Existence and Identity in Free Logic: A Problem for Inferentialism? Abstract Peter Milne (2007) poses two challenges to the inferential theorist of meaning. This study responds to both. First, it argues that the method of natural deduction idealizes the essential details of correct informal deductive reasoning. Secondly, it explains how rules of inference in free logic can determine unique senses for the existential quantifier and the identity predicate. The final part of the investigation brings out an underlying order in a basic family of free logics. 1. Background 1.1 Free v. unfree logic Standard, unfree logic has among its rules of natural deduction the following (see, for example, the system in the classic monograph Prawitz 1965). : ϕ where a does not occur in any Universal Introduction (∀-I) assumption on which ϕ depends a ∀xϕ x Universal Elimination (∀-E) ∀xϕ x ϕ t x Existential Introduction (∃-I) ϕ t ∃xϕ (i) x ϕ a where a does not occur in ∃xϕ, or in ψ, x Existential Elimination (∃-E) : or in any assumption, other than ϕ a, on ∃xϕ ψ (i) which the upper occurrence of ψ depends ψ Reflexivity of identity t=t Substitutivity of identity t=u ϕ (t) ϕ(u) Let t be any term, not necessarily closed. Then ∃!t is short for ∃x x=t, where x is alphabetically the first variable not free in t. In unfree logic we have, for every closed term t, – ∃!t. We also have – ∃x x=x. So unfree logic is committed to a denotation for every closed term, and to the existence of at least one thing. A logic is free when it is no longer based on the assumption that all singular terms denote. A free logic is universally free when in addition it is no longer based on the assumption that the universe is non-empty. In Tennant 1978, rules of natural deduction were provided for a system of universally free logic that will here be called FUNL. The salient rules for FUNL can be found in §7. (Rules for the connectives are omitted.) 1.2 Inferentialism An inferentialist theory of meaning holds that the meaning of a logical operator can be captured by suitably formulated rules of inference (in, say, a system of natural deduction). The problem to be addressed is: Can one be an inferentialist about the logical operators in free logic? Prima facie, the answer should be affirmative. For whatever reasonable system of free logic one chooses, there is a natural-deduction system for it, and the rules of inference involved ought to be ‘capturing’ the meanings of the logical operators involved. The problem is made sharper, however, by the proliferation of rules involved in free logic, and the formulation of some of them that involves so-called ‘existential presuppositions’. There are also many different equivalent formulations—different sets of rules of inference—for one and the same system of free logic. Is there an identification (in the context of free logic) of ‘the’ introduction rule for ∃ (for example), and of ‘the’ corresponding elimination rule, which enables one to show that they are indeed in harmony, and that they succeed in specifying a unique sense for ∃? 2. Milne’s challenge Peter Milne (2007) identifies what he believes is a problem for an inferentialist theory of meaning for logical operators in the context of a free logic. The problem is supposed to reveal a tension between, on the one hand, the present author’s invocation of free logic for constructive logicism (see Tennant 1987) and for abstractionist realism (see Tennant 2004) more generally; and, on the other hand, the inferentialist thesis that harmonious rules of inference for a logical operator successfully characterize its meaning. A diagnosis is offered below of the reason why the purported problem might be perceived as genuine; and a reasonably straightforward solution is offered. Milne’s investigations provide a welcome opportunity to clarify and amend an overall position developed at different stages over quite some time. §3 provides a defence of inferentialism about free logic. Milne also enters some misgivings about the extent to which systems of natural deduction faithfully capture the meanings of the quantifiers (especially the existential quantifier), as these are revealed in informal mathematical usage. In §5, the methods of natural deduction due to Gentzen and Prawitz are defended against Milne’s criticisms. 