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I don’t think a two-valued topos has to be Boolean, even classically. The page two-valued topos doesn’t think so.
Not in ordinary English, but in mathematics, ’true’ can be a noun, more precisely a name: we often pronounce the truth values and as ’true’ and ’false’ – and that was the usage the old title refers to.
Now the history page says that on Feb 2, Toby Bartels revised the truth.
More seriously, I agree with Todd about the intended way to read the page title “true”. But then I think the choice of title of a page should be viewed more as an internal book-keeping tool used to produce a URL. What really matters for the content of a page is that the Idea-section of any entry makes crystal clear what it is about, and under which set of different names and perspectives this is known.
@Mike #2: Of course, fixed.
@Todd #3: Yes, I'm sure that that's how the page names came to be what they were (and presumably it was me who originally named false, copying true that was presumably named by John Baez), but it just doesn't seem right to me now when written out as English words. It's like pronouncing ‘’ as ‘or’, which I might do even when appearing a phrase like ‘In classical logic, is commutative.’, where it is a noun, but you can't write ‘In classical logic, or is commutative.’ (although you could get away with ‘In classical logic, OR is commutative.’.) And appropriately, or is a redirect to disjunction.
@Urs #4: Actually, it says that I revised truth, which sounds even more impressive. (I did create a redirect from the truth, however.)
I do pronounce “In classical logic, is commutative” as “In classical logic, ’or’ is commutative”. (In what kind of logic isn’t commutative? Even the most straightforward kind of “ordered logic” that I know of has a non-commutative linear but still a commutative .)
One thing I don’t like about the title “truth” is that it sounds like it should be for a different topic, for example in philosophy of mathematics, or something metamathematical like Tarski’s undefinability of truth result, etc.
@Mike #6:
I do pronounce “In classical logic, is commutative” as “In classical logic, ’or’ is commutative”.
Yes, but would you write it that way? Maybe with the quotation marks or with some other typographical decoration, but not straight.
(Noncommutative linear logic has commutative but noncommutative . I don't know if anybody writes as , but relevant logics reportedly write as , so who knows what other variations exist? As far as the relation to classical logic, they both have equal claim to that symbol.)
@Todd #7: Yeah, that bothers me too. But I don't know what we'd write about those topics either.
Toby: as you know, I prefer the old title “true” for the present article. But as for what I’d (hypothetically) write about “truth”, I might say something like: it’s mathematicized by the notion of satisfaction of a proposition in a model, thus it has to do with the relation between syntax and semantics, etc. The Tarskian result on undefinability of truth could be made to fit within such an article. There could be a little about the philosophy and history as well; again if it were me writing, I can imagine cribbing from the SEP.
Not so unreasonable to model a noncommutative ’or’ if you consider how we use it to spell out consequences, e.g., “Do this, or (else) that will happen”. That’s why it appears in relevant logic.
In #6 I meant to write , not . I would argue that deserves the symbol more than does, because it behaves more like the “or” of classical logic. But sure, I suppose someone might do things differently.
I've moved the page to true proposition, keeping redirects from truth and similar terms until somebody writes the article outlined by Todd #9.
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