Investigations into the sentential calculus with identity.
Stephen L. Bloom, Roman Suszko · Notre Dame Journal of Formal Logic · 1972
The sentential calculus with identity (SCI) is obtained from the classical sentential calculus by adding a new "identity connective" = and axioms which say "p = q" means "p is identical to q".The second author was led to a study of this calculus by a desire to formalize part of the Ontology of Wittgenstein's Tracίatus (see [7], [8]). 1 Aside from this somewhat uncommon beginning, we think that there are independent reasons for studying the SCI.Firstly, it seems to be as general as a sentential logic can get: both classical and modal theories may bε developed in it and (by weakening an axiom) intuitionist theories as well.Furthermore, the study of its interpretations leads to interesting mathematical problems, (e.g.concerning topological Boolean algebras) and sheds light on why the classical sentential calculus is so well-behaved.Some people, upon discovering that the identity connective was not truth-functional, have thought that SCI is an intens tonal logic.We emphatically deny this.The essence of intensionality is that the rule "equals may be replaced by equals" fails.However, this rule does hold in the SCI (see the remarks following 1.3).The paper is divided into four sections.The first is a collection of most of the basic definitions and theorems.The second and third sections discuss the questions of decidability and adequacy.The last section presents a particular theory built in the logic of the SCI.We have omitted most proofs in 1 to keep the size of the paper within reasonable bounds. Definitions and Elementary Results.The formulas Fm of a language £ of the sentential calculus with identity are generated in the usual way from an infinite set VAR of sentential variables by the standard connectives Ί (negation) and -» (material implication) as well as the binary identity connective =.Considered as an abstract algebra, 8 = (Fm, Ί, -% = ) is free in 1.The relationship between the SCI and papers on the identity connective by other authors (M.J. Cresswell, H. Greniewski, A. N. Prior) is discussed in [8],