Proof Theoretic Harmony in the Substructural Era (draft)

Proof Theoretic Harmony in the Substructural Era∗ Ole Thomassen Hjortland† ole.hjortland@lmu.de February 7, 2014 Draft—do not cite without permission Abstract According to the logical inferentialist, the meaning of a logical connective is determined by the inference rules that govern its use. Proof theoretic semantics attempts to make this idea precise in a proof theoretic framework, using for example natural deduction or sequent calculus rules. Since Prior’s infamous connective tonk much of proof theoretic semantics have been occupied with formal anti-tonk conditions which rule out ill-behaved connectives (e.g. conservativeness, harmony). Common between them is that inference rules only succeed in determining the meaning of a connective if the proof theoretic conditions are fulfilled. On the traditional account, however, such conditions are insensitive to substructural dimensions of proof theory, e.g. the distinction between additive and multiplicative connectives. We argue that proof theoretic semantics ought to have the resources to attribute different meanings to substructurally distinct connectives. Subsequently we show how to develop a notion of proof theoretic harmony that preserves substructural distinctions from introduction to elimination rules. The substructural account of harmony can rule out cases of nonconservativeness that previous accounts have not dealt with. Keywords: Logical Inferentialism · Proof Theory · Proof Theoretic Semantics · Harmony · Substructural Logic · Natural Deduction · Sequent Calculus ∗ This research is supported by the Humboldt Foundation. I am very grateful for insightful comments on earlier drafts from Roy Dyckhoff, Bruno Jacinto, Jon Litland, Julien Murzi, Stephen Read, Greg Restall, Peter Schroeder-Heister, Florian Steinberger, and Luca Tranchini. † Munich Center for Mathematical Philosophy (MCMP), Ludwig-Maximilians-Universität (LMU), 80539 München, Germany. 1 Introduction A popular idea about the semantics of logical connectives is that their meanings are fixed by inferential roles. In other words, the meaning of a connective is determined exhaustively by the inference rules that govern its correct use. The position is sometimes called logical inferentialism or, in its formal guise, proof theoretic semantics (PTS). Both Carnap (1934) and Popper (1946) defended it in an early form, but their versions were met by a well-known objection. Prior (1961) rejected the thought that a connective’s meaning can be fixed by inference rules alone, e.g. introduction and elimination rules in a natural deduction system. He introduced the artificial connective ‘tonk’ (H) to show that it cannot be the case that inferential rules are sufficient to determine meanings: A AHB AHB B (HI) (HE) The problem is that in the presence of a transitive deducibility relation, the rules for H lead to triviality.1 On the assumption that an inconsistent connective cannot be meaningful, Prior invites us to conclude that basic inference rules are not meaning-determining. Since then a number of authors have spoken in defence of PTS. The problem, it is argued, is not with logical inferentialism in itself, but with the idea that there are no conditions on successful stipulation of new logical expressions. Inference rules can successfully determine the meaning of a logical connective, but not any old set of inference rules will do. Following Belnap’s (1962) suggestion that new inference rules should be conservative and unique extensions of the antecedent system, a host of proof theoretic conditions on inference rules have been proposed. The overarching idea is not simply to rule out tonk. More generally, with the proof theoretic condition in place, inference rules should succeed in determining the meaning of the associated connective. Dummett (1973, 1991) labelled such a condition ‘harmony’, alluding to the apparent lack of deductive equilibrium between the introduction and elimination rule for H. However, formulating the details of such a condition has proved challenging.2 In what follows we will assume that the inferentialist has got the big picture right. When the appropriate proof theoretic condition is satisfied, the meaning of a logical connective is fixed by its inference rules. The challenge will be to give the exact nature of the proof theoretic condition. We follow a promising proposal known as general elimination harmony (cf. Read 2000, 2010, 2013, Francez & Dyckhoff 2012), but argue that the proposal does not give the correct proof theoretic conditions for substructural connectives. In particular, substructural connectives can lead to nonconservativeness in certain cases. We 1 Specifically, with H alone we get that anything follows from a non-empty set of premises. In the presence of, say, a negation, we get full triviality. 2 The debate has to a large extent been part of a polemic between classical and intuitionistic logic. For the traditional revisionist arguments, see for example Prawitz (1977), Dummett (1991), Tennant (1987). For some recent approaches to harmony in classical logic, see Milne (1994, 2002), Read (2000), and Rumfitt (2000) 2 then introduce a new version of general elimination harmony that can handle a variety of substructural connectives. It can be shown that the new harmony conditions (in conjunction with some further restrictions on introduction rules) entail normalization and conservativeness. Section 1 gives a brief introduction to substructural connectives in natural deduction and sequent calculus. Sections 2-4 introduces the notion of general elimination harmony, before we in Section 5-6 show the problems that arise when substructural connectives are considered. Finally, in Sections 7-9 we present a new proof theoretic harmony condition, and show how it manages substructural connectives. 