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**Example text**

34 PURELY IMPLICATIONAL LOGIC Proof. Let the semantic tableau for the sequent Kj L, where K is (UI' U2, ... , Um) and L is (VI, V2 , ••• , Vn), be closed. Then, by Theorem 1, we can, by adding suitable applications PI, P2, ... , Pk, of Peirce's Law, enlarge K into a set K* such that a certain deductive tableau for K*jZ is closed. By Theorem 3, the representing formula: PI---+(P2 ---+( ... •• ---+(Um ---+ Z) .. » is an intuitionistic thesis and hence a thesis. Since PI, Pz, ... , Pk are axioms of (classical) logic, it follows by modus ponens that Y is a thesis.

True False z K' Y-+U X-+V (ijR) (ija) I (i) (iij) (iv) ~ II .... (ij) U Y (iv) (iij) V (ij) (i) X Now it may happen that the closure of the sequent (iij) results from the fact that the succedent of this sequent contains the formula Z. Then the corresponding deductive tableau for the given sequent (K', Y -+ U, X -+ V)/Z will not be closed. Premisses K' Z Y-+U X-+V (ijaI) (ijaI) Conclusions (i) (iij) (ij) (i) U Y (ij) Z (iv) (iij) (iv) V X Y For in this tableau the closure of the subordinate sequent (iij) is prevented by the fact that the formula Z is supplanted by the formulas Y and X.

As formulas/leA) and/2 (A) we may take, respectively, A and A; accordingly, we may take as a formula/(A, B): [A --+ B] --+ [A --+ B] . Therefore, the biconditional is clearly characterized by the following axiom-schemata: (XI) ( U ~ V) --+ {[ U --+ V] --+ [U --+ V]} , {[U --+ V] --+[0 --+ V]} --+(U~ V). (XII) We have the reduction schemata: True K' (ij'a) . False True L K U~V (i) U V I I (ij) (i) (ij) U V (ij'b) (i) I False L' U~V U 47 CHAPTER III THEORY OF QUANTIFICATION, EQUALITY, AND FUNCTIONALITY 8.