3. In defence of inferentialism about free logic 3.1 On existence Milne claims that the rules of FUNL for the existential quantifier (see §7) do not succeed in conferring upon it a unique sense. In order to justify this claim, he asks his reader to consider those rules with different notation for the quantifier (Σ in place of ∃): x (Σ-I) Σ!t ϕ t Σxϕ ___(i) ___(i) x (Σ-E) Σ!a , ϕ a : Σxϕ ψ (i) ψ To quote Milne: Let χ be any closed formula you like. These rules are truth-preserving when one reads Σxϕ as ∃x(ϕ∧χ). Let us call this technique the milning of a pair of introduction and elimination rules. Milne’s phrase ‘These rules’, however, refers to the doctored pair (Σ-I) and (Σ-E), respectively obtained from the rules (∃-I) and (∃-E) in FUNL by writing occurrences of ‘Σ’ in place of occurrences of ‘∃’. But what of the other two rules in which ∃ figures so prominently in FUNL—the rule of atomic denotation and the rule of functional denotation? Surely these rules, too, need their Σ-versions, if Σ is to be a genuine but deviant alternative to ∃? Let us therefore extend Milne’s competing rules by adding the following variants of the two denotation rules just mentioned: A(t) , where A(t) is atomic; and Σ!f(t) Σ!t Σ!t For the argument to follow, only the first of these two rules is needed. Consider now the following formal proofs in the expanded system. They show that, pace Milne, one will be able to deduce Σxϕ from ∃xϕ, and conversely. The proof of Σxϕ from ∃xϕ is as follows: ____(1) b=a (1) (2) Σ!b b=a (2) ∃x x=a Σx x=a (1) x ϕ a Σx x=a ∃xϕ Σxϕ (2) Σxϕ and the proof of the converse is obtained by interchanging ∃ and Σ. So we see that Σxϕ and ∃xϕ are synonymous, after all. Note that the rule allowing one to infer Σ!t from an atomic sentence A(t) prevents milning: for, in order for this rule to be truth-preserving when Σxϕ is read as ∃x(ϕ&χ), the sentence χ must be logically true—which is harmless. To be fair to Milne, it should be pointed out that the step from b=a to Σ!b in the proof above is one for which his selection of Σ-rules does not provide. But then (one might say) so much the worse for that selection of Σ-rules. Without an analogue of the rule of atomic denotation, they do not capture a genuine alternative to ∃. Lest the reader think that some genuine problem for ∃ is being downplayed here, it should be pointed out that one could milne the rules for &. Consider the following rules for a binary connective @: (@-I) ϕ ψ (@-E) ϕ@ψ ϕ@ψ ϕ@ψ ϕ ψ With ϕ@ψ interpreted as (ϕ&ψ)&χ, these rules are truth-preserving—if, but only if, χ is a logical truth. Milne observed, ingeniously, that the Σ-analogues of what had all along been called (∃-I) and (∃-E) in free logic allow for the deviant interpretation of Σxϕ as ∃x(ϕ&χ), where χ is any sentence. What Milne’s observation prompts is the realization that the rule of existential introduction in universally free logic should be understood as consisting not only of the usual part: x ∃!t ϕ t ∃xϕ but also of another part: A(t) , where A(t) is atomic, ∃!t which had escaped detection as a genuine part of the introduction rule for ∃. It escaped detection because it had been given a separate title of its own, namely ‘the rule of atomic denotation’. Once this rule is made part of the introduction rule for ∃, however, it prevents milning of the rules for ∃; for it is not truth-preserving on the deviant interpretation of ∃xϕ as ∃x(ϕ&χ). The inferential meaning-theorist, it turns out, had only a classification problem with the rules of natural deduction for free logic—a problem that, as we have just seen, can be solved. Milne goes to some lengths to stress an alleged difference between his own construal of the rule of atomic denotation, and what he takes to be the present author’s construal. This reply affords the opportunity to disavow various imputed views. … I suspect … Tennant and I differ fundamentally in how we construe the rule of atomic denotation. … Tennant does not tell us about the rule’s logical status. … I hazard the guess that Tennant … sees the rule as giving expression to an independent semantic principle. Tennant, I suspect, takes the sense of existence, of the existential quantifier, as antecedently given, as prior to the adoption of the rule. (Milne 2007, pp. 50–51) The textual evidence that Milne offers in support of these suspicions does not ground them at all. The claim that Milne quotes—‘an atomic claim will be true only if all its terms denote’—is merely a statement of the rule of atomic denotation, not some philosophically contentious ‘construal’ of it. Milne complains (p. 51) that he ‘can see no robust sense of reality reflected in use of the rule of atomic denotation’ if that rule is used ‘to bolster the proof-theoretic semantics credentials of free logic’.