Implications as rules

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1 ProDi Tübingen p. 1 Implications as rules In defence of proof-theoretic semantics Peter Schroeder-Heister Wilhelm-Schickard-Institut für Informatik Universität Tübingen

2 ProDi Tübingen p. 2 Two dogmas of standard semantics D. 1 The categorical is conceptually prior to the hypothetical the priority of the categorical over the hypothetical D. 2 Consequence is defined as the transmission of the basic categorical concept from the premisses to the conclusion the transmission view of consequence

3 ProDi Tübingen p. 3 Model-theoretic consequence A = B := ( M)(M = A M = B ) Every model of the premisses is a model of the conclusion Constructive consequence A = B := ( C)(C = A f(c) = B) (BHK, Lorenzen s admissibility interpretation of implication) We use truth-makers (constructions, proofs) and constructive transformations.

4 ProDi Tübingen p. 4 Material implication M = A B := ( M = A M = B ) Constructive material implication S = A B := ( C)(C,S = A f(c),s = B) There is a quantifier already in the material case. The transmission view already governs material implication. Although never formulated that way, the critique of the transmission view has fostered dialogical / game-theoretical semantics.

5 ProDi Tübingen p. 5 Critique of the transmission view Global view of deductive reasoning: Cannot deal with local (partial) meaning and non-wellfounded phenomena Non-definiteness of notion of proof or construction: Lack of proper meaning explanation Iteration of implication in Lorenzen s admissibility concept (improper meta-calculi ) Realizability: Not decidable of whether e is an index with certain properties Impredicativity of implication f : (A A) A λx.fx as argument of f Beyond monotone inductive definitions

6 ProDi Tübingen p. 6 Counterargument: Validity can be established By giving a derivation in a meta-calculus By providing a construction according to the BHK explanation By giving a realizing index The only problem is completeness. But is this a problem? The essential argument is an epistemological one: A speaker cannot grasp the meaning when it is explained according to the transmission view. Therefore a combinatorial way of explaining meaning is needed. Lorenz: The notion of proposition remains unexplained otherwise.

7 ProDi Tübingen p. 7 Dialogical logic and definiteness Non-definiteness of standard constructive semantics has been used as an argument in favour of dialogical logic. Plays as the level of meaning explanations, leading to a constructive notion of proposition. Strategies correspond to the level of proofs. Important is not so much the difference between plays and strategies, but the fact that even at the level of strategies, we have a strict codification of constructions. (Some game-theoretic semanticists dispute this.) Unlike proofs, the concept of strategy is not iterated.

8 ProDi Tübingen p. 8 In defence of proof-theoretic semantics The problem is implication We can do without the transmission view Implications as rules Only the applicative behaviour if implication is relevant Implication is treated separately from the other logical constants but not in the intuitionistic/constructive sense

9 ProDi Tübingen p. 9 Left-iterated implications Observation: Iteration of implication only relevant on the left side: A (B C) is A B C without conjunction, written in sequent-style: A (B C) is A,B C From a sequent-style perspective, this means that implications are only relevant in antecedent position (at least in a purely implicational system) (A B) (C (D E)) becomes (A B),C,D E or (A B),C,D E

10 ProDi Tübingen p. 10 Proposal: Implications as rules Claim: Implication is different from other constants. It is to be viewed as a rule, which operates essentially on the left (assumption) side. Symmetry / harmony does not apply to implication. Rather, implications-as-rules are presupposed for the dealing with harmony principles. Conclusion: The (purported) arguments against proof-theoretic semantics are no longer valid. This is a defence of proof-theoretic semantics, not an argument against game-theoretic semantics. (In fact, our rule-based reading of implications gives rise to a certain game-theoretic treatment. )

11 ProDi Tübingen p. 11 Left-iterated implications as rules Rule ::= Atom (Rule,...,Rule Atom) Intended meaning of ((Γ 1 A 1 ),...,(Γ n A n ) B) : If each A i has been derived from Γ i, respectively, then we may pass over to B. Γ 1 A 1... B Γ n A n In a sequent-style framework:,γ 1 A 1...,Γ n A n B

12 ProDi Tübingen p. 12 Schema for rule application,γ 1 A 1...,Γ n A n, ((Γ 1 A 1 ),...,(Γ n A n ) B) B This generalizes the schema Γ A Γ, (A B) B This is not a definition of implication based on some sort of harmony, but gives implication an elementary meaning.

13 ProDi Tübingen p. 13 Right-iteration as abbreviation Γ A (B C) understood as Γ,A,B C i.e., we are dealing with list structures. Initial sequents: R R This means: R, (R) 1 (R) 2 For example: (Γ A),Γ A This involves the reading of implications as rules. Not simply: Right and left side are identical.

14 ProDi Tübingen p. 14 Justification of cut Γ R,R C Γ, C Example: Γ,A B, (A B) C Γ, C Justification: The left premiss eliminates the application of A B in the right premiss. This yields an elementary Frege calculus.

15 Implications-as-rules from the database perspective: resolution Suppose the implication A B is available in our database. Then the goal B can be reduced to the goal A. More generally: Given a database (or logic program) B A 1. B A n then the goal B can be reduced to any of the goals A i. This reduction is called resolution. Reasoning with respect to a database of implications means reading them as rules. ProDi Tübingen p. 15

16 Generalization: Clausal definitions and common content Given a clausal definition D A :- 1. A :- n then A is intended to express the common content of 1,..., n : For all R: A R iff 1 R,..., n R This gives the usual right- and left rules: Γ i Γ A Γ, 1 C... Γ, n C Γ,A C At this level we have symmetry / harmony! ProDi Tübingen p. 16

17 ProDi Tübingen p. 17 Result Implication has a non-symmetric primordial meaning, other constants are symmetrically defined. We can define A B in terms of the rule A B. This allows us to interpret a nested implicational formula such as (A B) (C D).

18 ProDi Tübingen p. 18 Remarks on cut Better option in the spirit of the rule-interpretation: Use a weaker background logic, based only on rule application A, (A B) B and its generalization,γ 1 A 1...,Γ n A n, ((Γ 1 A 1 ),...,(Γ n A n ) B) B without having cut as primitive.

19 Summary Our case for proof-theoretic semantics: By giving implication an elementary combinatorial meaning (implications-as-rules) we avoid the problems that have led Lorenzen, Lorenz and (some of their) followers to abandon proof-theoretic in favour of dialogical semantics Symmetry / harmony comes into play only after implications-alias-rules are already available The critique of the transmission view of consequence speaks against certain types of proof-theoretic semantics (BHK, Lorenzen, Dummett-Prawitz), but not against proof-theoretic semantics as such This is no case against game-theoretical semantics! Personally, as a proof-theoretic semanticist, I favour Lorenzen I over Lorenzen II. ProDi Tübingen p. 19

20 ProDi Tübingen p. 20 References Implications-as-rules vs. implications-as-links: An alternative implication-left schema for the sequent calculus, JPL 40 (2011), See psh s homepage. Generalized elimination inferences, higher-level rules, and the implications-as-rules interpretation of the sequent calculus, in: E. H. Haeusler, L. C. Pereira and V. de Paiva, eds., Advances in Natural Deduction. See psh s homepage. Thomas Piecha: Implications as rules in dialogues. Next talk at this conference. Thomas Piecha & P. S.-H.: Implications as rules in dialogical semantics. Submitted for the LOGICA conference, Hejnice 2011.

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