arXiv:submit/5729843 [math.LO] 13 Jul 2024 Preconditionals W ESLEY H. H OLLIDAY1 Abstract: In recent work, we introduced a new semantics for conditionals, covering a large class of what we call preconditionals. In this paper, we undertake an axiomatic study of preconditionals and subclasses of preconditionals. We then prove that any bounded lattice equipped with a preconditional can be represented by a relational structure, suitably topologized, yielding a single relational semantics for conditional logics normally treated by different semantics, as well as generalizing beyond those semantics. Keywords: conditionals, Heyting algebras, ortholattices, orthomodular lattices, Sasaki hook, indicatives, counterfactuals, flattening, relational frames 1 Introduction Conditionals in their different flavors—material, strict, indicative, counterfactual, probabilistic, constructive, quantum, etc.—have long been of central interest in philosophical logic (see Egré and Rott 2021 and references therein). In this paper, based on a talk at Logica 2023, we further investigate a new approach to conditionals introduced in our recent work on the representation of lattices with conditional operations (Holliday 2023, § 6). We define a preconditional → on a bounded lattice to be a binary operation satisfying five natural axioms, which we show to be independent (Section 2.1). We also consider the properties of the associated negation defined by ¬a = a → 0 (Section 2.2). Familiar examples of bounded lattices equipped with a preconditional include Heyting algebras (Section 2.3), ortholattices with the Sasaki hook (Section 2.4), and Lewis-Stalnaker-style conditional algebras satisfying the so-called flattening axiom (Section 2.5). We characterize these classes axiomatically in terms of additional independent axioms they satisfy beyond those of preconditionals. We then show (Section 3) that every bounded lattice equipped with a preconditional can be represented using a relational structure (X, ⊳), suitably topologized. This yields a single relational semantics for conditional logics 1 Forthcoming in The Logica Yearbook 2023, ed. Igor Sedlár, College Publications. Thanks to Yifeng Ding, Matt Mandelkern, Guillaume Massas, and Snow Zhang for helpful comments. 1 Wesley H. Holliday normally treated by different semantics, as well as a generalization beyond those semantics. We conclude (Section 4) with some suggested directions for further development of this approach to conditionals. 2 Preconditionals 2.1 The axioms and their independence The definition of a preconditional from Holliday 2023 was discovered through an attempt to axiomatize the class of lattices with an implication operation amenable to a relational representation described in Section 3. However, here we will begin with axiomatics and turn to representation only at the end. Definition 1 Given a bounded lattice L, a preconditional on L is a binary operation → on L satisfying the following for all a, b, c ∈ L: 1. 1 → a ≤ a; 2. a ∧ b ≤ a → b; 3. a → b ≤ a → (a ∧ b); 4. a → (b ∧ c) ≤ a → b; 5. a → ((a ∧ b) → c) ≤ (a ∧ b) → c. Arguably all of the axioms are intuitively valid for both indicative conditionals and counterfactual conditionals in natural language, but we will not make that case here. Instead, let us begin with the following easy check. Fact 1 The axioms of preconditionals are mutually independent. Proof. For each axiom, we provide a lattice with a binary operation → in which the axiom does not hold but it is easy to check that the other axioms do. For axiom 1, consider the two-element lattice on {0, 1} with → defined by a → b = 1. Since 1 → 0 = 1 6≤ 0, axiom 1 does not hold. For axiom 2, consider the lattice on {0, 1} with → defined by a → b = 0. Since 1 ∧ 1 = 1 6≤ 0 = 1 → 1, axiom 2 does not hold. For axiom 3, consider the lattice on {0, 1} with → defined by a → b = b. Since 0 → 1 = 1 6≤ 0 = 0 → 0 = 0 → (0 ∧ 1), axiom 3 does not hold. For axiom 4, consider the lattice with → on the left of Figure 1. Since 0 → (1 ∧ 0) = 0 → 0 = 1 6≤ 1/2 = 0 → 1, axiom 4 does not hold. Finally, for axiom 5, consider the lattice with → on the right of Figure 1. Since 1/2 → ((1/2 ∧ 0) → 0) = 1/2 → (0 → 0) = 1/2 → 0 = 1 6≤ 0 = 0 → 0 = (1/2 ∧ 0) → 0, axiom 5 does not hold. 2 Preconditionals 1 → 0 1/2 1 1/2 0 1 0 0 1/2 1/2 1/2 1/2 1 1/2 1/2 1 → 0 1/2 1 0 0 1 0 1/2 0 1 1/2 1 0 1 1 0 Figure 1: Left: lattice L and → demonstrating independence of axiom 4. Right: → demonstrating independence of axiom 5. 