1 Substructural Logic The traditional inferentialist has been chiefly occupied with proof theoretic conditions for intuitionistic and classical logic. More recent work in PTS, however, has had a different focus. PTS has now expanded to include work in a range of different systems, both classical extensions such as modal logics, and various non-classical logics. The systems that will be the main concern in what follows are so-called substructural logics. Roughly put, these are logics that result from omitting structural rules from sequent calculus systems, or, alternatively, restricting discharge policies for assumptions in natural deduction systems.3 In philosophical logic, such systems are already familiar in the forms of relevant logics, paraconsistent logics, and non-monotonic logics, while others such as linear logics and affine logics are becoming more popular. Applications range from linguistics (e.g. Barker 2010) to truth-theoretic paradoxes (e.g. Shapiro 2010, Beall & Murzi 2013), from reasoning with vague predicates (e.g. Zardini 2008) to formal epistemology (e.g. Sequoiah-Grayson 2009). What makes a logic substructural? In sequent calculus systems there are two types of rules: operational rules and structural rules. The operational rules introduce a connective on the right or the left, and correspond to introduction and elimination rules respectively. The structural rules, on the other hand, do not govern a particular connective. Instead they reflect inferences that are permissible because of the properties of the deducibility relation (e.g. transitivity, reflexivity). For example, in Gentzen’s original (intuitionistic) sequent calculus we have the following structural rules: (Id) A⇒A Γ⇒C Γ, A ⇒ C (K) Γ, A, A ⇒ C Γ, A ⇒ C Γ ⇒ A Σ, A ⇒ C Γ, Σ ⇒ C (W ) Γ, A, B, Γ′ ⇒ C Γ, B, A, Γ′ ⇒ C (Ex) (Cut) In addition to the identity axiom (reflexivity), we have the structural rule of weakening (K). Weakening reflects the fact that the deducibility relation in 3 Linear logic is the most well-known of the substructural logics. It is first introduced in Girard (1987). Standard introductions include Restall (2000) and Paoli (2002). 3 question is monotonic. Since the antecedents and succedents are ordered sets, we also have the structural rule of contraction (W ) (allowing us to identify copies of the same formula) and the structural rule of exchange (Ex) (allowing us to permute the order of premises in the antecedent). In what follows, however, we will be interested in formulating a harmony condition in natural deduction systems. In natural deduction there are no explicit structural rules, but the structural properties can nevertheless be identified.4 For example, an open assumption A at a leaf node of a derivation tree corresponds to an instance A ⇒ A of the identity axiom. The structural rules of weakening and contraction are a bit more tricky. They correspond to global discharge policies for closing assumptions. Consider the standard introduction rule for the conditional: Γ, [A]u .. .. B A→B (→I)(u) An application of → I introduces the formula A → B from a subderivation of B from A (together with open assumptions Γ), while simultaneously discharging the assumption A. The structural rules manifest themselves in the policies governing the discharge of assumptions. In standard natural deduction systems, e.g. classical and intuitionistic logic, there are two special discharge policies: multiple discharge and vacuous discharge. The former—corresponding to structural contraction—says that multiple copies of an assumption A can be discharged with a single application of a hypothetical rule (e.g. → I). The latter—corresponding to structural weakening—says that discharging an assumption at all is optional in an application of a hypothetical rule. Whether or not the special discharge policies are permissible in a system make a difference to what is derivable. In both the classical and intuitionistic →-fragment the policies are essential to deriving for example the positive paradox for the conditional and the law of absorption: [A → (A → B)]2 A→B 1 [A] B→A A → (B → A) (1) [A]1 [A]1 B (1) A→B (A → (A → B)) → (A → B) (2) In the leftmost derivation of the positive paradox we first apply → I without discharging an assumption. In other words, it does not matter how A was derived, we can always conclude B → A, for any B.5 In the rightmost derivation of the law of contraction, the penultimate step is an application of → I where 4 See Troelstra and Schwichtenberg (2000) for a full translation between natural deduction and sequent calculus. 5 Obviously, relevantists who object to the positive paradox of classical logic will therefore typically reject vacuous discharge. 4 two copies of the assumption A are discharged, both indicated by the index 1. Again, if we drop multiple discharge, the law of absorption is not derivable.6 Structural rules and discharge policies also make a difference for the project of proof theoretic semantics. First, there is a general question of inferential meaning: Does the presence or absence of structural rules affect the meaning of a connective, or is the meaning determined by the operational rules (or introduction and elimination rules) alone? Second, there is a question about harmony in a substructural setting: How can we make proof theoretic harmony sensitive to substructural distinctions? The first question will not be answered in what follows, but there is an ongoing debate. Restall (2002) and Paoli (2003) have argued that structural rules can change the logic (i.e. what is derivable) without changing the meaning of the connectives, and a number of objections are discussed in Hjortland (2013) and Paoli (2013). Instead, the focus in the present paper will be on giving an adequate answer to the second question. The observation that proof theoretic ‘anti-tonk’ conditions are sensitive to structural rules is not new. In fact, when Belnap suggests conservativeness as the right measure to prevent tonk and tonk-like connectives, he remarks that the condition will depend on what he calls ’the antecedent context of deducibility’: It seems to me that the key to a solution lies in observing that even on the synthetic view, we are not defining our connectives ab initio, but rather in terms of an antecendently given context of deducibility, concerning which we have some definite notions. By that I mean that before arriving at the problem of characterizing connectives, we have already made some assumptions about the nature of deducibility. That this is so can be seen immediately by observing Prior’s use of the transitivity of deducibility in order to secure his ingenious result. But if we note that we already have some assumptions about the context of deducibility within which we are operating, it becomes apparent that by too careless use of definitions, it is possible to create a situation in which we are forced to say things inconsistent with those assumptions. (Belnap 1962, 131) There is an important lesson in Belnap’s remark. Conservativeness and other proof theoretic conditions give different results depending on the structural properties of the deducibility relation.7 As we move into a substructural settings, therefore, connectives which previously lead to inconsistency might become consistent.8 What is more, we will see examples where being insensitive to structural rules lead to the nonconservative introduction of new connectives. Worse, the nonconservativeness can result in inconsistency. 