Beth’s Nonclassical Valuations

We describe E. W. Beth’s use of nonclassical valuations (in his own terminology pseudo-valuations) in propositional logics. Three periods are distinguished. In the first period (1954) he develops the idea of pseudo-valuation intending to apply it to obtain a subformula theorem for arbitrary propositional logics. When this fails, he obtains in the second period (1958-1961) some simple but elegant applications of the idea, mainly with regard to proofs of independance of axioms systems. The thirs period (1961-1964) is the application of the idea towards the introduction of a semantics (his second one) for intuitionistic logics. We will show that it is highly likely that Beth discovered this version of “possible worlds semantics” for intuitionistic and some modal logics essentially independently from Kripke. The history of the concept of semantic tableaux is strongly bound to the birth of the concept of semantic tableaux, but we will touch the latter subject only in so far as is necessary for our considerations.