2.2 Precomplementation The preconditional axioms settle some basic properties of the negation operation defined from → by ¬x := x → 0. Proposition 1 Let L be a bounded lattice with a preconditional →. Then defining ¬x := x → 0, we have: 1. a ≤ b implies ¬b ≤ ¬a; 2. ¬1 = 0. Proof. For part 1, if a ≤ b, then we have b→0 ≤ b → ((b ∧ a) → 0) by axiom 4 of preconditionals ≤ (b ∧ a) → 0 by axiom 5 of preconditionals ≤ a→0 since b ∧ a = a from a ≤ b. Part 2 is immediate from axiom 1 of preconditionals. Following Holliday 2023, we call a unary operation ¬ satisfying parts 1 and 2 of Proposition 1 a precomplementation. Given a precomplementation, we can induce a preconditional as follows—an idea to which we will return in the context of ortholattices in Section 2.4. Proposition 2 Let L be a bounded lattice equipped with a precomplementation ¬. Then the binary operation → defined by a → b := ¬a ∨ (a ∧ b) is a preconditional. 3 Wesley H. Holliday Proof. Using the definition of →, the axioms of preconditionals become: 1. ¬1 ∨ (1 ∧ a) ≤ a; 2. a ∧ b ≤ ¬a ∨ (a ∧ b); 3. ¬a ∨ (a ∧ b) ≤ ¬a ∨ (a ∧ (a ∧ b)); 4. ¬a ∨ (a ∧ (b ∧ c)) ≤ ¬a ∨ (a ∧ b); 5. ¬a ∨ (a ∧ (¬(a ∧ b) ∨ ((a ∧ b) ∧ c))) ≤ ¬(a ∧ b) ∨ ((a ∧ b) ∧ c). Axiom 1 holds given the assumption that ¬1 = 0. Axioms 2-4 follow from the axioms for lattices. Axiom 5 holds given the assumption that a ∧ b ≤ a implies ¬a ≤ ¬(a ∧ b). 2.3 Heyting implication As suggested in Section 1, several familiar conditional operations are examples of preconditionals. Our first example is the Heyting implication in Heyting algebras. Consider the following axioms: • modus ponens (MP): a ∧ (a → b) ≤ b; • weak monotonicity: b ≤ a → b. Fact 2 1. Modus ponens is independent of the axioms of preconditionals plus weak monotonicity. 2. Weak monotonicity is independent of the axioms of preconditionals plus modus ponens. Proof. For modus ponens, consider the lattice with → in Figure 2. We have b ∧ (b → a) = b ∧ 1 = b 6≤ a, so modus ponens does not hold, but one can check that the other axioms do. For weak monotonicity, consider the two-element lattice on {0, 1} with → defined by a → b = a ∧ b. Since 1 6≤ 0 = 0 → 1, weak monotonicity does not hold, but one can check that the other axioms do. We can characterize Heyting implications as preconditionals satisfying the two axioms above. 4 Preconditionals 1 b a 0 → 0 a b 1 0 1 0 0 0 a 1 1 1 a b 1 1 1 b 1 1 1 1 1 Figure 2: (L, →) demonstrating independence of modus ponens in Fact 2.1. Proposition 3 For any bounded lattice L and binary operation → on L, the following are equivalent: 1. → is a Heyting implication, i.e., for all a, b, c ∈ L, a ∧ b ≤ c iff a ≤ b → c; 2. → is a preconditional satisfying modus ponens and weak monotonicity; 3. → satisfies axioms 3 and 4 of preconditionals, modus ponens, and weak monotonicity. Proof. The implication from 1 to 2 is straightforward and the implication from 2 to 3 is immediate. From 3 to 1, supposing a ≤ b → c, we have a ∧ b ≤ (b → c) ∧ b ≤ c by modus ponens. Conversely, supposing a ∧ b ≤ c, we have a ≤ ≤ b → a by weak monotonicity b → (a ∧ b) by axiom 3 of preconditionals ≤ ≤ b → (a ∧ b ∧ c) by our assumption that a ∧ b ≤ c b → c by axiom 4 of preconditionals. Fact 3 Axioms 3 and 4 of preconditionals, modus ponens, and weak monotonicity are mutually independent. Proof. For axiom 3, we can again use the two-element lattice on {0, 1} with → defined by a → b = b, as in the proof of Fact 1. For axiom 4, consider the three-element lattice on {0, 1/2, 1} with → defined as follows: x → y = 1 if x = y and otherwise x → y = y. Then 0 → 0 = 1 6≤ 1/2 = 0 → 1/2, so axiom 4 does not hold. However, one can check that axiom 3, modus ponens, and weak monotonicity hold. 5 Wesley H. Holliday For modus ponens, consider again the two-element lattice on {0, 1} with → defined by a → b = 1. Since 1 ∧ (1 → 0) = 1 ∧ 1 = 1 6≤ 0, modus ponens does not hold, but the other axioms clearly do. Finally, for weak monotonicity, consider again the two-element lattice on {0, 1} with → defined by a → b = 0. Since 1 6≤ 0 = 1 → 1, weak monotonicity does not hold, but the other axioms clearly do. A natural weakening of modus ponens to consider is that the derived ¬ operation is a semicomplementation: a ∧ (a → 0) = 0. Let us say that a proto-Heyting implication is a preconditional satisfying weak monotonicity and semicomplementation. The implication used in the proof of Fact 2.1 is a proto-Heyting implication that is not a Heyting implication. 