your reduction of material implication to set theory. I'm not sure how to understand that, really — Moliere
if the moon is made of green cheese then 2 + 2 = 4. That's the paradox, and we have to accept that the implication is true. How is it that the empirical falsehood, which seems to rely upon probablity rather than deductive inference, is contained in "2 + 2 = 4"? — Moliere
Leon and Hanover are more of an inspiration for Tones. They bring forth his best work.Only if you agree to write the preface. — Srap Tasmaner
I find the visualization helpful. We're just doing Venn diagram stuff here. — Srap Tasmaner
Ask yourself this: would "George will not open tomorrow" be a good inference? And we all know the answer: deductively, no, not at all; inductively, maybe, maybe not. But it's still a good bet, and you'll make more money than you lose if you always bet against George showing up, if you can find anyone to take the other side.
"George shows up" may be a non-empty set, but it is a negligible subset of "George is scheduled to open", so the complement of "George shows up" within "George is scheduled", is nearly coextensive with "George is scheduled". That is, the probability that any given instance of "George is scheduled" falls within "George does not show up" is very high. — Srap Tasmaner
The (probability) space of A is entirely contained within the (probability) space of not-A.
Well, of course it is. That's almost a restatement of the probability of P v ~P equals 1. — Moliere
"is contained within", i.e. determined by — Moliere
P can be empty set, which is a member of every set. — Moliere
P can be empty set, which is a member of every set. — Moliere
Your probability exploration is interesting. I think there's probably (pun intended) been a lot of work on it that you could find. — TonesInDeepFreeze
So, as far as I can tell, category theory does not eschew set theory but rather, and least to the extent of interpretability (different sense of 'interpretation' in this thread) it presupposes it and goes even further. — TonesInDeepFreeze
subset v member — TonesInDeepFreeze
replace set theory entirely — Srap Tasmaner
One other tiny point of unity: I always thought it was interesting that for "and" and "or" probability just directly borrows ∩ and ∪ from set theory. These are all the same algebra, in a sense, logic, set theory, probability. — Srap Tasmaner
0 subset of 0 holds by P -> P. — TonesInDeepFreeze
Peter Smith offers some nice content. — TonesInDeepFreeze
Oh, yes, the duals run all through mathematics. — TonesInDeepFreeze
Writers often used the word 'contained'; it is not wrong. But sometimes I see people being not clear whether it means 'member' or 'subset' — TonesInDeepFreeze
are you sure this is right? — Srap Tasmaner
do we want to say it's because of the proof that it is so? — Srap Tasmaner
"holds by" ― rather than, "is proved using" — Srap Tasmaner
Unforgiving when authors were too slapdash or handwavy about this, which I thought showed good philosophical sense. — Srap Tasmaner
motivates category theory — Srap Tasmaner
What does mathematics get out of pretending it's importing logic from elsewhere? — Srap Tasmaner
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