};
2. R and ⊑ satisfy the following conditions:
(a) R-rule: if for all y ∈ R(x), y ⊥ z, and x′ ⊑ x, then for all y ′ ∈ R(x′ ),
y ′ ⊥ z;
(b) R⇒win: if xRy, then ∀y ′ ⊑ y ∃x′ ⊑ x ∀x′′ ⊑ x′ ∃y ′′ ∈ R(x′′ ): y ′′ ≬ y ′ .
The R-rule condition says if x has “ruled out” the possibility z, then z remains ruled out by every refinement x′ of x. Contrapositively, if z is compatible
with some possibility accessible from x′ , then z must be compatible with some
possibility accessible from x, as shown in Figure 10:
if x′ ⊑ x and x′ Ri y ′ ≬ z, then ∃y: xRi y ≬ z.
The R⇒win condition has a natural game-theoretic interpretation.
Possibility Semantics
x
x
x′
y′
z
⇒
∃
x′
y
z
y′
Figure 10: the R-rule condition.
Definition A.2. Given a poset (S, ⊑), x, y ∈ S, and binary relation R on S, the
accessibility game G(S, ⊑, R, x, y) for players A and E has the following rounds,
depicted in Figure 11:
1. A chooses a y ′ ⊑ y;
2. E chooses an x′ ⊑ x;
3. A chooses an x′′ ⊑ x′ ;
if R(x′′ ) = ∅, then A wins, otherwise play continues;
4. E chooses a y ′′ ∈ R(x′′ );
if y ′′ ≬ y ′ , then E wins, otherwise A wins.
One can think of A and E as arguing about whether y is accessible to x: if
it is, then for any way y ′ of further specifying y, there should be some way x′ of
further specifying x that “locks in” access to possibilities compatible with y ′ , i.e.,
such that all refinements x′′ of x′ have access to some possibility y ′′ compatible
with y ′ . If refinements of x cannot keep up with refinements of y in this way, then
y is not accessible to x. Thus, player A is trying to show that y is not accessible
to x, while player E is trying to block A’s argument.
Lemma A.3 ([96]). For any poset (S, ⊑) and binary relation R on S, the following
are equivalent:
1. R and ⊑ satisfy R⇒win;
2. for any x, y ∈ S, if xRy, then E has a winning strategy in G(S, ⊑, R, x, y).
A natural strengthening of R⇒win entails both R⇒win and R-rule and can
be assumed without loss of generality. There are two obvious ways to consider
Wesley H. Holliday
y
x
1. A chooses
2. E chooses
y′
x′
≬?
3. A chooses
x′′
y ′′
4. E chooses
Figure 11: the accessibility game G – if y ′′ ≬ y ′ , E wins, otherwise A wins.
strengthening R⇒win: change the final condition from y ′′ ≬ y ′ to y ′′ ⊑ y ′ , and
change the “if. . . then” to and “if and only if”. Making both of these modifications,
we obtain:
• R⇔win – xRi y iff ∀y ′ ⊑ y ∃x′ ⊑ x ∀x′′ ⊑ x′ ∃y ′′ ∈ R(x′′ ): y ′′ ⊑ y ′ .
Changing the winning condition of the game G(S, ⊑, R, x, y) to y ′′ ⊑ y ′′ gives us
the game G(S, ⊑, R, x, y), as in Figure 12, for which the following holds.
Lemma A.4 ([96]). For any poset (S, ⊑) and binary relation R on S, the following
are equivalent:
1. R and ⊑ satisfy R⇔win;
2. for any x, y ∈ S, xRy iff E has a winning strategy in G(S, ⊑, R, x, y).
y
x
2. E chooses
1. A chooses
y′
x′
?
3. A chooses
x′′
y ′′
4. E chooses
Figure 12: the accessibility game G – if y ′′ ⊑ y ′ , E wins, otherwise A wins.
Finally, we can relate R⇔win to the notion of a strong possibility frame from
Definition 5.3.12.
Possibility Semantics
Lemma A.5 ([96]). R⇔win is equivalent to the conjunction of up-R, R-down,
R-refinability, and R-dense. Hence a possibility frame is strong iff it satisfies
R⇔win.
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Last revised: Friday 10th May, 2024
Wesley Holliday