Proof-theoretic Validity. Stephen Read University of St Andrews. Analytic Validity Harmony GE-Harmony Justifying the E-rules

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1 Boğaziçi University Arché: Philosophical esearch Centre for Logic, Language, Metaphysics and Epistemology Foundations of Logical Consequence Project Funded by 4 April 2012 GE- In 1960, Arthur Prior introduced a new connective tonk with the rules: tonk tonk-i tonk tonk-e His target was the idea that there are inferences whose validity arises solely from the meanings of certain expressions occurring in them However, by chaining together an application of tonk-i to one of tonk-e, we can apparently derive any proposition from any other This is clearly absurd and disastrous How can one possibly define such an inference into existence? We may agree with Prior that tonk had not been given a coherent meaning by these rules ather, whatever meaning tonk-introduction had conferred on the neologism tonk was then contradicted by Prior s tonk-elimination 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? slr@st-andacuk GE- 1 / 36 2 / 36 Michael ummett introduced the term harmony for this constraint: in order for the rules to confer meaning on a term, two aspects of its use must be in harmony Those two aspects are the grounds for an assertion as opposed to the consequences we are entitled to draw from such an assertion Those whom Prior was criticising, ummett claimed, committed the error of failing to appreciate the interplay between the different aspects of use, and the requirement of harmony between them If the linguistic system as a whole is to be coherent, there must be a harmony between these two aspects ummett 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 slr@st-andacuk GE- Justification ummett s aim is the proof-theoretic justification of logical laws In this, he was following the lead of ag Prawitz In a series of articles on the foundations of a general proof theory published in the 1970s, Prawitz had set out to find a characterization of validity of argument independent of model theory, as typified by Tarski s account of logical consequence Following Gentzen s idea in the passage cited above, Prawitz accounts an argument or derivation valid by virtue of the meaning or definition of the logical constants encapsulated in the introduction rules Take the introduction-rules as given Then any argument (or in the general case, argument-schema) is valid if there is a justifying operation ultimately reducing the argument to the application of introduction-rules to atomic sentences: 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 slr@st-andacuk GE- 3 / 36 4 / 36

2 General-Elimination What Prawitz does, in fact, is frame his E-rules in such a way that such a reduction is possible Given a set of introduction-rules for a connective (in general, there may be several, as in the familiar case of ), the elimination-rules (again, there may be several, as in the case of ) which are justified by the meaning so conferred are those which will permit a reduction operation of Prawitz kind Each E-rule is harmoniously justified by satisfying the constraint that whenever its premises are provable (by application of one of the I-rules), the conclusion is derivable (by use of the assertion-conditions framed in the I-rule) In other words, the E-rules are admissible rules oy yckhoff and Nissim Francez introduced the name General-Elimination for the general procedure by which we obtain the E-rule from the I-rule Prawitz based his constraint on Lorenzen s inversion principle, which Francez reformulates as follows: Let ρ be an application of an elimination-rule with consequence ψ from open assumptions Γ = Γ 1 Γ 2 Then, the derivation justifying the introduction of the major premiss φ of ρ from open assumptions Γ 1, together with the derivations of minor premisses of ρ from open assumptions Γ 2, contain already a derivation of ψ from the same Γ, without the use of ρ slr@st-andacuk GE- Suppose there are m I-rules for a connective δ, each with n i premises (0 i m): π i1 π ini δ Here δ 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 wff from certain assumptions which are discharged by the rule (as in I) This set of I-rules justifies m i=0 n i E-rules, each of the form: δ [π 1j1 ] δ-i i [π mjm ] Each minor premise derives from one of the grounds, π iji, in the i-th rule for asserting δ How does the justification work? δ-e slr@st-andacuk GE- 5 / 36 6 / 36 The Inversion Principle The GE-procedure ensures that one can infer from δ whenever one can infer from one of the grounds for assertion of δ Consequently, the actual assertion of δ is an unnecessary detour: π i1 δ π ini [π ] 1j1 δ-i i [π mjm ] δ-e j converts to π iji 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 (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 slr@st-andacuk GE- What can do for us? ummett and Prawitz (and others) actually make a stronger claim: that an inference is not justified if the rules are not harmonious For example, ummett claims that classical logic, with classical negation, is incoherent since -E goes beyond what is justified by -I In my view, this asks too much of harmony and the constraints on the rules it invokes For example, consider the Curry-Fitch rules for (possibility): [] -I and -E provided that in the case of -E, every assumption on which the minor premise depends, apart from (the so-called parametric formulae), is modal, that is, has the form and is co-modal, that is, has the form These rules are not harmonious: the (unrestricted) rule -I does not justify the restriction put on -E -I appears to say that just means 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 slr@st-andacuk GE- 7 / 36 8 / 36

