Perhaps it is worth taking continuing the conversation just a bit further. The idea that a proposition (statement, belief) is true if and only if it "corresponds to reality" is -- as I'm sure William would agree -- not entirely transparent. What does it commit us to, exactly? The deflationist about truth of course says that the proposition that snow is white is true if and only if reality is such that snow is white -- i.e. just if snow is white. So if the correspondence theorist is to be distinctively saying more than that, she needs to spell out what "correspondence" here comes to, over and above what the weak kind of correspondence that is already built into the deflationist view. Now, there are indeed metaphysicians who do claim to have an "industrial strength" version of the correspondence theory, who postulate the existence of facts as ingredients of the world, facts which are truth-makers whose existence is required to make propositions true (where the worldly...

Jack says "The next train to London is at 11.15"; Jill adds "That's true". Jill's remark in effect just repeats Jack's message. To say it is true that the next train to London is at 11.15 tells us no more about the world than that the next train to London is at 11.15. Dora witnesses a crime. She gives quite a long statement. "Three boys in jeans and hooded tops came into the shop just before 12. They ... etc., etc. etc., ... And finally they jumped into a red car and sped off." Dick adds "That's all true." Again, Dick is in effect just repeating Dora's statement, but saving breath. You can see why we should have use for such a very handy device in our language. Someone says something, or we read something in a book; saying "that's true" has the effect of saying the same, without all the bother of repeating what is said or written. And the same handy device is just as useful when what is said or what is written is not so common-or-garden but more theoretical. Alice says "The atomic weight...

Some talk about facts is just a stylistic variant of truth-talk. For example, in ordinary discourse, to say 'It's a fact that the fast train to London from Cambridge takes less than an hour' is to say no more than that 'It is true that the fast train to London from Cambridge takes less than an hour'. And, arguably, both those in turn say no more than that the fast train to London to Cambridge does take less than an hour. However there is also a more substantive notion of fact that has a long history in philosophy and has in recent years made something of a comeback -- this is the notion of facts as not truths but truth-makers . The proposition that the fast train to London from Cambridge takes less than an hour is true. And there is, plausibly, something worldly that makes it true, something that has to exist if the proposition is to be true, a truth-maker in short. And what kind of thing is a truth-maker? It isn't enough for London, Cambridge and trains to exist. And adding in the property of...

Isn't it the other way about? Only a small number of philosophers would not maintain the distinction! For as you remark, lots of truths aren't useful in any ordinary sense (e.g. there is a fact of the matter about whether the number of grains in the rice jar yesterday was odd or even -- but the truth one way or the other is no use now to man or beast); and lots of claims which are strictly false can be useful (quick and dirty approximations that do us well enough. So to tie the ideas of what is true and what is useful together will need, for a start, bending the idea of the useful into something fairly remote from its common-or-garden sense, and we will also probably have to be pretty revisionary about what counts as true, in a way that few philosophers will be happy with.

As it happens, I recently had to update the reading list for the logic paper of the first-year of the Cambridge Philosophy Tripos . One of the topic-headings is "Necessity" (see foot of p. 8 to top of p.10). That's a modest introductory list, concentrating on the notions of logical necessity and analyticity. Unfortunately, however, this might not be a terribly helpful response, as access to most of the references given will involve you using a university library. Perhaps others will know of useful and reliable free online resources of an introductory kind (and I'd be glad to hear of them to add to the reading list too!).

Evidently something is going pretty badly wrong here. Here's a truth: my laptop computer is right now on my lap as I'm typing this . It doesn't need "precise accuracy without error for an infinite amount of time" to establish that . It's a rough-and-ready proposition about the here-and-now: precise accuracy and infinite amounts of time just don't come into it. Likewise for many common-or-garden truths. Something else is going badly wrong. For here's another truth: it rained here today . Nothing there about the human condition and human behaviour. Just a local meteorological fact. Getting serious about philosophy is nothing to do with "loving a good argument", or trying to make up definitions off the top of your head, any more than getting serious about physics is. It's hard work, and you need to do your homework first in either case. Try this article on truth as a starting point, or Simon Blackburn's Truth .

It is a requirement for something to be genuinely known to be true that it is true. So knowledge implies truth (in the sense that if X knows that so-and-so, then it is the case that so-and-so). But that doesn't make knowledge the same as truth. The implication the other way around doesn't hold. There are truths that you don't know, that I don't know, and indeed that no one right now knows (maybe because nobody has bothered to find them out, maybe because the time has past when anyone could check, or because the truths are about far-off events like meteorite strikes on the far side of the moon, or for other kinds of reason). Leaving ominiscient deities out of it, not every truth is known. But we might wonder whether every truth is knowable , in principle, e.g. by a suitably placed and sufficiently smart observer. The trouble with that idea is in spelling out the "in principle".

What does it meant to say that a belief is true if it corresponds to a fact? A low-calorie reading of that slogan is that my belief e.g. that snow is white is true if and only if it is a fact that snow is white, i.e. if and only if snow is white. Similarly for other beliefs. And there doesn't seem to be anything very mysterious about this claim. (Nor is there much reason to suppose that "most people" are committed to any more.) Now, it's not that there aren't interesting problems hereabouts. But as the great Cambridge philosopher Frank Ramsey noted, the serious problems are about the nature of "intentionality" or "aboutness". For how can a state of me somehow be about something else, in this case, the colour of snow? (Note, however, the problem isn't in general one about relating separate realms -- I and my states are part of the same natural world as colours and snow! And there are naturalistic stories on the market which tell us how the link-up is made.) But once we've explained how it...

Perhaps there are two different questions here. There's a very general question about truth and proof; and there's a much more specific question about the sort of case exemplified by Euler, where a mathematician claims to know (or at least have good grounds for) a proposition even in the absence of a demonstrative proof. Let's take the specific question first, using a different and perhaps more familiar example. We don't know how to prove Goldbach's conjecture that every even number greater than two is the sum of two primes. Yet most mathematicians are pretty confident in its truth. Why? Well, it has been computer-verified for numbers up to the order of 10 16 . But so what? After all, there are other well-known cases where a property holds of numbers up to some much greater bound but then fails. [For example, the logarithmic integral function li ( n ) over-estimates the number of primes below n but eventually under-estimates, then over-estimates again, flipping back and forth, with...

Surely the question whether pentagons are more like hexagons than circles just invites the riposte: "more like in what respect?". If we are interested in whether figures have straight sides and vertices or lack them, then of course pentagons will get put in the same bucket as hexagons, while circles will go in another bucket (with e.g. elipses and parabolas). It's an objective fact that pentagons are like hexagons (and not circles) in having straight sides and vertices. If we interested in whether we can tile a plane with (regular) figures of a certain kind, then pentagons will be classed with circles (no, you can't tile a plane with those), and hexagons will belong in the other bucket along with e.g. squares and triangles. It's an objective fact that pentagons are like circles (and not hexagons) in that you can't tile a plane with them. So we might say that the bald question "are pentagons more like hexagons than circles?" is incomplete. It needs to filled out (either explicitly or by...