2.4 Sasaki hook For our second example, an ortholattice is a bounded lattice L equipped with a unary operation ¬, called an orthocomplementation, satisfying • antitonicity: a ≤ b implies ¬b ≤ ¬a, • semicomplementation: a ∧ ¬a = 0, and • involution: ¬¬a = a. From these properties, one can derive excluded middle (a ∨ ¬a = 1)2 and De Morgan’s laws (¬(a ∨ b) = ¬a ∧ ¬b and ¬(a ∧ b) = ¬a ∨ ¬b). In an ortholattice, the Sasaki hook is the binary operation defined by s a → b := ¬a ∨ (a ∧ b) = ¬(a ∧ ¬(a ∧ b)). The following is immediate from Proposition 2. Corollary 1 In any ortholattice, the Sasaki hook is a preconditional. Next we add axioms on a preconditional → to characterize the Sasaki hook. First, note that one half of the equation a → b = ¬a ∨ (a ∧ b), where ¬ is now defined from →, already follows from the preconditional axioms. Lemma 1 For any preconditional → on a bounded lattice, we have (a → 0) ∨ (a ∧ b) ≤ a → b. 2 Since a ≤ a ∨ ¬a and ¬a ≤ a ∨ ¬a, we have ¬(a ∨ ¬a) ≤ ¬a ∧ ¬¬a = 0, so ¬0 ≤ ¬¬(a ∨ ¬a) = a ∨ ¬a. Finally, 1 ≤ ¬¬1 and ¬1 = 1 ∧ ¬1 = 0, so 1 ≤ ¬0, which with the previous sentence implies 1 ≤ a ∨ ¬a. 6 Preconditionals Proof. We have a → 0 ≤ a → b by axiom 4 of preconditionals and a ∧ b ≤ a → b by axiom 2 of preconditionals, so (a → 0) ∨ (a ∧ b) ≤ a → b. To prove the reverse inequality, we assume that the negation defined by → is an involutive semicomplementation. Proposition 4 For any bounded lattice L and binary operation → on L, the following are equivalent: 1. → is a preconditional with a ∧ (a → 0) = 0 and (a → 0) → 0 = a; 2. L equipped with ¬ defined by ¬a := a → 0 is an ortholattice and → coincides with the Sasaki hook: a → b = ¬a ∨ (a ∧ b). Proof. From 2 to 1, that → is a preconditional follows from Corollary 1. That a ∧ (a → 0) = 0 and (a → 0) → 0 = a follows from the definition of ¬ and the assumption that ¬ is an orthocomplementation. From 1 to 2, first we show that ¬ is an orthocomplementation. Both a ∧ ¬a = 0 and ¬¬a = a follow from our assumptions on → and the definition of ¬. That a ≤ b implies ¬b ≤ ¬a is given by Proposition 1.1. Finally, we show a → b = ¬a ∨ (a ∧ b). The right-to-left inclusion is given by Lemma 1, so it only remains to show a → b ≤ ¬a ∨ (a ∧ b): x ∧ ¬y ≤ ¬y ⇒ ¬¬y ≤ ¬(x ∧ ¬y) by antitonicity for ¬ ⇒ y ≤ ¬(x ∧ ¬y) by involution for ¬ ⇒ x → y ≤ x → ¬(x ∧ ¬y) by axiom 4 of preconditionals ⇒ x → y ≤ x → ((x ∧ ¬y) → 0) by definition of ¬ ⇒ x → y ≤ (x ∧ ¬y) → 0 by axiom 5 of preconditionals ⇒ x → y ≤ ¬(x ∧ ¬y) by definition of ¬ ⇒ x → y ≤ ¬x ∨ y by De Morgan’s law and involution for ¬ ⇒ a → (a ∧ b) ≤ ¬a ∨ (a ∧ b) substituting a for x, a ∧ b for y ⇒ a → b ≤ ¬a ∨ (a ∧ b) by axiom 3 of preconditionals. An ortholattice is orthomodular if a ≤ b implies b = a ∨ (¬a ∧ b). In fact, as observed by Mittelstaedt (1972), orthomodularity is equivalent to the Sasaki hook satisfying modus ponens. Lemma 2 (Mittelstaedt) An ortholattice L is orthomodular if and only if a ∧ (¬a ∨ (a ∧ b)) ≤ b for all a, b ∈ L. 7 Wesley H. Holliday Combining Lemma 2 with Proposition 4 yields the following. Proposition 5 For any bounded lattice L and binary operation → on L, the following are equivalent: 1. → is a preconditional satisfying modus ponens and (a → 0) → 0 = a; 2. L equipped with ¬ defined by ¬a := a → 0 is an orthomodular lattice and → coincides with the Sasaki hook: a → b = ¬a ∨ (a ∧ b). Figure 3 summarizes the relations between the classes of preconditionals covered so far (OL and OML stand for ortho- and orthomodular lattices). We also add the classical material implication of Boolean algebras, which is equivalent to Heyting implication with involution of ¬ and to Sasaki hook in orthomodular lattices with weak monotonicity (by Proposition 3, Lemma 2). classical material implication (a → 0) → 0 = a b≤a→b Heyting implication Sasaki hook in OML (a → 0) → 0 = a b≤a→b a ∧ (a → b) ≤ b a ∧ (a → b) ≤ b proto-Heyting implication preconditionals with MP Sasaki hook in OL a ∧ (a → b) ≤ b (a → 0) → 0 = a b≤a→b preconditionals with semicomp a ∧ (a → 0) = 0 preconditionals Figure 3: classes of preconditionals. 