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New woodcuts by Goedele Peeters

reduction woodcut and stencil on Zerkall paper
27.5 inch x 34.5 inch | 70 cm x 95 cm

water reflections (morning, afternoon and evening) of a courtyard inside a residence in Tripoli, Libya, the work created during recent artist residence sponsored by the Belgian Embassy in Tripoli.

I am excited that Antwerp-based artist Goedele Peeters will be visiting Stamps School of Art and Design as a Roman J. Witt Visiting Artist this coming fall 2014 semester. Goedele was a Roman J. Witt Artist-in-Residence  at Stamps School in 2009.  -Endi

Analyticity and Kant and Gödel I

I’m not a Kant scholar but I take it that there have been a few headaches over how Kant meant the term “analytic.” It has been claimed that two notions of analyticity are afoot, which Kant mistakenly takes to be equivalent. The first we see in quotes from the first Critique such as:

In the analytic judgment we keep to the given concept, and seek to extract something from it. If it is to be affirmative, I ascribe to it only what is already thought in it.

On this account, a statement such as “all rectangles have four sides” is analytic because to confirm that the subject has the property one need only look to the concept of rectangularity itself. One will see that the property of four-sidedness is “contained” within it (“is already thought in it”).

Kant, however, being motivated to counter some of Hume’s theses, also employs the principle of contradiction- that no proposition and its negation may be simultaneously true- as a ready to hand technique to decide the truth of an analytic proposition:

[I]f the judgment is analytic… its truth can always be adequately known in accordance with the principle of contradiction.

Some of the debate, I think, centers around whether the two- analytic in virtue of “containment” and analytic in virtue of noncontradiction- are indeed one and the same. I think it’s pretty clear that Kant intended them as equivalents; while the second quote indicates that the latter is necessary for the former, Kant explicitly suggests that it is “sufficient” as well. Kant’s arithmetical examples betray a style of doing mathematics quite a bit different than the modern, axiomatic approach, so I don’t think it is crazy to conflate the two. But I don’t have a dog in the fight with respect to the historical question. (Though I will add that irrespective of their equivalence or inequivalence, I’ve always thought that Gödel’s Theorems demonstrated quite nicely the existence of the synthetic a priori.)

I’d soon like to outline some strategies for the development of a formal approach to the notion of analyticity-as-containment and how this can be reconciled with a classical framework. For now, I’ll leave with the below blatant appeal to authority. 

In Parry’s 1933 dissertation (as relayed by Anderson and Belnap’s Entailment), Gödel is quoted as saying:

”\(p\) implies \(q\) analytically” can be perhaps interpreted as “\(q\) is derivable from \(p\) and the logical axioms and does not include any other concepts than \(p\)” and, after this definition is more precise, it would be [wise?] to seek a completeness theorem [Vollständigkeitsbeweis] for the axioms of Parry, in the sense that all sentences valid for the above interpretation of \(\rightarrow\) are derivable. 

So, yes, Gödel is just as fallible as Kant. Nevertheless, the approach of retaining one logic (classical logic) to handle synthetic truths while employing a fragment thereof (in this case, Parry’s \(\mathsf{PAI}\)) to track the analytic has a pretty distinguished provenance. 

Gödel, Escher, Bach: An Eternal Golden Braid (pronounced [ˈɡøːdəl ˈɛʃər ˈbax]) is a 1979 book by Douglas Hofstadter, described by his publishing company as “a metaphorical fugue on minds and machines in the spirit of Lewis Carroll”.[1]

On its surface, GEB examines logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach, discussing common themes in their work and lives. At a deeper level, the book is an exposition of concepts fundamental to mathematics, symmetry, and intelligence.

Through illustration and analysis, the book discusses how self-reference and formal rules allow systems to acquire meaning despite being made of “meaningless” elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of “meaning” itself.

In response to confusion over the book’s theme, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms. In the book, he presents an analogy about how the individual neurons of the brain coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony of ants.[2][3]

Logical "Preambles"

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.

I think that this picture leads to some concrete conclusions that I’ll leave aside for now. But I think it is important to point out that this isn’t an observation pertaining only to the deviant logician. It is well known that the Goedel embedding into the modal logic \(\mathsf{Triv}\) (the modal logic characterized by one-point, reflexive frames) determines precisely the classical propositional calculus itself. But this picture absolutely captures the preamble “this proposition is true in the actual world” when that single point is construed as the actual world. Even if it doesn’t change the logic, the embedding still provides a clear semantical picture underlying why classical logic behaves the way it does. 

Wansing's Constructive Logics

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.

Why is this the case? Why is it the least obvious that works? Why should Lukasiewicz and Nelson negation converge at this point? It’s a very odd situation, I think.

 

 

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