A herbrandized functional interpretation of classical first-order logic

Abstract

We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ * of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand's theorem on the extraction of terms from classically provable existential statements.

Interpreting weak Kőnig's lemma in theories of nonstandard arithmetic

“…We rely on the complete axiomatization of classical logic described in [13, § §2.6 & 8.3]. We omit the study of the law of excluded middle and of the propositional rules since the checking is essentially the one done in [6] (the property of monotonicity is used in this part). The situation with the axiom of substitution and the rule of ∀-introduction is different because the clause for the interpretation of the universal quantifier is unusual in nonstandard arithmetic.…”

“…The other reason is that the properties of finiteness needed to pull through the herbrandized interpretation are very minimal. The herbrandized functional interpretation can actually be given a sense for pure classical logic, as shown in [6]. Of course, in the context of arithmetic, the notions of finiteness and (natural) number must be closely related on pain of straining one (or both) of them.…”

A parametrised functional interpretation of Heyting arithmetic

“…In fact, these interpretations are closely related with the interpretations given in [3] for WE-HA ω st . They are also closely connected with the interpretation for "pure logic" considered by Gilda Ferreira and Fernando Ferreira in the paper [9]. Moreover, Fernando Ferreira has a recent paper [8] where he considers essentially the Herbrand Diller-Nahm interpretation for WE-HA ω as well as an extension to second-order arithmetic.…”

“…However, the unifying functional interpretation programme has so far been unable to capture the two more recent families of functional interpretations, namely the bounded functional interpretations [6,10,11,12], and the Herbrandized functional interpretations [3,9].…”

On extracting variable Herbrand disjunctions

“…• ϕ ∨ ϕ ⊢ ϕ (the 'contraction rule'); 1 We mention that other higher-order treatments of Herbrand's theorem have recently been given by Ferreira and Ferreira [4] -using a 'star-combinatory' calculus which has later seen arithmetical use in [2,3] -and by Afshari, Hetzl and Leigh [1] -using higher-order recursion schemes.…”