8 Preconditionals 2.5 Lewis-Stalnaker-style conditionals The third example of lattices with preconditionals that we will consider are Boolean algebras with Lewis-Stalnaker-style conditionals (Stalnaker 1968, Lewis 1973) satisfying the axiom of flattening (Mandelkern in press, §§ 6.4.16.4.2, citing Cian Dorr, p.c.): a → ((a ∧ b) → c) = (a ∧ b) → c. Axiom 5 of preconditionals is simply the left-to-right inclusion. Lewis-Stalnaker-style (set-)selection function semantics in effect treats a conditional a → b as the result of applying an a-indexed normal modal operator a to b. That is, there is a binary relation Ra between worlds, and a → b is true at w iff all Ra -accessible worlds from w make b true. Further constraints are imposed so that the relations Ra can be derived from wellfounded preorderings of the set of worlds: wRa v iff v is one of the closest a-worlds to w according to the well-founded preorder 6w associated with w. But for our purposes here, the key aspects of the Lewis-Stalnaker (set)selection function semantics are captured by the following definition. Definition 2 A selection frame is a pair (W, {RA }A⊆W ) where W is a nonempty set and each RA is a binary relation on W satisfying the following for all w, v ∈ W and A ⊆ W : 1. success: if wRA v, then v ∈ A; 2. centering: if w ∈ A, then wRA v iff v = w. Such a frame is functional if it satisfies the following: 3. if wRA v and wRA u, then v = u. Such a frame is strongly dense if it satisfies the following: 4. if wRA∩B v, then ∃u: wRA u and uRA∩B v. Strong density says that instead of conditioning on a stronger proposition, one can first condition on a weaker one, then condition on the stronger one, and end up in the same state as one would reach by conditioning on the stronger proposition straightaway. Though not all Lewis-Stalnaker-style frames that satisfy success, centering, and functionality are strongly dense, the following are, as observed by Boylan and Mandelkern (2022). 9 Wesley H. Holliday Example 1 Given a well-ordered set (W, 6), for any w ∈ W and A ⊆ W , let wRA v iff v is the first world in A according to 6 such that w 6 v. Then (W, {RA }A⊆W ) is a strongly dense, functional selection frame. Proposition 6 For any strongly dense selection frame (W, {RA }A⊆W ), the operation →R defined by A →R B := A B = {w ∈ W | for all v ∈ W, wRA v ⇒ v ∈ B} is a preconditional on the Boolean algebra ℘(W ). Proof. We must check the following for → = →R : 1. W → A ⊆ A; 2. A ∩ B ⊆ A → B; 3. A → B ⊆ A → (A ∩ B); 4. A → (B ∩ C) ⊆ A → B; 5. A → ((A ∩ B) → C) ⊆ (A ∩ B) → C. Condition 1 follows from centering (in particular, the right-to-left direction of the biconditional in centering), as does condition 2 (but now the left-to-right direction of the biconditional in centering); condition 3 follows from success; and condition 4 is immediate from the definition of →. For condition 5, suppose w ∈ A → ((A ∩ B) → C) and wRA∩B v. Then by strong density, there is a u such that wRA u and uRA∩B v. Since w ∈ A → ((A ∩ B) → C) and wRA u, we have u ∈ (A ∩ B) → C, which with uRA∩B v yields v ∈ C. This shows that w ∈ (A ∩ B) → C. The key principles validated by selection frames beyond the axioms of preconditionals are modus ponens, • identity: a → a = 1, and • normality: (a → b) ∧ (a → c) ≤ a → (b ∧ c). Functional frames also validate • negation import: ¬(a → b) ≤ a → ¬b. Proposition 7 Let B be a finite Boolean algebra equipped with a preconditional → satisfying modus ponens, identity, normality, and negation import. c the isomorphism from B to ℘(W ). Let W be the set of atoms of B and (·) For a ∈ B, define wRba v iff for all b ∈ B, w ≤ a → b implies v ≤ b. Then (W, {RA }A⊆W ) is a strongly dense, functional selection frame, and (B, →) is isomorphic to (℘(W ), →R ). 