#goedel

@godelitonthemountain

Gödelovy dny 2016

v Galerii RUV na Pedagogické fakultě MU v Brně, Poříčí 9, úterý 26. dubna 2016 a pátek 29. dubna 2016. Všichni zájemci jsou srdečně zváni.

Nelson's Logic of Constructible Falsity demands that falsity be treated symmetrically with truth, that is, to say that a sentence is false requires more than mere reductiones, it requires a construction. E.g., to disprove a conditional requires a (constructive) proof of the antecedent and a (constructive) disproof of the consequent.

But this isn't the only way to go. While the falsity conditions for, say, disjunction can be read easily from the connective, implication (as well as coimplication) do not so readily suggest their falsity conditions. Wansing has introduced a sequence of sixteen "completions" of Nelson's logic; there is little reason--other than a purely philosophical motivation--to accept one over the other.

I suggest that the data concerning many-valued extensions of these logics do weigh in on this question.

The heredity property of intuitionistic logic--that truth persists through all accessible points--yields a tidy property on finite, linearly ordered frames. In an \(n\)-pointed frames, there are only \(n+1\) distributions for any formula. The familiar Goedel-Dummett logics \(\mathsf{G}_{n}\) abuse this to yield finitely-valued extensions of intuitionistic logic. A truth value in the interval \((0,1)\) can be read as the proportion of points at which the sentence is true with respect to the number of points in the frame; this can be read as a "degree of truth."

Nelson's logic--as well as Wansing's logics--likewise have this property, save for the fact that not only truth persists but falsity as well. Thus, rather than a single ratio, we must track a pair of points in \((0,1)\). Call such a pair \((x_{0},x_{1})\).

A very attractive, semantical constraint is that the degree of truth and the degree of falsity ought to add to \(1\). On, e.g., an epistemic or doxastic reading, belief in the truth of a proposition with probability \(1\) ought to imply that the belief in its falsity is \(0\). If one is 75% sure that a sentence is true, one is 25% sure that it is false, etc.

A property of the Goedel-Dummett logics is that their operations preserve this property, as the degree of falsity is implicit in the degree of truth. When one segues to the finitely-valued extensions of Wansing's logics, however, only one of the sixteen can be maintained under this constraint. This is to say that for the set of pairs of positive rationals \((x_{0},x_{1})\) such that \(x_{0}+x_{1}=1\) is not closed under the operations of fifteen of the sixteen logics, including that suggested by Nelson.

The only logic under whose operations this set of truth values is closed is Wansing's \(\mathsf{I}_{4}\mathsf{C}_{4}\) and this is very unusual. Whereas the other fifteen have rather straightforward conditions for the falsity of a conditional, those discovered in the logic \(\mathsf{I}_{4}\mathsf{C}_{4}\) seem slightly artificial. For example, the logics \(\mathsf{I}_{1}\mathsf{C}_{i}\) treat a negated conditional as a conjunction between the antecedent and the negation of the consequent (cf. Nelson's reading). \(\mathsf{I}_{2}\mathsf{C}_{i}\) treat it connexively, so that \(\neg(A\rightarrow B)\) is defined simply as \(A\rightarrow\neg B\). And \(\mathsf{I}_{3}\mathsf{C}_{i}\) treats the negation of \(A\rightarrow B\) simply as the corresponding coimplication \(A\leftarrow B\). \(\mathsf{I}_{4}\mathsf{C}_{i}\), on the other hand, defined the negation as contraposed coimplication, which increases the complexity of the formula. (\(\mathsf{I}_{j}\mathsf{C}_{4}\) on the other hand treats negated coimplication as contraposed implication.)

But if we calculate the values of the functions associated with these logics on finite, linearly ordered frames, all of the others go berserk under the above constrain; only \(\mathsf{I}_{4}\mathsf{C}_{4}\) is left standing. From a Goedel-Dummet perspective, the negation of this logic can be seen to be the extension of Goedel-Dummet logics with Lukasiewicz negation--a strange convergence.

Just unpacked another 66 books. Enough for today, time to head out to the beach!

Recall the famous embeddings from Goedel, McKinsey, and Tarski (generating intuitionistic logic) and Thomason (generating Nelson's Logic of Constructible Falsity) into \(\mathsf{S4}\). As an example, the former is:

-\(A^{G}=\Box A\) for atoms \(A\)

-\( (\neg A)^{G}=\Box\neg(A^{G})\)

-\( (A\vee B)^{G}= A^{G}\vee B^{G}\)

-\( (A\wedge B)^{G}= A^{G}\wedge B^{G}\)

-\( (A\rightarrow B)^{G}= \Box(A^{G}\rightarrow B^{G})\)

The Thomason embedding is similar. We may also recall the Fregean Principle of Compositionality that complex propositions are constructed from simpler propositions.

I think that these embeddings capture a very correct compositional principle and reveal a picture that ought to be considered by even a classical logician. We can ask what these embeddings in general mean. I think the answer to this is that they make clear that propositions necessarily bear "preambles." 

What is a "preamble"? Well, the underlying thought is that a proposition is empty without taking some stand on how it is to be understood. The classical logician prefaces her utterances with the understanding that the propositions are to be interpreted as true in the actual world. The constructivist prefaces his utterances with a different claim, namely, that the propositions are provable. 

Thinking about the embeddings purely syntactically, prior to considering their ranges as modal logics, a clear picture of compositionality upon this background emerges. The symbol "\(\Box \)" simply represents a preamble. If there indeed are such implicit preambles, then it is clear, e.g., that the meaning of a formula \( A\rightarrow B\) is not a function of \(A\) and \(B\), but rather \(\Box A\) and \(\Box B\). And this is precisely what is captured by such embeddings.