The Fraenkel-Carnap question has today come to my notice, and I suspect this may be the source of the matters to which Speranza referred in an earlier post to Carnap Corner on categoricity.
The Fraenkel-Carnap question is (according to Weaver and George):
whether every finitely axiomatizable semantically complete second-order theory is categorical
I offer an "explication" of this as follows.
A theory is a set of sentences in some logical system.
A second order theory is a theory in a second order logic.
A theory is finitely axiomatisable if all the sentences in the theory are formally derivable from some finite subset if the sentences.
A theory is semantically complete if it "determines" the truth value of every sentence in the language, however this means semantically, not syntactically.
It means that every sentence in the language has the same truth value in every model of the theory, so semantically it is either true or false in the context of the theory, which does not mean that it is provable or disprovable, since second order logic is not complete.
Categorical has two meanings, a syntactic and a semantic meaning.
We know that the semantic meaning is the relevant one here since the answer to the question is otherwise too easy.
A theory is syntactically categorical if every sentence or its negation is in the theory.
A theory is semantically categorical if it has only one model up to isomorphism, this is sometimes qualified by cardinality, since a first order theory with an infinite model will have models of every infinite cardinality and no two models of different cardinality will be isomorphic. So, often, categorical should be read: all models of the same cardinality are isomorphic.
One might naively suppose that a semantically complete theory will be semantically categorical, but the theory of "true arithmetic" provides a counter-example. "true arithmetic" is the set of true sentences of first order arithmetic.
It is semantically complete, because every sentence or its negation is true, but it is not semantically categorical, it has non-standard countable models.
The fact that first order logic is complete tells us that any semantically complete axiomatic theory will be syntactically categorical, but the above counterexample shows that this does not entail semantic categoricity.
In the case of second order logic, we forgo completeness of the deductive system for the sake of greater expressiveness in the semantics (leaving the syntax behind).
The consequence of the more expressive semantics is that second order arithmetic becomes semantically categorical, but the semantic expressiveness is not matched by any greater syntactic strength, so we don't have syntactic categoricity. Consequently we now have a categorical "true second order arithmetic", and its no longer obvious where to look for a counter-example to the general thesis considered in the Fraenkel-Carnap question.
This seems to me to be a somewhat recondite problem in mathematical logic.
I cannot myself see that it has any philosophical significance.
Carnap apparently offered a proof of the conjecture which was flawed.
Some partial results have been proven, the unqualified conjecture remains unsolved.
RBJ
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Jones: "I cannot myself see that it has any philosophical significance."
ReplyDeleteThanks for an interesting post, Jones.
Jones: "I cannot myself see that it has any philosophical significance". I will play Gricean devil's advocate here - or is it Devil's Gricean advocate -- whatever. NEVER say that! Just implicate it! Only kidding. But I was reading Grice, WoW:i, and he considers Searle's example, "The man in the next table is NOT lighting his cigarrette with a 20-dollar bill" as "not having, really, philosophical significance" (or words to that effect). This is GRICE *NOT* finding 'philosophical significance' in something! So beware! Of course Searle's example does have it. Grice is actually more cautious. His words run along the lines that Searle's example has failed to 'attract the attention of philosophers' (first joke) and then Grice goes on to add that Searle is UNjustified from deriving a point that applies to his "unphilosophical example" to a philosophical one. But note that at the end of WoW:i -- I am particular to quotes because I know Jones has easy access to that volume -- Grice ends up by commending Searle into bringing the GENERAL picture into the philosophical picture, as it were. I.e. Grice, following Searle, there, wants to trace the issue to GENERAL 'points' that apply to both philosophy and other fields of inquiry.
So, mutatis mutandis with 'categoricity'. Since I see you have written another post on this after the one I'm commenting, I will read that and see if I can further comment.
The idea of 'category' is TOO monumental. Everything that relates to it should have a philosophical significance -- or other!