3 Let us now turn to consider some more and less familiar connectives, and how to treat them harmoniously, by the GE-procedure: Given I we obtain two generalized -E rules, assuming -I to exhaust the grounds for asserting (so m = 1 and n 1 = 2): [] [] and -E 1 -E 2 slr@st-andacuk GE- The Simplified -E The generalised -E rules yield the more familiar -E rules of Simp(lification) immediately, by letting be and respectively: which reduce to [] -E 1 and Simp 1 and [] -E 2 Simp 2 given that we can always derive from, for any Conversely, -E 1 follows from Simp 1, that is, that if there is a derivation of from, then follows from : slr@st-andacuk GE- 9 / 36 and the same for -E 2 10 / 36 Generalised -E yckhoff and Francez, Schroeder-Heister and others, have a single form of the generalised rule: [] [] }{{} -GE To see that -GE is equivalent to the conjunction of -E 1 and -E 2, let us replace the two-dimensional representation of the derivation of from and by the linear form, Then we can derive each of -E 1 and -E 2 from -GE: and the same for Conversely,, K (Weakening) -GE, -E 1 -E 2 What this shows is,, and -GE follows by Contraction (W) slr@st-andacuk GE- Thus we have two competing forms of -E, though they are equivalent, given Contraction and Weakening But in the absence of W and K, which is the right form? ecall the additive and multiplicative rules for and in linear logic:, Γ Θ, Γ Θ, Γ Θ, Γ Θ,, Γ Θ, Γ Θ Clearly, -GE gives the multiplicative rule for, whereas -E 1 and -E 2 give the correct rules for additive In the presence of W and K, the additive/multiplicative distinction is erased, but to give the rules in their proper form, we should give separate E-rules for, each corresponding to one premise in -I -E 1 and -E 2 confirming the correctness of the GE-procedure slr@st-andacuk GE- 11 / / 36

4 As a second example, consider the I-rule for implication: [] -I that is, -I inferring (an assertion of the form) from (a derivation of), permitting the discharge of (zero or more occurrences of) To find the form -E should have, there should be the appropriate justificatory operation of which Prawitz spoke That is, we should be able to infer from an assertion of no more (and no less) than we could infer from whatever warranted assertion of We can represent this as follows: [] -E that is, [ ] -E That is, if we can infer from assuming the existence of a derivation of from, we can infer from slr@st-andacuk GE- [ ] What does mean? It says that, assuming we have a derivation of from, we can obtain a derivation of Hence, if we have a derivation of, we may assume we are able to derive, from which we derive That is, [ ] -E means -E [] which consequently justifies this schema: -E oy yckhoff was the first to propose this formulation, in 1988 during the MacLogic project at the slr@st-andacuk GE- 13 / / 36 Modus Ponendo Ponens Another way to think of this move appeals to the sequent formulation The minor premise of -E reads: ( ) Using Gentzen s -left rule, we have Thus our generalised -E rule reads: ( ) -left -E -E Other things being equal, we can now permute the derivation of from with the application of the elimination-rule, to obtain the familiar rule of Modus Ponendo Ponens (MPP): [] MPP slr@st-andacuk GE- Often, is treated by definition as, where is governed solely by an elimination-rule, from infer anything: E In the MS, Gentzen treated as primitive As introduction-rule, he took reductio ad absurdum: [] [] What elimination-rule does this justify? We can infer from whatever (all and only that which) we can infer from its grounds There is one I-rule with two premises (m = 1, n 1 = 2), so there will be two E-rules, one for each premise of the I-rule: [ [] ] E 1 and [ ] [] E 2 slr@st-andacuk GE- 15 / / 36

