Abstract

Wittgenstein's conception of the general form of a truth function given in thesis 6 can be presented as a sort of a trade-off: the author of the Tractatus is unable to reconcile the simplicity of his original idea of a series of forms with the simplicity of his generalisation of Sheffer's stroke; therefore, he is forced to sacrifice one of them. As we argue in this paper, the choice he makes – to weaken the logical constraints put on the concept of a series of forms, thus effectively metaphorising that concept for the sake of upholding the N-operation's role of generating the series – is unfortunate. An actual expansion of a series of truth functions as defined in 6 would require either making decisions at each step (Anscombe) or outwardly rejecting the concept of a series (Sundholm). However, neither is faithful to Wittgenstein's own fundamental intuitions regarding the nature of logic. For this reason, a different trade-off that prioritises upholding the basic features of a series of forms over the simplicity of the operation that generates that series seems to be much more reasonable. We offer such a trade-off by developing the schema already present in the Tractatus (5.101). The key element of our alternative solution is the construction of an operation that can perform the task of producing all consecutive truth functions of a given collection of atomic propositions as an invariant difference between any base and its successor throughout the series. We define it as a composition of three functions (in an algebraic sense): the first turns a symbol of a given truth function into a binary number, the second increments that number, the third turns the result back into a symbol of another truth function. We also show how the operation can be defined by means of a different notation without invoking binary arithmetic.