[FOM] How much of math is logic?

[Cristian Cocos]

Quoting "Panu Raatikainen":
 
> Let me try again. For me, and I presuppose for many others, logic is
> primarily about logical consequence, or entailment, and the validity of
> arguments. The inference from A to B is logically valid  iff "A -> B"
> is logically true (as I understand "logically true": true in all models;
> i.e., it has no counter-example).

These, and many such "definitions," or circumscribing efforts involving
references to deductive powers or whatnot, are, and always remain, circular.
Not since the times of the psychologists have I seen (the germs of) an
acceptable (i.e. non-circular) definition of logic, and I suspect will never
see for as long as the myth of Logic as having no specific subject matter is
being perpetuated. This dispute over what axioms to count as logical and
what axioms not to will go on indefinitely unless (a) either everybody
agrees to put an end to it by fiat, as it were--hence, in Quinean manner, to
simply decree that this or that is LOGIC just because I/we say so--or (b)
one reverts to finding a subject matter worthy of such an honor. But, of
course, we all "know" that Frege dealt a mortal blow to 19th century
psychologism, so (b) has since fallen into disrepute.

I myself feel tempted to reevaluate the (presumably obsolete) "laws of
thought" project not as a *normative* enterprise (which is how Boole may
have construed it), but as an actual *factual* investigation, through the
prism of an updated science of the brain and mind. What I am, ultimately,
driving at, is that answering the "What is LOGIC?" question might help us
answer the "How much of math is logic?" question; and the answer to the
former should not be looked for in any such types of (internal)
considerations (deductive powers, ontological commitment to actual
infinities etc.).

Christian

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