5 Ex Falso Quodlibet Flattening as before, where we infer from assuming the existence of derivations, respectively, of and of from, we obtain: and [] [ ] and so and so The second of these is simply a special case of the first, and so we have justified Gentzen s form of Ex Falso Quodlibet as the matching elimination-rule for : slr@st-andacuk GE- eduction We need to check, however, that this rule does accord harmoniously with I and permit a reduction of Prawitz kind So suppose we have an assertion of justified by, immediately followed by an application of : [] [] If we now close the open assumptions of the form in and with the derivation, we obtain: A worry, recognized by Gentzen, is that we still have an occurrence of the wff, major premise of an application of and possibly inferred by Indeed, since and are independent, the degree of may be greater than that of How can we ensure that a reduction been carried out? slr@st-andacuk GE- 17 / / 36 Gentzen s Solution Gentzen s solution, described in the MS, is first to perform a new kind of permutative reduction on the original derivation of, so that it concludes in a single application of Suppose otherwise, that is, that the derivation of concludes in successive applications of : [] [], [] [], [] The detour through is unnecessary The derivation can be simplified as follows by inferring directly from and : [], [] [], [] By successive simplifications of this kind, we can ensure that does not conclude in an application of I and so in the original application of is not a maximum formula slr@st-andacuk GE- Classical The account of negation given by and is intuitionistic But similar arguments extend this account to classical negation, by setting it in a multiple-conclusion framework generalizes to a multiple-conclusion rule as: [] [],,, m from which the inversion principle yields the pair of higher-level E-rules:, Γ Γ, [ ] which flatten and simplify as before to and, Γ Γ,, Γ, Γ, m [ ] From m and m we can derive double-negation elimination, and so justify C, as derived rules: (1), K (1), K m (1), m [ ] [ ] N slr@st-andacuk GE- 19 / / 36

6 Multiple- easoning The intelligibility of multiple-conclusion reasoning has been challenged What does it mean to draw more than one conclusion from a set of premises? The idea behind natural deduction formulations of logic is to capture the idea of reasoning from assumptions But so-called natural deduction systems are often seen as very unnatural In particular, many students find the rule of -E very unintuitive and difficult to master: [] [] What students want to do is go forward from and infer That s invalid! But what is reasonable is to go forward from and infer O reasoning by cases In other words, the rule they want to use is:, or better: -E -E M Of course, there need to be constraints on how one can proceed from each conclusion here we mustn t apply -I! But this is multiple-conclusion reasoning, and it s quite natural (more natural than -E) slr@st-andacuk GE- Equivalence Now consider the obvious introduction rule for the biconditional,, which requires both that be derivable from and vice versa: [] [] -I Then m = 1 and n 1 = 2, so there are two E-rules each with one minor premise: [ ] -E 1 [ ] -E 2 Each simplifies by flattening of the rules and moves similar to those with the generalised rule for -E: -E 1 -E 2 slr@st-andacuk GE- 21 / / 36 Oot Next, suppose we introduce a novel connective which instead disjoins the grounds for asserting instead of conjoining them: [] -I 1 [] -I 2 Now we have two I-rules each with one premise (m = 2, n 1 = n 2 = 1), so there will be one E-rule with two minor premises: Flattening now yields: [ ] [] [ ] [] -E -E slr@st-andacuk GE- Tautology This may seem puzzling, but in fact, reflection shows that it is to be expected, at least from a classical perspective says that either follows from or follows from That is a classical tautology: ( ) ( ) In fact, even with the intuitionistic negation rules, we can prove ( ): ( ) (1) (2) -I 1(3) -I 2(2) ( ) ( ) (1) (1) Intuitionistically, a third possibility is never ruled out, but nor can it be denied, on pain of contradiction slr@st-andacuk GE- and the major premise seems redundant We can prove directly from the minor premises (indeed, twice over) 23 / / 36