10 Preconditionals Proof. First, we check the following: 1. success: if wRba v, then v ≤ a; 2. centering: if w ≤ a, then wRba v iff v = w; 3. functionality: if wRba v and wRba u, then v = u; 4. strong density: if wR[ v, then ∃u: wRba u and uR[ v. a∧b a∧b For success, given w ≤ a → a from identity, wRba v implies v ≤ a. For centering, assume w ≤ a. Modus ponens for → yields wRba w. Then the rest of centering follows given functionality, which we prove next. For functionality, if wRba v, then we claim w ≤ a → v. For if w 6≤ a → v, then since w is an atom, we have w ≤ ¬(a → v) and hence w ≤ a → ¬v by negation import, contradicting wRba v. Then since w ≤ a → v, if wRba u, then u ≤ v, which implies u = v given that v is an atom. For strong density, assume wR[ v. Let a∧b ^ x = {y ∈ B | w ≤ a → y}. We claim that x 6= 0. Otherwise there are y1 , . . . , yn such that w ≤ (a → y1 ) ∧ · · · ∧ (a → yn ) and y1 ∧ · · · ∧ yn = 0. But then by normality and axioms 4-5 of preconditionals, w ≤ a → (y1 ∧ · · · ∧ yn ) = a → 0 ≤ (a ∧ b) → 0, contradicting wR[ v. Hence x 6= 0, so there is an atom u ≤ x, and by a∧b construction of x, wRba u. To show uR[ v, suppose u ≤ (a ∧ b) → c. a∧b Then w ≤ a → ((a ∧ b) → c), for otherwise w ≤ a → ¬((a ∧ b) → c) using negation import, in which case u ≤ ¬((a ∧ b) → c) by construction of x, which contradicts u ≤ (a ∧ b) → c given modus ponens. Then since w ≤ a → ((a ∧ b) → c), we have w ≤ (a ∧ b) → c by axiom 5 of preconditionals, which with wR[ v implies v ≤ c. Thus, uR[ v. a∧b a∧b Finally, the proof that a\ →b=b a →R bb is just like the usual proof for a normal modal box. This kind of result can be generalized V beyond finite algebras (e.g., V to complete and atomic algebras, assuming {a → bi | i ∈ I} ≤ a → {bi | i ∈ I}) and beyond Boolean algebras, but we will not do so here. Finally, let us return to the Heyting and Sasaki examples of Sections 2.3 and 2.4, respectively, with identity, normality, and negation import in mind. 11 Wesley H. Holliday Proposition 8 1. Heyting implications satisfy identity, normality, and negation import. 2. Proto-Heyting implications satisfy identity and negation import, but not necessarily normality.3 3. In orthomodular lattices, Sasaki hook satisfies normality, but not necessarily negation import. 4. In ortholattices, Sasaki hook satisfies identity but not necessarily normality.4 Proof. Part 1 is standard. For part 2 and identity, by weak monotonicity and axiom 3 of preconditionals, 1 ≤ a → 1 ≤ a → (a ∧ 1) = a → a. For negation import, by weak monotonicity, b ≤ a → b, so ¬(a → b) ≤ ¬b by Lemma 1.1, and ¬b ≤ a → ¬b by weak monotonicity again, so indeed ¬(a → b) ≤ a → ¬b. For a proto-Heyting implication that does not satisfy normality, consider the lattice on the left of Figure 4, and define → such that for any elements x and y: 1 → x = x; x → 0 = 0 if x 6= 0; 0 → 0 = 1; x → y = 1 if x 6= 1, y 6= 0, and (x, y) 6= (a, d); and a → d = d. Then (a → b) ∧ (a → c) = 1 ∧ 1 = 1 but a → (b ∧ c) = a → d = d, so normality does not hold, but one can check that the other axioms hold. For part 3, in an orthomodular lattice, ¬x ≤ y implies y ≤ ¬x ∨ (x ∧ y). Then since ¬a ≤ (a → b) ∧ (a → c), we have (a → b) ∧ (a → c) ≤ ≤ = ¬a ∨ (a ∧ (a → b) ∧ (a → c)) ¬a ∨ (a ∧ b ∧ c) by Lemma 2 a → (b ∧ c) by definition of Sasaki hook. To see that negation import is not necessarily satisfied, consider the modular lattice M4 with elements {0, a, b, c, d, 1} such that a, b, c, d are incomparable in the lattice order, ¬a = b, and ¬c = d. Then for the Sasaki hook we have ¬(a → c) = ¬(¬a ∨ (a ∧ c)) = ¬(¬a ∨ 0) = ¬¬a = a, whereas a → ¬c = ¬a ∨ (a ∧ ¬c) = ¬a ∨ (a ∧ d) = ¬a ∨ 0 = ¬a. For part 4, identity for Sasaki hook is just excluded middle. For a failure of normality, consider the ortholattice in Figure 4. Then for the Sasaki hook, 3 Moreover, there are normal proto-Heyting implications that are not Heyting implications, such as the implication used in the proof of Fact 2.1. 4 Moreover, there are non-orthomodular lattices in which the Sasaki hook satisfies normality, such as the lattice O6 (the “benzene ring”). 