6 This is true for the →-fragment but not true in general. Note that in the presence classical ¬, ∨ (with the law of excluded middle, A ∨ ¬A as an axiom), the law of contraction is derivable without multiple discharge. 7 Some examples are explored in Restall (2007). 8 Cook (2005), for example, has explored tonk in a non-transitive context. 5 2 General Elimination Rules There are several reasonable formulations of proof theoretic harmony in the literature, e.g. definitional reflection in Schroeder-Heister (2003, 2007, forthcoming), and harmony-as-deductive-equilibrium in Tennant (1997, forthcoming). In what follows, the focus will be on a proposal known as general elimination harmony (GE harmony) as developed in Read (2000, 2010, 2013) and Francez & Dyckhoff (2012).9 GE harmony is based on a unification of elimination rules known as general elimination rules.10 The general elimination rules themselves are formulated to capture Prawitz’s inversion principle. Prawitz sums it up as follows: ‘By an application of an elimination rule one essentially restores what had already been established if the major premiss of the application was inferred by an application of an introduction rule’ (Prawitz 1965, 33). For example, consider the standard rules for disjunction: A A∨B (∨I,i) B A∨B (∨E,ii) [A]u .. .. A∨B C C [B]u .. .. C (∨E)(u) The inversion principle tells us that whatever is derivable from each of the grounds for introducing a complex formula, is derivable directly from the complex formula. Thus, since disjunction has two introduction rules (and therefore two possible sufficient grounds, A and B) the elimination rule says that a formula C can be derived from A ∨ B if there are already two independent derivations from A to C and B to C respectively. Formally, the inversion principle is encapsulated in the reduction steps of normalization proofs. For the disjunction rules we have the following well-known reduction conversions (we call the derivation after the conversion the redux ): Π0 A A∨B [A] Π1 C C [B] Π2 C Π0 B A∨B Π0 A Π1 C [A] Π1 C C [B] Π2 C Π0 B Π2 C The first conversion shows that we can find a derivation Π1 of C directly from a derivation Π0 of A given that the derivation of A was used to introduce A ∨ B, and that C can be derived from both possible grounds A and B as assumptions. The importance of these reduction conversions is not only that they lead to normalization for intuitionistic logic, but that the normalization theorem in turn entails conservativeness. The inversion principle is therefore a candidate 9 Note that Read’s theory of harmony in Read (2000) is different from the one given in Read (2013). The earlier theory gave a formulation of harmony that produced a single elimination rule from a set of introduction rules. We follow the latter formulation where a set of introduction rules induces a set of elimination rules. 10 The main development of general elimination rules is in Schroeder-Heister (1984). Other attempts at formulating unified elimination rules can be found in Kutschera (1968), Prawitz (1978), Martin-Löf (1984). 6 for a local condition on inference rules that leads to a global condition on the proof system—conservativeness, and hence relative consistency. Ultimately, the aim is a local harmony condition based on the inversion principle that entails normalization and conservativeness. The advantage of a local condition is that individual connectives can be proved to yield conservative extensions of an antecedent theory, merely because they satisfy a harmony condition. The challenge is to provide a formal constraint on inference rules which encapsulates the inversion principle and guarantees the existence of reduction conversions. That is precisely the motivation for general elimination rules. The inversion principle can be used as a guideline to formulate general elimination rules for other connectives than disjunction. For example, the standard introduction rule for conjunction gives two general elimination rules: A B A∧B (∧I) A∧B C [A]u .. .. C (∧GE,i) A∧B C [B]u .. .. C (∧GE,ii) The duality between disjunction and conjunction now becomes salient. The introduction rule has two grounds A and B that are jointly sufficient, so a formula C can be derived from A ∧ B only if it can be derived either from A or from C. Accordingly, there will be two reduction conversions: Π0 Π1 A B A∧B C [A] Π2 C Π0 Π1 A B A∧B C Π0 A Π2 C [B] Π2 C Π1 B Π2 C Unlike for disjunction, the general elimination rules for conjunction are not the same as the standard elimination rules. However, it is easy to see that the general elimination rules are equivalent to the standard rules. 3 Higher-Order Rules If the grounds for introducing a complex formula is hypothetical, i.e. discharges an assumption, then the inversion principle becomes slightly more complicated. Consider the standard introduction rule for the intuitionistic conditional: [A]u .. .. B A→B (→I)(u) The inversion principle tells us that the general elimination rule should allow us to derive a formula C from A → B only if there is a derivation of C from the grounds for introducing A → B. But the ground specified by → I is a derivation, not a formula. The result is a general elimination rule that includes a rule as 7 assumption. We therefore say that the general elimination rule is a higher-order rule:11  u A ..  ..  B .. .. A→B C (→GE)(u) C The rule → GE discharges an assumed rule from A to B and concludes C directly from A → B, if C is derivable from the higher-order assumption. It is in other words an application of the inversion principle in which the ground is a rule rather than a formula. There is now a corresponding reduction conversion: [A]u Π0 B A→B (u) C  Π2  v A .  ...  B Π1 C (v) Π2 A Π0 B Π1 C The subderivation Π2 yields A as conclusion (a special case is where A is an assumption), hence giving a derivation of B on the (higher-order) assumption that there is a rule that gives B directly from A. From B in turn there is a derivation Π1 to C, and we subsequently conclude C directly from A → B and discharge the higher-order assumption.12 Note that A is not discharged by the application of → GE. But that is as we should expect, as A is a minor premise in the standard modus ponens rule: A→B B A (→E) The standard rule → E is equivalent to the higher-order rule → GE. 11 The higher-order rules are first developed in Schroeder-Heister (1984). (1987) suggested an alternative general elimination rule for the intuitionistic conditional, but without the higher-order assumption: 12 Dyckhoff A→B A C Γ, [A]u . . . . C The rule was subsequently studied by Tennant (2002) and von Plato (2001). Olkhovikov and Schroeder-Heister (forthcoming) have later shown that the higher-order assumptions are necessary in order to proof-theoretically represent all intuitionistic connectives. We therefore stick with higher-order rules throughout the presentation. 