7 Higher-level vs Flattened The example of illustrates a general problem affecting the flattening procedure In the case of and, the flattened rule is easily shown to be as strong as the higher-level rule But in the case of and other connectives, and in general, this is not true Schroeder-Heister raises the issue, identifying two kinds of problem case Take the general case of δ-e: δ [π 1j1 ] 1 [ 1 2 ] i [π mjm ] m δ-e where discharged assumption π iji is of higher level, assuming a derivation of 2 from 1 First, in any application, the assumption may have been discharged vacuously; that is, there may be a proof of not depending on the assumption of a derivation of 2 from 1 at all slr@st-andacuk GE- Lost Assumptions Secondly, in the derivation of from the higher-level assumption of a derivation of 2 from 1, there may be some assumption made in the derivation of 1 which is only discharged subsequent to the use of the higher-level assumption: δ [ɛ] 1 2 (ɛ) δ-e does not justify δ [ɛ?] 1 [ 2 ] (ɛ?) That is, 1 may depend on some assumption ɛ on which 2, but not, also depends In the higher-level derivation, ɛ is discharged in the course of In the flattened scheme, however, ɛ is left undischarged in and is not available for use in, which consequently is no longer a derivation δ-e slr@st-andacuk GE- 25 / / 36 Another Novel Connective It therefore seems that, in general, the higher-level rule is stronger than its flattened version At least, this is clearly so in intuitionistic logic Take the rules for the novel connective c 2 (introduced by Schroeder-Heister): [ 1 ] 2 c 2 ( 1, 2, 3, 4 ) c 2-I 1 [ 3 ] 4 c 2 ( 1, 2, 3, 4 ) c 2-I 2 Considerations of GE-harmony yield a single higher-level E-rule with two minor premises: c 2 ( 1, 2, 3, 4 ) and the corresponding flattened rule: [ 1 2 ] [ 3 4 ] c 2 -E [ 2 ] [ 4 ] c 2 ( 1, 2, 3, 4 ) 1 3 c 2 -E slr@st-andacuk GE- Higher-Order Equivalence Let c 2 ( ) abbreviate c 2 ( 1, 2, 3, 4 ), and ( ) abbreviate ( 1 2 ) ( 3 4 ) With the higher-level rule, c 2 -E, we can show by intuitionistically acceptable means that c 2 ( ) ( 1 2 ) ( 3 4 ): c 2 ( ) Conversely, ( 1 2 ) ( 3 4 ) (1) (2) (3) (4) I(1) 4 -I(3) ( ) ( ) c 2 -E(2, 4) ( 1 2 ) ( 3 4 ) 1 (1) 1 2 (2) 2 c 2 ( ) c 2-I 1 (1) c 2 ( ) 3 (3) (4) c 2 ( ) c 2-I 2 (3) -E(2, 4) slr@st-andacuk GE- 27 / / 36

8 c2( ) Classical ecapture With the flattened rules, however, it is not possible to derive ( 1 2 ) ( 3 4 ) from c 2 ( ) using intuitionistically valid rules But with classical reductio, C, it is possible: 1 (2) ( ) 1 (3) -I(3) 1 (4) (1) 2 ( ) C(2) 1 2 ( ) ( ) 3 (5) ( ) 3 (6) -I(6) 3 4 (7) (1) ( ) 3 4 C(5) ( ) c2-e (4, 7) (1 2) (3 4) (1) ( ) C(1) There is more than the consequentia mirabilis role of C in play here (that is, to infer from a demonstration that leads to contradiction) There is also much use of K and W in the multiple and vacuous discharge of assumptions in C and -I slr@st-andacuk GE- Multiple- easoning The reason is classically derivable, and that c 2 ( ) ( 1 2 ) ( 3 4 ) is that the classical negation rules yield the full classical theory of implication In the multiple-conclusion sequent calculus, the classical left-implication sequent calculus rule -left is invertible, that is, if Γ, is derivable, so are Γ, and Γ, The classical negation rules of natural deduction and sequent calculus allow single-conclusion systems to mimic multiple-conclusion (at least to this extent) by parking the negations of the parametric succedent formulae as antecedent fomulae (ie, assumptions) Consider the following multiple-conclusion sequent calculus proof that c 2 ( ) ( ): , 2 1, 1 2 1, ( ) 2 2 2, ( ) The rule c 2 -left used here reads: c 2 ( ) ( ) , 4 3, 3 4 3, ( ) Γ 1, Γ, 2 Γ 3, Γ, 4 Γ, c 2 ( ) 4 4 4, ( ) c 2 -left c 2 -left slr@st-andacuk GE- 29 / / 36 Classical The rules for negation in Gentzen s LK, his classical sequent calculus, are: Γ,, Γ -left, Γ Γ, -right With these rules, within the multiple-conclusion system LK we can establish the following derived rule:, Γ Γ, C with proof:, Γ, Γ, Cut Γ, If we now move to a single-conclusion sequent calculus, we can use C (with empty) to derive two further negation rules, CT (ie, contraposition) and CM (ie, consequentia mirabilis): Γ, Γ, CT with proof: Γ, Γ CM with proof: Γ, Γ,, -left Γ, C slr@st-andacuk GE- Γ, Γ,, -left Γ, W Γ 31 / 36 The Single- erivation Then we can establish c 2 ( ) ( ) in single-conclusion sequent calculus using CT and CM: 1 1 2, 1 1 1, 1 2 CT ( ) ( ) 1 CT 2 2 2, ( ) 3 3 4, 3 3 3, 3 4 CT ( ) ( ) 3 CT ( ), c 2 ( ) ( ) c 2 ( ) ( 1 2 ) ( 3 4 ) CM 4 4 4, ( ) c 2 -left This proof exhibits essentially the same proof architecture as the earlier proof of the same result in classical natural deduction slr@st-andacuk GE- 32 / 36

