On The Substitutivity of Variables

Abstract

This paper examines the constraints for the decidability of a logical system byanalyzing the theories of undecidability in formal logic and recursive func-tions. The proof of the diagonal lemma requires the implicit premise that allformulas are closed under avariable composition, i.e., the composition of vari-ables in the formulas signatures. A variable composition is representable infirst-order logic if∀ψ∀k(⊢ψ(k)↔∃y(ψ∧y≡k))can be derived, e.g., froma semantic definition of the quantifiers. A further condition is the assump-tion of an interpretation providing the existence of what will be defined as anindeterminate variable signaturewith aconstant interpretation.A recursion theorem for formulas in analogy to the recursion theorem ofKleene (1943, pp.52-53) will be proved which covers the diagonal lemma as aspecial case. The chosen notation and reasoning intend to make the necessaryconditions for its provability explicit. It will be proved that the representabilityof variable composition and the existence of an indeterminate variable signa-ture to represent a constant interpretation are consequences of the recursiontheorem, i.e., equivalent to the existence of fixed points∀ψ∃φ(⊢φ↔ψ(⌜φ⌝)).These results give a reason why the negation of the diagonal lemma canbe proved for a predicative logic that contradicts these premises but holdsthe explicit condition of the diagonal lemma, i.e., a language of this logic iscapable of representing all computable functions, as has been shown as anon-expectable result in Solte 2020. The paper concludes with an outline ofa decidable structure of computable functions. I.e., it is possible to provide aninterpretation of a predicative logic without undecidability in this structure