8 4 General Elimination Harmony The inversion principle and the general elimination rules have been used to formulate a harmony condition—so-called GE harmony. The idea is to let an arbitrary (finite) set of introduction rules, I, induce a set of general elimination rules, E. This approach has at least two advantages: First, it is a local condition on inference rules, rather than a global condition on a proof system. Global conditions such as conservativeness and consistency should in turn follow from the local conditions when systematically applied to the proof rules. Second, GE harmony provides a method for constructing the set of elimination rules directly from the introduction rules. In what follows, we use the formulation in Read (2013). Suppose the n-place connective ⊙(β1 , ..., βn ) has m ≥ 0 introduction rules, ⊙I1 , ..., ⊙Im , each 1 ≤ i ≤ m of which has ki ≥ 0 premises, ρi1 , ..., ρiki . That is, I⊙ =  ρ , ..., ρ  ρm1 , ..., ρmkm 11 1k1 (⊙I1 ) (⊙Im ) ⊙β~ ... ⊙β~ We write ⊙β~ for ⊙(β1 , ..., βn ). Each premise ρij may be either categorical, i.e. a formula, or hypothetical, i.e. a subderivation discharging (first-order) assumptions.13 Qm The set of introduction rules I ⊙ induces a set with i=1 ki general elimination ⊙ ⊙ ⊙ rules E (I GE E ) via the following schema:    ~   ⊙β [ρ11 ]u .. .. γ ... γ [ρm1 ]u .. .. γ (⊙GE) ⊙β~ ... [ρ1k1 ]u .. .. γ ... γ [ρmkm ]u .. .. γ (⊙GE)      There is one general elimination rule, ⊙GE for every m-tuple of premises ρij (1 ≤ i ≤ m, 1 ≤ j ≤ ki ). Every elimination rule will therefore have, for each introduction rule 1 ≤ i ≤ m, exactly one minor premise with an assumption corresponding to a premise in the ith introduction rule. The only other premise ~ is the major premise ⊙β. We can then define GE harmony as follows: Definition 4.1 (GE Harmony). A set of elimination rules E ⊙ are in GE harmony with a set of introduction rules I ⊙ if I ⊙ GE E ⊙ . If a connective ⊙ has rule sets that are in GE harmony we will say that the connective is GE harmonious. That Def 4.1 is in accordance with the inversion principle can then be seen by producing a schematic reduction conversion. Consider a derivation with an application of the ith introduction rule, and then an immediately succeeding application of an elimination rule: 13 There are, in other words, no higher-order assumptions in introduction rules. If that was allowed the general elimination rules would take more complicated assumptions. We will bracket that option for now. 9 (Πi1 ) ρi1 (Πiki ) ρiki ... ⊙β~ [ρ1j1 ]u Π∗1j1 γ γ ... [ρmjm ]u Π∗mjm γ (u) (Πiji ) ρiji Π∗iji γ (Π) The notation ρ indicates that a premise in an introduction rule can either be a formula A which labels the end node of the subderivation Π (e.g. in ∧I), or a subderivation Π from some set of assumptions Γ to a conclusion B (e.g. in → I). We will follow Pfenning and Davies (2001) and call the existence of such a reduction conversion local soundness. It is not in general the case, however, that local soundness entails normalization. Whether or not the reduction conversions lead to a normal form proof (in the presence of other permutation conversions) depends crucially on the relationship between the formula β~ and the formula occurring in the premises ρij . Note that the introduction rules for standard intuitionistic connectives are such that every formula occurring in a ρij is a strict ~ We will return to this issue in Section 9 below. subformula of β. One desirable consequence of the definition of GE-harmony is that H is disharmonious: Fact 4.1. The introduction and elimination rules for tonk H are not in GE harmony. The introduction rule H harmoniously induces the following general elimination rule: A AHB AHB C GE [A]u .. .. C (u) The GE harmonious elimination rule for H can be seen to normalize, and is therefore not equivalent to the standard elimination rule (which leads to trivialization). 5 Substructural Connectives Let us now turn to substructural connectives. The question we want to address is whether GE harmonious connectives in the presence of substructural deducibility relations produce the appropriate reduction conversions. It turns out that they do not, unless we make critical modifications to the definition. Consider again the above GE rules for conjunction ∧: A∧B C [A]u .. .. C (∧GE,i) A∧B C 10 [B]u .. .. C (∧GE,ii) These rules are closely related to rules for conjunction in intuitionistic sequent calculus: Γ, A ⇒ C Γ, A ∧ B ⇒ C Γ, B ⇒ C Γ, A ∧ B ⇒ C (L∧,i) (L∧,ii) Informally, we can read both sets of rules as saying that if something can be derived from the assumption of A (B), then it can be derived directly from A ∧ B. The above sequent rules for conjunction are known as additive rules. The corresponding additive right-rule is the context-sharing version of conjunction introduction: Γ⇒A Γ⇒B Γ⇒A∧B (R∧) In contrast, the multiplicative or context-free conjunction, ⊗, has the following sequent rules: Γ, A, B ⇒ C Γ, A ⊗ B ⇒ C Γ⇒A Σ⇒B Γ, Σ ⇒ A ⊗ B (L⊗) (R⊗) In classical and intuitionistic sequent calculus, the additive and multiplicative conjunction rules are equivalent. However, ∧ and ⊗ come apart in logics where either of the structural rules of weakening and contraction are absent, e.g. relevant logics, affine logics, linear logics. In substructural logics, therefore, there are two conjunctions: one additive and one multiplicative. In fact, other connectives can also be bifurcated into two non-equivalent connectives. We should expect there to be an analogous distinction between additive and multiplicative conjunctions in natural deduction. It is not difficult to find a candidate for the multiplicative elimination rule:14 A∧B C [A, B]u .. .. C (⊗GE)(u) Strictly speaking, of course, the multiplicative elimination is not in GE harmony with the introduction rule. Def. 4.1 above requires the introduction rule to have two distinct elimination rules. Nevertheless, the multiplicative rule in natural deduction can also be used in a reduction conversions, but one which differs from that of ∧ in a crucial respect: [A, B]u | {z } Π0 Π1 A B Π2 A⊗B C (u) C Π0 A | Π1 B {z Π2 C } 14 The rule can be found both in Martin-Löf (1984) and Read (2000), but not in a substructural context. 