12 Preconditionals 1 1 a a c b d ¬a c b 0 ¬b ¬c 0 Figure 4: Left: lattice for the proof of Proposition 8.2. Right: an ortholattice in which the Sasaki hook violates normality for Proposition 8.4. (a → b)∧(a → c) = (¬a∨(a∧b))∧(¬a∨(a∧c)) = (¬a∨b)∧(¬a∨c) = 1 ∧ 1 = 1, whereas a → (b ∧ c) = ¬a ∨ (a ∧ b ∧ c) = ¬a ∨ 0 = ¬a. Though the conditionals in Proposition 8 are normal if they satisfy modus ponens, this is not the case for preconditionals in general.5 An instructive example comes from the following probabilistic interpretation. Given W = {0, . . . , 10}, define for each w ∈ W a measure µw : ℘(WP ) → [0, 1] by µw ({w}) = .9, µw ({v}) = .01 for v 6= w, and µw (A) = v∈A µw ({v}) for non-singleton A ⊆ W . Then for A, B ⊆ W with A 6= ∅, let A → B = {w ∈ W : µw (B | A) ≥ .9}, where as usual µw (B | A) = µw (A ∩ B)/µw (A), and ∅ → B = W . Proposition 9 The operation → just defined is a preconditional on ℘(W ) satisfying modus ponens but not normality. Proof. First observe that A ∩ (A → B) ⊆ B, because if w ∈ A, then since µw ({w}) = .9, we can have µw (B | A) ≥ .9 only if w ∈ B. This also shows that axiom 1 of preconditionals holds. For axiom 2, if w ∈ A ∩ B, then again since µw ({w}) = .9, we have µw (B | A) ≥ .9, so 5 Note that preconditionals satisfying normality but not modus ponens can easily be obtained from strongly dense selection frames that do not satisfy centering, as well as from the examples in Footnotes 3 and 4. 13 Wesley H. Holliday w ∈ A → B. Axioms 3 and 4 also clearly hold. For axiom 5, suppose w ∈ A → ((A ∩ B) → C), so µw ((A ∩ B) → C | A) ≥ .9. If w ∈ A, then by modus ponens, we have w ∈ (A ∩ B) → C, as desired. So suppose w 6∈ A. Further suppose for contradiction that w 6∈ (A ∩ B) → C, i.e., µw (A ∩ B ∩ C)/µw (A ∩ B) < .9, so |A ∩ B ∩ C| < |A ∩ B|. We claim that (A ∩ B) → C ⊆ A ∩ B. Consider any x ∈ W \ (A ∩ B) with x 6= w. Since x 6∈ A ∩ B, w 6∈ A ∩ B, x 6= w, and |W | = 11, we have |A ∩ B| ≤ 9, which with |A ∩ B ∩ C| < |A ∩ B| implies µx (C ∩ A ∩ B)/µx (A ∩ B) ≤ .08/.09 < .9, so x 6∈ (A ∩ B) → C. Thus, (A ∩ B) → C ⊆ A ∩ B, which implies (A ∩ B) → C ⊆ A ∩ B ∩ C by modus ponens, which with µw ((A ∩ B) → C | A) ≥ .9 implies µw (A ∩ B ∩ C | A) ≥ .9, which in turn implies µw (C | A ∩ B) ≥ .9 and hence w ∈ (A ∩ B) → C. Finally, for the failure of normality, 0 ∈ {1, . . . , 10} → {1, . . . , 9} and 0 ∈ {1, . . . , 10} → {2, . . . , 10}, but 0 6∈ {1, . . . , 10} → {2, . . . , 9}. 3 Relational representation of lattices with preconditionals Having hopefully shown the interest of the class of preconditionals from an axiomatic perspective, let us return to the semantic origin of preconditionals. Given a set X, a binary relation ⊳ on X, and A, B ⊆ X, define6 A →⊳ B = {x ∈ X | ∀y ⊳ x (y ∈ A ⇒ ∃z ⊲ y : z ∈ A ∩ B)}. (1) As shown in Holliday 2023, the operation c⊳ defined by c⊳ (A) = X →⊳ A is a closure operator, so its fixpoints ordered by inclusion W form a complete lattice L(X, ⊳) with meet as intersection and join as ⊳ {Ai | i ∈ I} = S c⊳ ( {Ai | i ∈ I}). If A, B ∈ L(X, ⊳), then A →⊳ B ∈ L(X, ⊳),7 so we may regard →⊳ as an operation on L(X, ⊳). The following is easy to check. Fact 4 The operation →⊳ on L(X, ⊳) is a preconditional. Example 2 In the relational frame shown at the top of Figure 5, the arrow from y to x indicates x ⊳ y, etc. Reflexive loops are assumed but not shown. 6 In Holliday 2023, we denoted the operation defined in (1) by ‘։ ’ in order to distinguish ⊳ it from a different operation denoted by ‘→⊳ ’. Since we do not need that distinction here, we will use the cleaner ‘→⊳ ’ for the operation defined in (1). 7 Obviously A → B ⊆ X → (A → B). For the reverse, suppose x 6∈ A → B. ⊳ ⊳ ⊳ ⊳ Hence there is some y ⊳ x such that y ∈ A and for all z ⊲ y, we have z 6∈ A ∩ B. It follows that there is no w ⊲ y with w ∈ A →⊳ B. Thus, if x 6∈ A →⊳ B, then there is a y ⊳ x such that for all w ⊲ y, w 6∈ A →⊳ B, which shows that x 6∈ X →⊳ (A →⊳ B). 