A Translation Characterizing the Constructive Content of Classical Theories

Abstract.

A simple syntactical translation of theories and of existential formulas – $\mathcal C^\forall$ and $\mathcal C^\exists$, respectively – is described for which the following holds: For any classical theory T and all formulas A(x),

$$T\vdash A(t) \mbox{for some term $t$} \ \Leftrightarrow \ \mathcal C^\forall(T) \vdash \mathcal C^\exists(\exists x A(x)) .$$

In other words, $\mathcal C^\forall (T)$ proves exactly those formulas $\mathcal C^\exists (\exists x A(x))$ for which T can prove ∃ x A(x)constructively and thus circumscribes the constructive fragment of T. The proof of the theorem is based on properties of the resolution calculus; which allows to extract a primitive recursive bound on the size of the witness term t, with respect to the size of a proof of $\mathcal C^\forall (T) \vdash \mathcal C^\exists (\exists x A(x))$. In fact, a generalization of the above statement, that takes into account a designation of certain function symbols as ‘constructor symbols’ is proved. Different types of examples are provided: Some formalize well known non-constructive arguments from mathematics, others illustrate the use of the theorem for characterizing classes of classical theories that are constructive with respect to certain types of existential formulas.