Constructive semantics and the validity of Peirce's law

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1 Introduction In Dummett-Prawitz-style proof-theoretic semantics the meaning of a proposition is given in terms of what conditions must be fulfilled in order to assert the proposition. If the condition to assert a proposition is the possession of a proof, then a constructive semantics requires a description of what are proofs of basic propositions and of what are proofs of logically complex propositions. The description is usually given as an inductive definition. Among others, Dummett [1] and Prawitz [5, 6, 7] are references for this approach (for an overview see [11, 13]). Recently, Sandqvist [10] proposed a semantics for logically complex propositions, which is closely related to constructive semantics, and which takes the form of an inductive definition too. It is of special interest as it is intended to yield a justification of classical logic without making use of the principle of bivalence. We focus on the logical constant of implication. The implicational fragment {⊃} of natural deduction for minimal logic NM (and also for intuitionistic logic NI) is given by the following introduction and elimination rules: [ϕ] . � (⊃I) ϕ ⊃ � ϕ ϕ ⊃ � (⊃E) � These rules also hold in the fragment {⊃} of natural deduction for classical logic NK, although they do not suffice to obtain all classical theorems in this fragment. Classical laws like Peirce’s law ((ϕ ⊃ �) ⊃ ϕ) ⊃ ϕ are provable when one of the following two versions of Peirce’s rule is added: [ϕ ⊃ �] . ϕ ϕ (ϕ ⊃ �) ⊃ ϕ ϕ Each version can be derived from the other by using (⊃I) and (⊃E). Let NP be NM plus (one version of) Peirce’s rule. The addition of ex falso quodlibet ⊥ ϕ to NM gives NI, and the addition of it to NP gives NK. Peirce’s rule is not justifiable constructively, since its addition to NI allows to prove tertium non datur, which is normally rejected as being non-constructive 1 . 2 Sandqvist’s semantics for classical logic Sandqvist [10] has proposed a semantic justification of classical logic, which does not make use of the principle of bivalence, for the rule of double negation elimination for the fragment {⊃, ⊥, ∀}. Once double negation elimination is established, the 1 See Heyting [2], p. 103f. 2
other logical constants can be defined, and the justification becomes a justification for classical logic without using the principle of bivalence. Sandqvist considers bases B which are sets of basic sequents, that is, relations between finite sets of basic sentences and basic sentences, where basic sentences are formulas containing neither logical constants nor free variables. In the following, basic sentences are also called atoms. We are not considering quantifiers, as they are not relevant to the arguments put forward here. This means that we can consider atoms to be propositional variables. The language of all bases is the same, that is, the set of all atoms. A basis B can thus be viewed as a set of atomic inferences, that is, of production rules p1 . . . pn pn+1 where the pi are atoms. For a given basis B Sandqvist then defines inductively a relation �B, called ‘valid inferability’, between finite sets 2 of sentences in general and sentences in general (ibid., p. 213, our numbering; we omit the clause for ∀): Definition 1. Where p is a basic sentence: �B p ⇐⇒ Every set of basic sentences closed under B contains p Where Γ is non-empty: Γ �B ϕ ⇐⇒ �C ϕ for every C ⊇ B such that �C � for every � ∈ Γ (1) (2) �B ϕ ⊃ � ⇐⇒ ϕ �B � (3) �B ⊥ ⇐⇒ �B p for every basic sentence p (4) An inference ϕ1 . . . ϕn ϕ holds for B if ϕ1, . . . , ϕn �B ϕ. The inference is considered logically valid if ϕ1, . . . , ϕn �B ϕ for every B. If Γ �B ϕ holds for every B, then we also speak of logical consequence, denoted by Γ � ϕ. A justification of the inference of double negation elimination consists thus in showing that this inference is a logical consequence. This is stated in his main lemma as follows (ibid., p. 216), where the choice of the basis B is arbitrary: Lemma 4 (Double negation). (ϕ ⊃ ⊥) ⊃ ⊥ �B ϕ. 3 The proof of Lemma 4 begins as follows (ibid., p. 216): Proof. It suffices to prove the result for basic ϕ; given Lemma 3, non-basic applications of double-negation elimination can be reduced to basic ones in the manner, mutatis mutandis, of Prawitz [4, pp. 39–40]. Lemma 3 (ibid., p. 216) states that the usual introduction and elimination rules for the fragment {⊃, ∀} hold for the inferential relation �B (as given in his Definition 1). Lemmas 3 and 4 establish that Γ �B ϕ whenever ϕ is a classical consequence of Γ (ibid., p. 217), from which Sandqvist’s Theorem 2 (expressing the justification of classical logic) follows directly (ibid., p. 214): 2 Sandqvist considers only finite sets of assumptions. 3 More precisely, he uses a consequence relation referring to substitutions for free variables, which is not significant in the context of propositional logic considered here. 3
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