14 Preconditionals Transitive arrows are not assumed. Note that y ∈ G but y 6∈ B →⊳ G, contra weak monotonicity. Also note that y ∈ (P →⊳ 0) →⊳ 0 = 0 →⊳ 0 = 1, but y 6∈ P , contra involution. So →⊳ is neither Heyting nor Sasaki. y w x z G B P R O R = {x} O = {z} G = {x, y} B = {w, z} P = {x, z} →⊳ 0 R O G P B 1 0 1 B G O 0 R 0 R 1 1 G P G R R O 1 B 1 O B P O G 1 1 G 1 G R G P 1 1 1 P 1 P P B 1 B 1 O B 1 B 1 1 1 1 1 1 1 1 Figure 5: A relational frame (top) giving rise to a lattice with preconditional (bottom). In fact, every complete lattice with a preconditional can be represented by such a frame (X, ⊳). More generally, we have the following. Theorem 1 (Holliday 2023, Theorem 6.3) Let L be a bounded lattice and → a preconditional on L. Then where P = {(x, x → y) | x, y ∈ L} and (a, b) ⊳ (c, d) if c 6≤ b, there is a complete embedding of (L, →) into (L(P, ⊳), →⊳ ), which is an isomorphism if L is complete. Let us take this a step further with a topological representation. Important precedents for the non-conditional aspects of this representation can be found in Urquhart 1978, Allwein and Hartonas 1993, Ploščica 1995, and Craig, Haviar, and Priestley 2013. Given a bounded lattice L and a preconditional →, define FI(L, →) = (X, ⊳) as follows: X is the set of all pairs (F, I) such that F is a filter in L, I is an ideal in L, and for all a, b ∈ L: if a ∈ F and a ∧ b ∈ I, then a → b ∈ I. 15 Wesley H. Holliday Call such an (F, I) a consonant filter-ideal pair. We define (F, I) ⊳ (F ′ , I ′ ) if I ∩ F ′ = ∅. Finally, given a ∈ L, let b a = {(F, I) ∈ X | a ∈ F }, and let S(L, →) be FI(L, →) endowed with the topology generated by {b a | a ∈ L} (cf. Bezhanishvili and Holliday 2020). Theorem 2 For any bounded lattice L and preconditional → on L, the map a 7→ b a is 1. an embedding of (L, →) into (L(FI(L, →)), →⊳ ) and 2. an isomorphism from L to the subalgebra of (L(FI(L, →)), →⊳ ) consisting of elements of L(FI(L, →)) that are compact open in the space S(L, →). Proof. For a ∈ L, let ↑a (resp. ↓a) be the principal filter (resp. ideal) generated by a. First we claim that for any a, b ∈ L, (↑a, ↓a → b) ∈ X. For suppose c ∈ ↑a and c ∧ d ∈ ↓a → b, so a ≤ c and c ∧ d ≤ a → b. Then c→d ≤ ≤ c → (c ∧ d) by axiom 3 of preconditionals c → (a → b) by axiom 4, since c ∧ d ≤ a → b = ≤ c → ((a ∧ c) → b) since a ≤ c (a ∧ c) → b by axiom 5 = a → b since a ≤ c, so c → d ∈ ↓a → b. Since b = 1 → b by axioms 1 and 2 of preconditionals, it follows that (↑1, ↓b) = (↑1, ↓1 → b) ∈ X as well. Using the above facts, the proof that b a belongs to L(FI(L, →)) and that a 7→ b a is injective and preserves ∧ and ∨ is the same as in the proof of Theorem 4.30.1 in Holliday 2023. Also note that b 1 = X and b 0 = c⊳ (∅). b \ → b, Next we show that a → b = b a →⊳ b. First suppose (F, I) ∈ a\ ′ ′ ′ ′ (F , I ) ⊳ (F, I), and (F , I ) ∈ b a, so a ∈ F ′ . Since (F, I) ∈ a\ → b, we have a → b ∈ F , which with (F ′ , I ′ ) ⊳ (F, I) implies a → b 6∈ I ′ , which with a ∈ F ′ and the definition of X implies a∧b 6∈ I ′ . Now let F ′′ = ↑a∧b and I ′′ = ↓(a ∧ b) → 0. Then (F ′′ , I ′′ ) ∈ X, (F ′ , I ′ ) ⊳ (F ′′ , I ′′ ), and → b, (F ′′ , I ′′ ) ∈ a[ ∧ b. Thus, (F, I) ∈ b a →⊳ bb. Conversely, if (F, I) 6∈ a\ ′ ′ so a → b 6∈ F , then setting (F , I ) = (↑a, ↓a → b), we have (F ′ , I ′ ) ∈ X and (F ′ , I ′ ) ⊳ (F, I). Now consider any (F ′′ , I ′′ ) such that (F ′ , I ′ ) ⊳ (F ′′ , I ′′ ), so a → b 6∈ F ′′ . Then by axiom 2 of preconditionals, a ∧ b 6∈ F ′′ , so (F ′′ , I ′′ ) 6∈ a[ ∧ b. Thus, (F, I) 6∈ b a →⊳ bb. 16 Preconditionals The proof of part 2 is the same as the proof of Theorem 4.30.2 in Holliday 2023. Finally, we can characterize the spaces equipped with a relation ⊳ that are isomorphic to S(L, →) for some (L, →). Let X be a topological space and ⊳ a binary relation on X. Let COFix(X, ⊳) be the set of all compact open sets of X that are also fixpoints of c⊳ . If COFix(X, ⊳) is a lattice with meet as ∩ and join as ∨⊳ , F is a filter in this lattice, and I is an ideal, then we can speak of (F, I) being a consonant filter-ideal pair as defined above Theorem 2, using →⊳ in the definition. Given x ∈ X, let F(x) = {U ∈ COFix(X, ⊳) | x ∈ U } I(x) = {U ∈ COFix(X, ⊳) | ∀y ⊲ x y 6∈ U } and note that (F(x), I(x)) is a consonant filter-ideal pair from COFix(X, ⊳). For if U ∈ F(x) and U ∩ V ∈ I(x), then x ∈ U but for all y ⊲ x, y 6∈ U ∩ V , which implies that for all y ⊲ x, y 6∈ U →⊳ V , so U →⊳ V ∈ I(x). Proposition 