Remarks on the Completeness of Logical Systems Relative to the Validity-Concepts of P. Lorenzen and K. Lorenz

Abstract

The so-called semantics of elementary logic (predicate logic of first order) has some peculiarities which may be considered as disadvantages, at least if looked at from a certain point of view. First of all, it makes use of a very strong set-theoretical apparatus which is highly non-constructive. Strict constructivism may not be obtainable for a semantic foundation of logic; but even if one does not subscribe to constructivism one should expect that a weaker apparatus would be sufficient to define logical validity or logical implication on the elementary level. Secondly, this set-theoretical approach is restricted to classical logic and can therefore not be used, e.g. to define a concept of validity for intuitionistic logic or other logical systems differing from the classical one. This fact will not be considered as a drawback by those for whom classical logic is the only ‘real’ logic. On the other hand ‘classicists’ as well as ‘intuitionists’ should welcome an account of logic with the help of which different logical systems can be compared on the basis of semantical concepts alone — an account in which the deviations of logical systems from each other would be mirrored by different concepts of validity. Thirdly there are two characteristics of this semantics based upon the Bolzano-Tarski approach which in various contexts have been the main points of attack in the arguments of intuitionists and constructivists against this approach (which are mistakenly thought to be arguments against classical logic): (a) the procedure of introducing logical connectives as truth-functions tacitly presupposes that every meaningful sentence is either true or false (true-false-alternative). In view of such unproved and unrefuted sentences as ‘there is at least one odd perfect number’ one can reasonably doubt whether this assumption is correct. Namely if one decides to identify in mathematical contexts ‘true with ‘provable’ and ‘false’ with ‘refutable’ it is certainly not justified. And if one does not accept this identification then if one doesn’t want to destroy the meaning of ‘true’ altogether it seems hardly possible to do without some kind of ‘ontological hypothesis’ about the pre-existence of numbers or other kinds of mathematical entities; but the best that can be said about