What does mathematics get out of pretending it's importing logic from elsewhere? — Srap Tasmaner
you have to have sets (or an equivalent) to do much of anything in the rest of mathematics, but so what? — Srap Tasmaner
Propositional logic deals in propositions. Your piece has the form of a modus ponens, but doesn't deal in propositions. That makes it interesting in several ways. But "not-a" is pretty well defined in propositional logic, in various equivalent ways. And by that I mean that the things we can do with negation in propositional logic are set. There are not different senses of "not-A" in propositional calculus. — Banno
Absolutely sure. — TonesInDeepFreeze
What pretending? — TonesInDeepFreeze
Someplace to start writing without having to explain yourself. — fdrake
I think of mathematical logic sub-subject of formal logic. — TonesInDeepFreeze
Set theory axiomatizes classical mathematics. And the language of set theory is used for much of non-classical mathematics That's one so what. — TonesInDeepFreeze
because there is nothing anyone can say to explain it. — Srap Tasmaner
There's your foundations, all in box, instead of logic coming from outside mathematics ― that's what I was questioning, am questioning. (I suppose, as an alternative to reducing it to something acknowledged as being part of mathematics, which I admit doesn't seem doable.) — Srap Tasmaner
natural language statements — fdrake
Don't know what you're driving at. — TonesInDeepFreeze
...there is life... — Janus
But then nothing is both alive and dead at the same time. — Banno
Isn't formal language a part of natural language? — Banno
logic as just "given" in toto — Srap Tasmaner
member and collection — Srap Tasmaner
there just is no way around ∈, no way to cobble it together from the other logical constants. — Srap Tasmaner
Just a tendentious turn of phrase, not important. — Srap Tasmaner
whether we could get away with thinking of its use elsewhere, not only in the sciences, but in philosophy and the humanities, as, in essence, applied mathematics. — Srap Tasmaner
I imagined Srap and I were talking about how the formalism in mathematics doesn't start at its "grounds", in the axioms. I imagine Srap and I are reacting to an imagined enemy of a formalist who thinks that mathematics is somehow "just" symbol manipulation. Or alternatively just awed at how the root of the formalism is in as something as messy as natural language, despite how set in stone - settable in stone - the concepts of mathematics seem to be. — fdrake
I'm not so sure of this, since Kripke's theory of truth contains it's own truth predicate, and there is considerable work around its relation to arithmetic, I don't think we can yet rule out a Kripke-style first order arithmetic. I might be mistaken.Consistent systems capable of first order arithmetic can't contain their own truth predicate — fdrake
Just that there's at least here a dependence of mathematics on natural language, which gives the appearance of being purely pedagogical, or unimportant "set up" steps (still closely related to the thing about logical schemata, from above). — Srap Tasmaner
the indefinability of "set" — Srap Tasmaner
Isn't formal language a part of natural language? — Banno
Isn't formal language a part of natural language? — Banno
Consistent languages capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. But natural language does contain its own truth predicate and behaves... well it doesn't disintegrate. That's at the very least a type distinction between consistent formal languages and natural language - one can contain its own truth predicate without being crap, one cannot. — fdrake
In one view, we have a formal object-language, and an informal or formal meta-language that includes the formal object-language. — TonesInDeepFreeze
Kripke's system plays with consistency, creating a formal language that contains it's own truth predicate.Consistent systems capable of first order arithmetic can't contain their own truth predicate, so we don't put the predicate in. — fdrake
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