11 Note that the redux now preserves both a subderivation ending with A and a subderivation ending with B. That will be important in what follows. So far only the elimination rules in the natural deduction framework are different for ∧ and ⊗. But following Negri (2002) we can give natural deduction introduction rules corresponding to both the additive and multiplicative sequent calculus rules. In order to do that we need to introduce auxiliary formulae (contexts) explicitly in the rule schemata. We will write the multiplicative ⊗I introduction rule as follows: Γ. Σ. .. .. . . A B A⊗B (⊗I) The contexts Γ, Σ are sets of (open) assumptions labelling the leaf nodes of the subderivations terminating with A and B respectively. It should then be clear that the standard introduction rule in both classical and intuitionistic natural deduction corresponds to the multiplicative right rule for ⊗ in sequent calculus. There is no restriction on the open assumptions used to derive A and B respectively. In the additive introduction rule, however, we will see precisely that the open assumptions come with a decisive restriction. The additive counterpart of the introduction rule for ∧ is as follows: Γ.α Γ.α .. .. . . A B A∧B A∧B C Γ, [A]u .. .. C A∧B C Γ, [B]u .. .. C In the additive rule the subderivations of A, B is adorned by the same contexts Γ in both subderivations. That indicates that the members of the assumption class Γ must be the same in each subderivation. Furthermore, the index α requires that each of the assumption classes must have members with the same assumption index. Only then can the additive conjunction rule be applied. (Recall that in sequent calculus an application of ∧R contracts the auxiliary formulae of the two premise sequents.) As a result, whenever a formula A in one copy of Γ is discharged in the derivation, a copy of A in the other occurrence of Γ is also discharged. The result is that in the presence of vacuous and multiple discharge policies, e.g. in intuitionistic logic, the natural deduction rules for the two conjunctions ∧ and ⊗ are equivalent. Assume the rules for ⊗, and suppose there is a derivation both from Γ to A and from Γ to B. Then we can apply ⊗I (in the special case where Γ = Σ): Γ. Γ. .. .. . . A B A⊗B 12 Whenever a succeeding rule application discharges an assumption in either the left-side Γ, multiple discharge permits the closing of the corresponding assumption in the right-side Γ. For the other direction: Suppose there is a derivation of A from Γ and of B from Σ. Then we substitute these derivations with derivations of A from Γ ∪ Σ and B from Γ ∪ Σ, and apply ∧I (indexing both copies of Γ ∪ Σ with a label α).15 For the elimination rules the role of the discharge policies are more prominent. First assume the ∧ rules, and suppose that there is a derivation of A ∧ B and from A, B to C. Two successive applications of ∧GE then discharges both assumptions in the minor premise: A∧B A∧B C C [A]1 , [B]2 .. .. C (1) (2) Multiple discharge now permits us to discharge both copies of A ∧ B simultaneously later in the derivation. In the other direction an application of ⊗E with vacuous discharge is sufficient to derive both ∧GE rules. 6 Substructural Insensitivity in GE Harmony Def. 4.1 of GE harmony does not include any explicit information about contexts. However, the standard introduction rule for conjunction in classical and intuitionistic logic is the multiplicative one. That should be clear as there is no restriction on the open assumptions required to derive the two premises. Thus the introduction rule is the multiplicative ⊗I rule. If we take this rule as our input, however, the definition of GE harmony induces the additive elimination rules for ∧. That is an unintended effect, and although innocuous in standard natural deduction, it leads to trouble in substructural settings. That is not to say that matching up additive and multiplicative rules necessarily have an adverse effect on inferential meanings. Unlike with the rules of H, for example, there is no reason to claim that mixing additive and multiplicative rules fail to determine the inferential meaning of the connective in question. But, if one is interested in substructural logics, it is important to notice that mixing the rules can lead to structural nonconservativeness. If conservativeness is a requirement on successful implicit definition of a logical connective, then GE harmony cannot be correct as it stands. We consider the cases of weakening and contraction in turn. 15 The derivations Γ ∪ Σ can be constructed by weakening in additional premises via ∧GE with vacuous discharge. 13 6.1 Weakening To see why nonconservativeness can follow in substructural settings, consider again the multiplicative sequent rules for ⊗.16 When an application of the cut rule is principal for the ⊗ conjunction we have the following reduction conversion which produces a derivation with two applications of cut on A and B: Γ ⇒ A Σ ⇒ B Θ, A, B ⇒ C Γ, Σ ⇒ A ⊗ B Θ, A ⊗ B ⇒ C Γ, Σ, Θ ⇒ C Σ⇒B Γ ⇒ A Θ, A, B ⇒ C Γ, Θ, B ⇒ C Γ, Σ, Θ ⇒ C In the conversion step for the additive ∧, on the other hand, only one cut application is required: Θ, A ⇒ C Γ⇒A Γ⇒B Γ⇒A∧B Θ, A ∧ B ⇒ C Γ, Θ ⇒ C Γ ⇒ A Θ, A ⇒ C Γ, Θ ⇒ C Notice that both the above reductions in sequent calculus produce the exact same sequent as the original derivation. In contrast, consider the case for a connective ⊓ which has a multiplicative introduction rule and two additive elimination rules: Γ, A ⇒ C Γ, A ⊓ B ⇒ C (L⊓)(i) Γ, B ⇒ C Γ, A ⊓ B ⇒ C (L⊓)(ii) Γ⇒A Σ⇒B Γ, Σ ⇒ A ⊓ B (R⊓) The appropriate conversion in the cut elimination strategy will now be as follows (the case for B is similar): Θ, A ⇒ C Γ⇒A Σ⇒B Γ, Σ ⇒ A ⊓ B Θ, A ⊓ B ⇒ C Γ, Σ, Θ ⇒ C Γ ⇒ A Θ, A ⇒ C Γ, Θ .⇒ C .. (K) . Γ, Σ, Θ ⇒ C Since the sequent Σ ⇒ B (alternatively, Γ ⇒ A) does not occur at all in the redux, the derivation requires a number of applications of weakening to add back the auxiliary formulae in Σ (Γ). The result, therefore, is that the conversion only works in the presence of weakening. Put differently, the redux proves something that could not be obtained with the original derivation. Ordinarily this is not a problem, but in a logic without weakening, say, a relevant logic, it leads to nonconservativeness. To see this, simply start with the redux derivation, and subsequently use the added ⊓ rules to expand to the left-most derivation. We can assume that the redux derivation does not include the connective ⊓ at all, and that it is therefore a derivation in the antecedent language. The expansion to the right-most derivation now allows us to weaken in additional premises in the antecedent, but without explicit use of a weakening rule. In other words, adding the mixed conjunction together with its left- and right-rule has allowed 16 See also Troelstra (1992) for a discussion of nonconservativeness. 