On GE-harmony Stephen Read Workshop in honour of Roy Dyckhoff Analytic Validity St Andrews 18-19 November 2011 Harmony GE-Harmony On GE-harmony Justifying the E-rules Conjunction Two E-Rules or One? Implication Stephen Read Dyckhoffization Modus Ponens Equivalence and ODot Arch´ e: Philosophical Re- Negation Reduction search Centre for Logic, Language, Metaphysics and Bullet Epistemology Inconsistency Summary Foundations of Logical Con- References sequence Project Funded by 19 November 2011 1 / 26
On GE-harmony Analytic Validity Stephen Read ◮ In 1960, Arthur Prior introduced a new connective ‘tonk’ with the rules: Analytic Validity α tonk β α Harmony α tonk β tonk-I tonk-E β GE-Harmony Justifying the E-rules However, by chaining together an application of tonk-I to one of Conjunction tonk-E, we can apparently derive any proposition ( β ) from any Two E-Rules or One? other ( α ). Implication ◮ Prior described such a commitment as “analytic validity”. Dyckhoffization Modus Ponens ◮ This is clearly absurd and disastrous. How can one possibly define Equivalence and ODot such an inference into existence? Negation Reduction ◮ We may agree with Prior that ‘tonk’ had not been given a Bullet coherent meaning by these rules. Inconsistency ◮ Rather, whatever meaning tonk-introduction had conferred on the Summary neologism ‘tonk’ was then contradicted by Prior’s tonk-elimination References rule. ◮ But we might respond to Prior by claiming that if rules were set down for a term which did properly confer meaning on it, then certain inferences would be “analytic” in virtue of that meaning. ◮ What constraints must rules satisfy in order to confer a coherent meaning on the terms involved? 2 / 26
On GE-harmony Harmony Stephen Read ◮ Dummett introduced the term ‘harmony’ for this constraint: in Analytic Validity order for the rules to confer meaning on a term, two aspects of Harmony its use must be in harmony. GE-Harmony ◮ Those two aspects are the grounds for an assertion as opposed Justifying the E-rules to the consequences we are entitled to draw from such an Conjunction Two E-Rules or One? assertion. Implication ◮ Those whom Prior was criticising, Dummett claimed, Dyckhoffization Modus Ponens committed the “error” of failing to appreciate Equivalence and ODot Negation “the interplay between the different aspects of ‘use’, and Reduction the requirement of harmony between them. Bullet If the linguistic system as a whole is to be coherent, Inconsistency Summary there must be a harmony between these two aspects.” References ◮ Dummett is here following out an idea of Gentzen’s, in a famous and much-quoted passage where he says that “the E-inferences are, through certain conditions, unique consequences of the respective I-inferences.” 3 / 26
On GE-harmony Proof-theoretic Justification Stephen Read ◮ But in fact, Dummett had loftier ambitions than this. He introduced Analytic Validity the idea of the proof-theoretic justification of logical laws. Harmony ◮ In this, Dummett was following the lead of Dag Prawitz GE-Harmony ◮ In a series of articles on the “foundations of a general proof theory” Justifying the E-rules published in the early 1970s, Prawitz had set out to find a Conjunction characterization of validity of argument independent of model theory, Two E-Rules or One? as typified by Tarski’s account of logical consequence. Implication ◮ Following Gentzen’s idea in the passage cited above, Prawitz accounts Dyckhoffization Modus Ponens an argument or derivation valid by virtue of the meaning or definition Equivalence and ODot of the logical constants encapsulated in the introduction rules. Negation ◮ Take the introduction-rules as given. Then any argument (or in the Reduction Bullet general case, an argument-schema) is valid if there is a “justifying Inconsistency operation” ultimately reducing the argument to the application of Summary introduction-rules to atomic sentences: References “The main idea is this: while the introduction inferences represent the form of proofs of compound formulas by the very meaning of the logical constants . . . and hence preserve validity, other inferences have to be justified by the evidence of operations of a certain kind.” 4 / 26
