An Investigation into Intuitionistic Logic with Identity

Abstract

<jats:p>We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.</jats:p>

Meaning is Use: the Case of Propositional Identity

“…The first axiom underlines that identity is reflexive; the second axiom shows identity as a stronger connective than implication; and the third axiom expresses the fact that ≡ is a congruence relation. The axioms are valid under the proposed interpretation of the identity connective, as it has been shown in [3]. The only rule of inference is modus ponens.…”

“…This derivation can easily be transformed in such a way that an occurrence of A ≡ A disappears from the derivation: 3 We choose this form of the introduction rule for ≡ to exhibit the similarity between BHK-interpretations of implication and identity: both the former and the latter denote a function, but in case of identity it is a very specific one. Other possibility, since the assumption is immediately discharged, is to consider a no-premiss rule:…”

Tableau-based Decision Procedure for Non-Fregean Logic of Sentential Identity

“…First, we will prove soundness of the tableau system T SCI . 4 A technical appendix to the paper with all omitted proofs can be found in [12] Let A, B be finite sets such that A ⊆ LF and B ⊆ Id. A set A ∪ B is said to be satisfied in an SCI-model M = U, D,¬,→,≡ by a valuation V in M and a function f : L −→ U if and only if the following hold: (1) V (ϕ) = f (w), for all w ∈ L and ϕ ∈ FOR such that w : ϕ ∈ A, (2) f (w) ∈ D iff w ∈ L + , for all labels w that occur in A ∪ B, (3) f (w) = f (v), for all w, v ∈ L such that w = v ∈ B, (4) f (w) = f (v), for all w, v ∈ L such that w = v ∈ B.…”

“…In particular, a logic obtained from SCI by adding propositional quantifiers is undecidable and can express many mathematical theories, e.g., Peano arithmetic, the theory of groups, rings, and fields [8]. Furthermore, non-classical and deviant modifications of SCI have been developed and extensively studied in the literature, in particular intuitionistic logics [17,14,4], modal and epistemic logics [15,16], logics with non-classical identity [13], paraconsistent [6,9]. The non-Fregean approach could turn out to be more adequate than the classical one in cognitive science or natural language processing.…”