14 us to derive a weakened sequent. In a system without weakening this gives us structural nonconservativeness. It is instructive to compare the situation in natural deduction. The connective ⊓ has the following (multiplicative) introduction and (additive) elimination rules: Γ. Σ. .. .. . . A B A⊓B (⊓I) A⊓B C Γ, [A]u .. .. C (⊓E)(i) A⊓B C Γ, [B]u .. .. C (⊓E)(ii) As pointed out above these are in fact the GE harmonious rules for ∧ in both intuitionistic and classical logic. Nevertheless, the problem of structural nonconservativeness is the same as in sequent calculus. The reduction conversion for normalization has the same feature as the sequent calculus counterpart: Γ Σ Θ, [A]u Π0 Π1 | {z } A B Π2 A⊓B C (u) C Γ Π0 A Θ | {z } Π2 C If we compare with the reduction conversion of the multiplicative ⊗ (in Section 5) we see that the subderivation Π1 from Σ to B now becomes redundant in the redux. Instead the derivation of C proceeds from the open assumptions in Γ ∪ Θ alone. In order to produce a derivation of C from Γ ∪ Θ ∪ Σ, we have to weaken the open premises in Σ back in. More directly, we can now show that weakening has become admissible. If we assume that C is already derivable from A, then we can use the above rules of ⊓ to produce a derivation of C from A,B. As was the case in sequent calculus, since the formulae can be from the ⊓-free fragment of the language, nonconservativeness follows. A B A ⊓ B [A]1 A .. .. C (⊓E)(1) That is, C is derivable from A, B, and there is no need for vacuous discharge. Note the contrast to the multiplicative conjunction elimination rule, an application of which would have required vacuous discharge.17 17 Note that the derivation is not possible if the introduction rule is additive. In that case open assumptions in the derivations of A and B must be the same. In the derivation, however, the application of the introduction rule has the premises A and B themselves as open assumptions, violating the context-sharing constraint of the additive rule. Correspondingly, in sequent calculus, the additive ∧R rule cannot be applied when the premise-sequents are the axioms A ⇒ A and B ⇒ B respectively. 15 6.2 Contraction A similar structural phenomenon arises when we consider a connective f which combines an additive introduction rule with a multiplicative elimination rule. First, in sequent calculus we have the following left- and right-rules: Γ, A, B ⇒ C Γ, A f B ⇒ C (Lf)(i) Γ⇒A Γ⇒B Γ⇒AfB (Rf) The conversion step for an application of cut turns out as follows: Γ ⇒ A Σ, A, B ⇒ C Σ, Γ, B ⇒ C Γ⇒B Γ, Γ, Σ. ⇒ C .. (W ) . Γ, Σ ⇒ C Γ ⇒ A Γ ⇒ B Σ, A, B ⇒ C Γ⇒AfB Σ, A f B ⇒ C Γ, Σ ⇒ C Now it is the structural rule of contraction that is required to produce a derivation of the same sequent in the redux. In other words, without structural contraction the derived sequent is too weak. If the connective f is added to an antecedent theory without contraction, therefore, its introduction and elimination rules can be applied to contract on the left. As in the case of weakening, there is an analogous result in natural deduction. The connective f has the additive introduction rule and the multiplicative elimination rule: Γ.α Γ.α .. .. . . A B AfB (fI) AfB C Γ, [A, B]u .. .. C (fE)(u) The reduction conversion then has the following shape: Γα Γα Θ, [A, B]u Π0 Π1 | {z } A B Π2 AfB C (u) C Θ | Γ Γ Π0 Π1 A B {z } Π2 C The redux derives C from Θ and two copies Γ. But unlike the original derivation it does not identify the two copies of Γ through the application of fI (and index α). Instead the redux derivation requires multiple discharge in order to derive what is already derivable without it in the original derivation. In other words, the detour through the additive conjunction rule introduces an element of contraction, even in a context where multiple discharge is not permissible. What we have, therefore, is a failure of local soundness: In a contraction-free context there is no reduction conversion of the original derivation (or even of anything stronger, as in the case of weakening). 16 As opposed to ⊓, which made weakening admissible, the mixed connective f makes structural contraction admissible. Consider the following derivation:18 2 2 A A AfA C Γ, [A, A]1 .. .. C (1) The derivation takes advantage of the fact that the additive introduction rule for f allows us to treat two copies of A as identical, and that the elimination rule can discharge two copies of A in the special case where the major premise is A f A. As a consequence, any formula C derivable from two copies of A is now derivable from one copy. The structural nonconservativness has given us back multiple discharge. The reintroduction of contraction through nonconservativeness can be particularly problematic. There are philosophically interesting formal theories that cannot include contraction on pain of contradiction. Unrestricted rules for a formal truth predicate and unrestricted set theoretic comprehension are examples of principles that are famously susceptible to paradox in classical logic.19 Both truth theoretic and set theoretic systems have therefore been developed in contraction-free logics (e.g. Zardini 2011, Petersen 2000). In such systems structural nonconservativeness leads to triviality, and so control over extensions in order to include new connectives (and new inference rules) is particularly critical. 7 Additive GE Harmony What do the examples tell us about GE harmony? GE harmony, as previously defined, is insensitive to substructural distinctions. As it stands the proposal induces additive elimination rules for conjunction from multiplicative introduction rules. The problem generalizes to other connectives as well: nonconservativeness can result from adding mixed—but GE harmonious—connectives to systems without weakening or contraction. In a substructural context, therefore, local soundness is not entailed by GE harmony. Local soundness only follows in the presence of special (structural) discharge policies, i.e. vacuous and multiple discharge. 