10 For any space X and binary relation ⊳ on X, there is a bounded lattice L with preconditional → such that (X, ⊳) and S(L, →) are homeomorphic as spaces and isomorphic as relational frames iff the following conditions hold for all x, y ∈ X: 1. x = y iff (F(x), I(x)) = (F(y), I(y)); 2. COFix(X, ⊳) contains X and c⊳ (∅), is closed under ∩, ∨⊳ , and →⊳ , and forms a basis for X; 3. each consonant filter-ideal pair from COFix(X, ⊳) is (F(x), I(x)) for some x ∈ X; 4. x ⊳ y iff I(x) ∩ F(y) = ∅. The proof is very similar to that of Proposition 3.21 in Holliday 2022, but we provide the adapted proof here for convenience. Proof. Suppose there is such an L. It suffices to show S(L, →) satisfies conditions 1–4 in place of (X, ⊳). That condition 2 holds for COFix(S(L, →)) and S(L, →) follows from the proof of Theorem 2. Let ϕ be the isomorphism a 7→ b a from L to COFix(S(L, →)) in Theorem 2, which induces a 17 Wesley H. Holliday bijection (F, I) 7→ (ϕ[F ], ϕ[I]) between consonant filter-ideal pairs of L and of COFix(S(L, →)). Conditions 1, 3, and 4 follow from the fact that for any x = (F, I) ∈ S(L, →), (ϕ[F ], ϕ[I]) = (F(x), I(x)). (2) First, b a ∈ ϕ[F ] iff a ∈ F iff x ∈ b a iff b a ∈ F(x). Second, b a ∈ ϕ[I] iff a ∈ I, and we claim that a ∈ I iff b a ∈ I(x), i.e., for all (F ′ , I ′ ) ⊲ (F, I), (F ′ , I ′ ) 6∈ b a, i.e., a 6∈ F ′ . If a ∈ I and (F, I) ⊳ (F ′ , I ′ ), then a 6∈ F ′ by definition of ⊳. Conversely, if a 6∈ I, let F ′ = ↑a and I ′ = ↓a → 0, so (F ′ , I ′ ) is consonant by the proof of Theorem 2, (F, I) ⊳ (F ′ , I ′ ) since a 6∈ I, and a ∈ F ′ . Thus, b a 6∈ I(x). This completes the proof of (2). Now for condition 1, given x, y ∈ S(L, →) with x = (F, I) and y = (F ′ , I ′ ), we have (F, I) = (F ′ , I ′ ) iff (ϕ[F ], ϕ[I]) = (ϕ[F ′ ], ϕ[I ′ ]) iff (F(x), I(x)) = (F(y), I(y)); similarly, for condition 4, (F, I) ⊳ (F ′ , I ′ ) iff I ∩F ′ = ∅ iff ϕ[I]∩ϕ[F ′ ] = ∅ iff I(x)∩F(y) = ∅. Finally, for condition 3, if (F , I) if a consonant filter-ideal pair from COFix(S(L, →)), then setting x = (ϕ−1 [F ], ϕ−1 [I]), we have x ∈ S(L, →) and (F , I) = (F(x), I(x)). Assuming X satisfies the conditions, COFix(X, ⊳) is a bounded lattice with a preconditional by Fact 4, and we define a map ǫ from (X, ⊳) to S(COFix(X, ⊳), →⊳ ) by ǫ(x) = (F(x), I(x)). The proof that ǫ is a homeomorphism using conditions 1–3 is analogous to the proof of Theorem 5.4(2) in Bezhanishvili and Holliday 2020. That ǫ preserves and reflects ⊳ follows from condition 4. 4 Conclusion We have seen that preconditionals encompass several familiar classes of conditionals, including Heyting implication, the Sasaki hook, and LewisStalnaker style conditionals satisfying flattening. Lattices with these implications are therefore covered by the general representation in Theorem 2. A natural next step is to try to obtain nice characterizations of the relationaltopological duals of these kinds of algebras, as well as of more novel kinds— such as lattices with normal preconditionals—not to mention going beyond representation to categorical duality. We hope that the delineation of preconditionals and their relational-topological representation may help to provide a unified view of a vast landscape of conditionals arising in logic. 18 Preconditionals References Allwein, G., & Hartonas, C. (1993). Duality for bounded lattices. (Indiana University Logic Group, Preprint Series, IULG-93-25 (1993)) Bezhanishvili, N., & Holliday, W. H. (2020). Choice-free Stone duality. The Journal of Symbolic Logic, 85(1), 109-148. Boylan, D., & Mandelkern, M. (2022). Logic and information sensitivity. (ESSLLI 2022 lecture notes, https://mandelkern.hosting.nyu.edu/ESSLLI2022Day3.pdf) Craig, A. P. K., Haviar, M., & Priestley, H. A. (2013). A fresh perspective on canonical extensions for bounded lattices. Applied Categorical Structures, 21, 725-749. Egré, P., & Rott, H. (2021). The Logic of Conditionals. https://plato.stanford.edu/archives/win2021/entries/logicconditionals/. (In The Stanford Encyclopedia of Philosophy (Winter 2021 Edition)) Holliday, W. 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Holliday University of California, Berkeley USA E-mail: wesholliday@berkeley.edu 19