9 Finally, we might take a brief look at my favourite connective, Take as a one-place connective, whose single introduction-rule has one hypothetical premise: [ ] -I GE-harmony yields as E-rule in the usual way: [ ] satisfies the inversion principle: which flattens to [ ] -I converts to slr@st-andacuk GE- is an inferential Curry paradox introduces inconsistency, in fact, triviality, since we can prove, for any : 1 1 -I [1] 2 2 -I [2] The proof fails to normalize, since clearly, if we try to remove the maximum formula in the major premise of the final use of, we obtain just the same proof again How can we prevent this? Should it be prevented? One proposal is ummett s complexity constraint: The minimal demand we should make on an introduction rule intended to be self-justifying is that its form be such as to guarantee that, in any application of it, the conclusion will be of higher complexity than any of the premisses and than any discharged hypothesis We may call this the complexity condition Although this rules out, and classical reductio, it also rules out apparently innocuous rules such as Gentzen s above, and even ummett s own -I rule for minimal negation: [] -I The moral I draw is that GE-harmony is not designed to rule out anything, but to ensure that the E-rules add no more (and no less) to whatever meaning is given by the assertion-conditions encapsulated in the I-rule(s) slr@st-andacuk GE- 33 / / 36 Prawitz wrote: An argument that is built up of other arguments or argument schemata is thus valid by the very meaning of the logical constants it is valid by definition so to speak However, analytic truth, and analytical validity, does not guarantee truth or validity What is good about the notion of proof-theoretic validity is that it recognises that what rules one adopts determines the meaning of the logical terms involved and commits one to accepting certain inferences as valid What is bad is to infer from this that those inferences really are valid Michael ummett introduced the notion of harmony in response to Arthur Prior s tonkish attack on the idea of proof-theoretic justification of logical laws (or analytic validity) ag Prawitz had already articulated an idea of Gerhard Gentzen s into a procedure whereby elimination-rules are in some sense functions of the corresponding introduction-rules oy yckhoff, in a joint paper with Nissim Francez, A note on harmony, coined the term general-elimination harmony for the relationship created by this procedure should ensure that the E-rule(s) add no more and no less to whatever meaning is given by the assertion-conditions encapsulated in the I-rule(s) However, GE-harmony cannot guarantee normalization, or prevent inconsistency or triviality What GE-harmony does do is ensure that meaning is given solely, and hence transparently, by the assertion-conditions encapsulated in the I-rule(s) slr@st-andacuk GE- M ummett, Logical Basis of Metaphysics (uckworth, 1991) and H Lewis (Leeds 1988), yckhoff and N Francez, A Note on, Journal of Philosophical Logic, Online First 8 July 2011 G Gentzen, Investigations concerning logical deduction In M Szabo ed (1969) The Collected Papers of Gerhard Gentzen North- Holland, Amsterdam (1969), G Gentzen, Untersuchungen über das logische schliessen, Manuscript 974:271 in the Bernays Archive, Eidgenössische Technische Hochschule Zürich Prawitz, Towards the foundation of a general proof theory, in Logic, Methodology and Philosophy of Science I: Proceedings of the 1971 International Congress, ed P Suppes, Henkin, L, A Joja and G Moisil (North-Holland, 1973), S ead, General-elimination harmony and the meaning of the logical constants, Journal of Philosophical Logic 39 (2010), P Schroeder-Heister, Natural extension of natural deduction, Journal of Symbolic Logic 49 (1984), P Schroeder-Heister, Generalized elimination inference, higher-level rules, and the implications-as-rules interpretation of the sequent calculus, forthcoming J von Plato, Natural deduction with general elimination rules, Archive for Mathematical Logic 40 (2001), slr@st-andacuk GE- 35 / / 36