For Wittgenstein, however, it is a defining feature of a mathematical proposition that it is either decided or decidable by a known decision procedure in a mathematical calculus.
Thus, the later Wittgenstein agrees with the intermediate Wittgenstein that the only sense in which an undecided mathematical proposition (RFM VII, §40, 1944) can be decidable is in the sense that we know how to decide it by means of an applicable decision procedure.
Informally, being decidable means that there is a mechanical procedure which enables one to decide whether an arbitrary given sentence (of the language of the theory) is a theorem or not.
First, they are all effectively decidable.
And Tarski has pointed out that the set V¹ of valid sentences in the first-order language of equality (with no non-logical symbols at all) is decidable.
Moreover, if Wittgenstein held this position, he would claim, contra (RFM V, §9), that a question or proposition does not become decidable since it simply (always) is decidable.
Furthermore, as we have just seen, Wittgenstein rejects PIC as a non-proposition on the grounds that it is not algorithmically decidable, while admitting finitistic versions of PIC because they are algorithmically decidable.
He thus rejects restrictions of the notion of function to cases presented by familiar formulas, representable in intuition (via their graphs), or decidable by a formal procedure.
Formal specifications of these models can provide the basis for exploring the formal features of the system (such as whether it is decidable, or whether the consequence relation for the system is compact).
Notable extensions include QPTL, the propositionally quantified extension of LTL, which is decidable, though with non-elementary complexity, as well as CTL* under some special semantic restrictions, see French (2001).
We note that all logics mentioned in this section have the finite model property and are decidable.