18 The premises Aα in the application of fI are assumptions. Since an assumption of A is a special case of a derivation of A, we have derived A from A. This permits us to index the two copies of A with α. 19 Natural deduction rules for a unrestricted truth predicate and unrestricted comprehension: A T (pAq) Φ(t) t ∈ λxΦ(x) T (pAq) A (T I) (T E) t ∈ λxΦ(x) (λE) Φ(t) 17 (λE) Even if it does not follow that there is anything semantically wrong with mixed connectives, we have a motivation to give a modified definition of GE harmony that also delivers local soundness for substructural systems. The strategy will be a bifurcation of GE harmony according to the additive-multiplicative divide. Let us first consider the case for additive rules. The set of additive introduction rules, ⊙ , is specified in a manner similar to the previous definition of GE harmony: IA   α G.α G. G.α G.α     .. .. ..   .. . . . . ρ . . . ρ . . . ρ ρ m1 1k1 mkm   11 (⊙I1 ) (⊙Im )     ~ ~ ⊙β ... ⊙β Γ. G. .. .. . . The notation ρ is slightly abusive, as it can be instantiated both as A, in the Γ, [A] .. .. case where the premise is categorical, or as B , in the case where the premise is hypothetical. In both cases the members of G are the open assumptions labelling leaf nodes of the subderivation. (In the categorical case where A is an assumption, Γ = {A}.) ⊙ The GE harmoniously induced set of additive general elimination rules, EA , will now have minor premises adorned with (shared) contexts G:   G α , [ρ1j1 ]u G α , [ρmjm ]u   .. ..   .. .. ~ γ ... γ    ⊙β (⊙GE)(u)  γ ⊙ ⊙ ⊙ We write IA are induced GE EA to indicate that the elimination rules E ⊙ from the introduction rules I via the above templates. We can then give a modified version of GE harmony: ⊙ are in Definition 7.1 (Additive GE Harmony). A set of elimination rules EA ⊙ ⊙ ⊙ additive GE harmony with a set of introduction rules IA if IA GE EA . If a connective ⊙ has rule sets that are in additive GE harmony we will say that the connective is additively harmonious. Since there is a reduction conversion for rules satisfying the condition in Def. 7.1, additive GE harmony entails local soundness. Gα (Πi1 ) ρi1 ... ⊙β~ Gα (Πiki ) ρiki β H , [ρ1j1 ] Π∗1j1 γ γ u β ... H , [ρmjm ] Π∗mjm γ (u) u G Πiji H ρiji | {z } Π∗iji γ The above conversion is different from the conversions of the mixed connectives, and therefore also different from those given by the pre-substructural Def. 4.1 of 18 GE harmony. The difference is that the redux is a derivation of γ from the very same open assumptions as the original derivation. Hence, there is no need for structural rules (special discharge policies) in order to either weaken or contract. What we have, therefore, is a definition of GE harmony for additive connectives that is sensitive to the absence of weakening and contraction. 8 Multiplicative GE Harmony There is also a multiplicative definition of GE harmony, but one that is restricted in the natural deduction framework. Because natural deduction, unlike sequent calculus, typically is a single conclusion framework, we cannot have the full range of multiplicative connectives. What we can have, however, is a definition of GE harmony for all multiplicative connectives, ⊙, with only one introduction ⊙ rule in IM . For example, the conjunction and the biconditional can be given multiplicative natural deduction rules. In contrast, the multiplicative disjunction rules would require multiple conclusion (or at least a more general deducibility relation). Multiple conclusion in natural deduction is explored in a number of places, e.g. Boričić (1985), Ungar (1992) and Read (2000). Precisely how multiple conclusion natural deduction can be used to include the full range of multiplicative connectives will not be explored here. Instead we will give only a restricted form of multiplicative GE harmony. Thus, let the set of multiplicative (context-free) introduction rules be simply ⊙ = {⊙I}. Then we have the following schema: IM G.1 .. . ρ1 ... ⊙β~ G.k .. . ρk (⊙I1 ) The first index now becomes redundant since there is only one introduction rule, but as opposed to the additive rules, the auxiliary formula in each premise can now have distinct sets of open assumptions. ⊙ The single introduction rule in IM induces a single general elimination rule ⊙ ⊙ ⊙ in EM (IM GE EM ): ⊙β~ G, [ρm1 , ..., ρmkm ]u .. .. γ (⊙GE) γ The distinct feature of the multiplicative elimination rule is that its single minor premise takes every premise of the introduction rule as an assumption. These are jointly discharged by the application of the elimination rule, as seen, for example, in the multiplicative elimination rule for ⊗ above. Definition 8.1 (Restricted Multiplicative GE Harmony). A set of elimination ⊙ rules EM are in multiplicative GE harmony with a singleton set of introduction 19 ⊙ ⊙ ⊙ rules IM if IM GE EM . If a connective ⊙ has rule sets that are in additive GE harmony we will say that the connective is multiplicatively harmonious. ⊙ It follows immediately from the schemata that the set EM will also be a singleton set. We can then proceed to give a reduction conversion for the multiplicatively harmonious connectives: G1 Π1 ρ1 ... ⊙β~ Gk Πk ρk G γ | [ρ1 , ...ρk ]u {z } Π γ G | (u) Gk G1 Π1 Πk ρ 1 . . . ρk {z } Π γ Again, we can now see that local soundness has been restored. The redux derivation requires exactly the same open assumptions as the original derivation. Let us consider an example slightly more complicated than the multiplicative conjunction. The following is a context-free introduction rule for a multiplicative 4-ary connective, ։, together with its single general elimination rule: u  A .. C.. ..  .. Θ  B D | {z } Γ, [A]u Σ, [C]u .. .. .. .. .. .. E ։ (A, B, C, D) D (u) B (u) ։ (A, B, C, D) E We can then find the following reduction conversion for the new connective: Γ, [A]u Σ, [C]u Π1 Π0 D B ։ (A, B, C, D) E  Π3  A Θ  ... . B | {z Π2 E  Π4 C.  ..  . D Π3 A | {z } Π0 B {z Π2 E Γ } Θ | Π4 C | {z } Π1 D Σ } The restriction on multiplicative harmony raises the following question: Why develop notions of harmony in a natural deduction setting when it leads to artificial constraints on the connectives definable? The answer, we believe, is simply that sequent calculus is the preferable framework for proof theoretic harmony and PTS in general. However, there is a lot of resistance towards sequent calculus and, in particular, multiple conclusion relations in the PTS literature (cf. Dummett 1991, Rumfitt 2008, Steinberger 2011). Although we do not share these views, the guiding idea in the present paper has been to explore harmony for substructural connectives in a natural deduction framework. That 20 has the advantage that it is a more direct continuation of the debates about harmony in the literature, but the disadvantage that the expressive power is somewhat curtailed. 