On GE-harmony General-Elimination Harmony Stephen Read ◮ What Prawitz does, in fact, is frame his E-rules in such a way that such a reduction is possible Analytic Validity ◮ Given a set of introduction-rules for a connective (in general, there Harmony GE-Harmony may be several, as in the familiar case of ‘ ∨ ’), the elimination-rules Justifying the E-rules (again, there may be several, as in the case of ‘ ∧ ’) which are Conjunction justified by the meaning so conferred are those which will permit a Two E-Rules or One? reduction operation of Prawitz’ kind Implication ◮ Each E-rule is harmoniously justified by satisfying the constraint Dyckhoffization Modus Ponens that whenever its premises are provable (by application of one of Equivalence and ODot the I-rules), the conclusion is derivable (by use of the assertion- Negation conditions framed in the I-rule) Reduction ◮ Roy Dyckhoff and Nissim Francez introduced the name Bullet Inconsistency “General-Elimination Harmony” for the general procedure by which Summary we obtain the E-rule from the I-rule References ◮ They reformulate Prawitz’ “inversion principle” as follows: Let ρ be an application of an elimination-rule with consequence ψ . Then, the derivation justifying the introduction of the major premiss φ of ρ , together with the derivations of minor premisses of ρ “contain” already a derivation of ψ , without the use of ρ . 5 / 26
On GE-harmony Justifying the E-rules Stephen Read Here is my formulation of the procedure for generating the set of generalised E-rules. This differs to some extent from the form proposed Analytic Validity by Dyckhoff and Francez, as we will see: Harmony GE-Harmony ◮ Suppose there are m I-rules for a connective ‘ δ ’, each with n i Justifying the E-rules premises (0 ≤ i ≤ m ): Conjunction Two E-Rules or One? π i 1 . . . π in i δ -I i Implication δ� α Dyckhoffization Modus Ponens ◮ Here δ� Equivalence and ODot α is a formula with main connective ‘ δ ’ ◮ Each π ij , 0 ≤ j ≤ n i , may be a wff (as in ∧ I), or a derivation of a Negation Reduction wff from certain assumptions which are discharged by the rule (as Bullet in → I). Inconsistency ◮ This set of I-rules justifies � m i =0 n i E-rules, each of the form: Summary References [ π 1 j 1 ] [ π mj m ] . . . . . . . . . . . δ� α γ γ δ -E γ ◮ Each minor premise derives γ from one of the grounds, π ij i , in the i -th rule for asserting δ� α . 6 / 26
On GE-harmony The Inversion Principle Stephen Read ◮ The GE-procedure ensures that one can infer γ from δ� α whenever one can infer γ from one of the grounds for assertion Analytic Validity Harmony of δ� α GE-Harmony ◮ Consequently, as Dyckhoff and Francez say, the actual assertion Justifying the E-rules of δ� α is an unnecessary detour: Conjunction Two E-Rules or One? . . . . . . [ π 1 j 1 ] [ π mj m ] Implication . . . . . . . . . Dyckhoffization π i 1 π in i . . . . Modus Ponens δ -I . . . Equivalence and ODot δ� α γ γ δ -E γ Negation Reduction . . Bullet . . Inconsistency π ij i converts to . Summary . . . References γ ◮ Having one minor premise in each E-rule drawn from among the premises for each I-rule ensures that, whichever I-rule justified assertion of δ� α (here it was the i -th), one of its premises can be paired with one of the minor premises to remove the unnecessary application of δ -I immediately followed by δ -E. 7 / 26
On GE-harmony What can Harmony do for us? Stephen Read ◮ Dummett and Prawitz (and others) actually make a stronger claim: that an inference is not justified if the rules are not harmonious Analytic Validity ◮ For example, Dummett claims that classical logic, with classical negation, Harmony is incoherent since ¬ -E goes beyond what is justified by ¬ -I GE-Harmony ◮ In my view, this asks too much of harmony and the constraints on the Justifying the E-rules rules it invokes Conjunction ◮ For example, consider the Curry-Fitch rules for ♦ (possibility): Two E-Rules or One? Implication [ α ] Dyckhoffization Modus Ponens . . Equivalence and ODot . Negation α ♦ α γ and ♦ α ♦ -I Reduction ♦ -E γ Bullet Inconsistency provided that in the case of ♦ -E, every assumption on which the minor Summary premise γ depends, apart from α (the so-called parametric formulae), is References modal and γ is co-modal , that is, has the form ♦ β ◮ These rules are not harmonious: the (unrestricted) rule ♦ -I does not justify the restriction put on ♦ -E. ♦ -I appears to say that ♦ α just means α ◮ But the model theory shows that the rules do define possibility. Quite how they interact to do so is far from obvious ◮ What harmony can do for us is ensure that the I- and E-rules confer the same meaning. 8 / 26
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