9 Normalization and Consistency It was pointed out in section §4 that the original Def. 4.1 of GE harmony does not entail normalization. The existence of a reduction conversion, i.e. local soundness, is guaranteed, but that alone is not sufficient for normalization. Read (2000) gives a counterexample, a harmonious connective that leads to inconsistency (and therefore nonconservativeness). Consider the following connective, •, read ‘bullet’:20 " • #u .. .. u ⊥ [•] .. .. .. .. • C ⊥ (•I)(u) (•GE)(u) • C The •I rule harmoniously induces the •GE rule. Moreover, there exists a reduction conversion for the • rules as prescribed by the above schema, but it does not lead to normalization: [•]u Π0 ⊥ (u) • C Π " • 2 #v .. .. ⊥ Π1 C (v) Π2 • Π0 ⊥ Π1 C The redux potentially produces a new maximum segment of the same complexity as the original derivation. In the right-most derivation the copy of • could have been introduced by an application of •I immediately followed by an application of •GE. Worse, the rules for • lead to contradiction (assuming anything is derivable from ⊥). Thus • is harmonious, yet its rules cannot be normalized and it is nonconservative over any consistent proof system.21 Moreover, the substructural definition of additive GE harmony Def. 7.1 does not prevent this problem, so another condition is required for normalization and conservativeness to follow in general. A flat-footed condition that would do the trick is the following: any formula occurring in the premises ρij of any introduction rule ⊙I must be a strict 20 For an earlier proof-theoretic study of normalization in the context of paradoxical expressions, see Tennant (1982). 21 See Read (2000) for details on how triviality follows from the rules for •. The derivation that leads to triviality is precisely an example of a derivation that cannot be put into normal form through reduction conversions. 21 ~ It follows immediately that every ρij occurring subformula of the conclusion β. in an elimination rule ⊙E (i.e. the assumptions in the minor premises) will also be strict subformulae of the major premise. Obviously, purity is not satisfied by •I, but it is satisfied by the other GE harmonious rules we have considered. The advantage is simple. GE harmony and purity will jointly be sufficient for normalization, the subformula property, and conservativeness. That is already an achievement. Nevertheless, it is not necessarily a desirable condition. First, note that although it is sufficient for normalization of GE harmonious rules, the purity condition is not a necessary condition for normalization. There are, in other words, connectives whose rules normalize despite failing to satisfy the purity condition. One example is the following rules for intuitionistic negation: [A]u .. .. B [A]u .. .. ¬B ¬A (u) A ¬A B The left-most rule includes a formula B in both subderivations that can be of any complexity, and is therefore not pure. Nevertheless, the rules can be shown to normalize. Hence, imposing purity as an additional constraint on introduction rules excludes some perfectly well-behaved connectives. A more ambitious approach is to aim for an exact (necessary and sufficient) condition on introduction and elimination rules that lead to normalization. That is still an open problem. A conjecture is that there is at least a weaker purity condition that entails normalization. Let us call it negative purity: every introduction rule in I⊙ should include only premises ρij in which every formula ~ A occurring in negative position is a strict subformula of the conclusion β. formula A is in negative position in a premise ρij just in case it occurs as an assumption. It should be immediately clear that •I fails the negative purity condition, whereas the intuitionistic negation introduction rule satisfies it. As mentioned above, the move to substructural (additive and multiplicative) modifications of GE harmony will not change the need for a further condition to ensure normalization. But it is interesting to note that substructural systems do allow for normalization of rules that are not normalizable in proof systems with weakening and contraction. The introduction and elimination rules for • is one example, albeit somewhat artificial. Schroeder-Heister (1992) shows that cut-elimination holds for contraction-free systems even in the presence of a connective like •. In the absence of multiple discharge of assumptions, normalization can be proved by an induction on the size of the derivation tree, rather than on the complexity of maximum formulae. That this is not possible when multiple discharge is allowed as can be seen from the following example: 22 u u [•] , ..., [•] {z } | Π0 ⊥ (u) • C Π " • 2 #v .. .. ⊥ Π1 C (v) Π2 • | ... {z Π0 ⊥ Π1 C Π2 • } Since the left-most derivation requires a number of copies • to derive ⊥ via Π0 , the same number of subderivations Π2 to • are required for the redux derivation. The subderivation Π2 can be of any length, and therefore rules out an induction on the size of the derivation. In general, the proof theoretic conditions for normalization and consistency are significantly weaker in the absence of contraction. Proofs of cut elimination already exists for contractionfree theories of unrestricted set comprehension and unrestricted truth predicates (cf. Petersen 2000, Zardini 2011). These and similar results makes the demand for structural conservativeness all the more pressing. If we were to introduce a mixed additive and multiplicative connective into such a system, the result might be nonconservativeness. In particular, if a connective makes contraction admissible, the entire theory will become inconsistent. Substructural sensitivity is therefore critical. The modified account of GE harmony is at least a partial solution. Conclusion Proof theoretic harmony was conceived as a formal condition that excludes trivializing connectives such as tonk. More generally, harmony conditions were meant to ensure conservativeness once a new connective is introduced into an antecedent theory. GE harmony is a good candidate for such a formal condition, with the caveat that some form of purity condition is also required. But we have seen that GE harmony in its original form can yield nonconservativeness in substructural systems. 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