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Can you please suggest some good or essential readings on necessity as a concept

Can you please suggest some good or essential readings on necessity as a concept? Or where it is useful to start as a beginner?

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!).

I attempted to define 'Truth' today and so far the best I can come up with is:

I attempted to define 'Truth' today and so far the best I can come up with is: In order to really understand and analyse exactly what truth is; we first need to explore the idea of truth in its purest form. The Compact Oxford English dictionary suggests that Truth is 'that which is fact or can be accepted as true.' In this sense, I would first suggest that, philosophically, truth falls more aptly into the area of faith and belief as opposed to anything definitive. This is due to the fact that nothing can be proven to be precisely accurate without error for an infinite amount of time. In fact, even if something were theoretically created at a point in time that was, at that point in time, precisely accurate it cannot be proven to be accurate for an infinite amount of time as, by definition, you would need to test the theory or creation infinitely. We can thus resolve that, despite common definition, truth is a label given to an abstract, repetitive belief specifically in relation to the human condition...

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.

This has been bugging me for quite some time now. Is knowledge truth? Is truth

This has been bugging me for quite some time now. Is knowledge truth? Is truth knowledge? Are these concepts the same?

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".

Most people believe that a belief is true if it corresponds to a fact. But facts

Most people believe that a belief is true if it corresponds to a fact. But facts and ideas are very different things. They exist in completely separate realms. How can they "correspond" to each other?

You write that facts and ideas are very different things. (You also contrast beliefs and facts to make the same point, so perhaps you believe that beliefs are ideas.) From this you infer a difficulty about the possibility of ideas and facts corresponding to one another. 'Facts and ideas are very different things', you write, so 'how can they "correspond" to each other?' Consider, though. Written notes on a stave are very different from the sounds that we hear, but why should that stop them "corresponding" to sounds? Aunts and nephews are very different kinds of beings, but that need not stop them corresponding. You put "correspond" in scare quotes, and here you seem to me to be on the right track. We need to know what correspondence is. What is say a 1:1 correspondence?

I'm a mathematician looking at some of the work of Leonhard Euler on the

I'm a mathematician looking at some of the work of Leonhard Euler on the "pentagonal number theorem". My question is about how we can know some statement is true. Euler had found this theorem in the early 1740s, and said things like "I believed I have concluded it by a legitimate induction, but at the same time I haven't been able to find a demonstration" (my translation), and that it is "true even without being demonstrated" (vraies sans etre demontrees). This got me thinking that "knowing" something is not really a mathematical question. A proof lets us know a statement is true because we can work through the proof. But a mathematical statement is true whether we know it or not, and if you tell me you know that a statement is true, and then in fact someone later proves it, I can't show mathematically that you didn't know it all along. This isn't something I have thought about much before, and my question is are there any papers or books that give some ideas about this that would be approachable by...

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 1016. 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 the first tipping point now thought to be in the order of 10316. See here.] We know, then, that extrapolation from (merely!) the first 1016 cases is dangerous -- for that "sample" is biased towards relatively tiny numbers. Yet, as I said, mathematicians do all the same tend to be confident in Goldbach's conjecture. Which suggests that they have other non-demonstrative grounds for their belief, grounds better than mere extrapolation from the initial cases. For a discussion of what these might be, and some evaluative comments, see e.g. Alan Baker's paper "Is there a problem of induction for mathematics?" in M. Leng, A. Paseau & M. Potter (eds.) Mathematical Knowledge. That's a pretty accessible read, and one of the few recent papers I know about the interesting question of the role of non-demonstrative reasoning in mathematical thought.

Now, suppose NM (for "naive mathematician"!) does believe Goldbach's conjecture just on the basis of extrapolation from small cases. Then, if a proof were discovered, could NM claim to have known the conjecture to be true already? Surely not so. NM's mathematical reason for his belief was (demonstrably!) a bad reason; and a true belief held for a bad reason isn't a case of genuine knowledge (it's more like a lucky guess). So actually, I think it is wrong in general to say "if you tell me you know that a statement istrue, and then in fact someone later proves it, I can't showmathematically that you didn't know it all along." NM didn't know Goldbach's conjecture all along, and that's because his grounds don't pass muster as good mathematical reasons.

Now a nod towards the much more general question here. Certainly, it is very plausible to say that when we prove a mathematical proposition we aren't (so to speak) creating a new truth, but discovering something that was true all along -- "a mathematical statement is true whether we know it or not". Even when mathematicians invent a new branch of mathematics -- as Eilenberg and Mac Lane did in founding category theory -- it remains tempting to say that they are discovering pre-existent patterns and structures. But is this right? For what is the nature of this supposed realm of pre-existing mathematical structures, and how do we get epistemic access to it? Well, they are two of the Big Questions in the philosophy of mathematics. And here, I can probably do no better than just point to Stewart Shapiro's fine introductory book, Thinking About Mathematics.

Do you think it's possible, even theoretically, for there to exist a substantive

Do you think it's possible, even theoretically, for there to exist a substantive belief (any kind, about anything) that is impervious to any argument, cannot be debunked, etc., and yet is false?

Yes, at least theoretically. An example of how this might be is given in the first of Descartes' Meditations on First Philosophy. Descartes asks us to consider a world that is governed by a kind of evil god who delights in nothing more than making us believe what is false. In such a world, we would be able to find no evidence at all to debunk the falsehoods to which the god inclined us. Descartes challenges us to see if we can be absolutely sure that we do not actually inhabit such a world!

Modern popular culture has taken up this scenario in various entertaining ways. I think it is fair to say that the worlds imagined in "Total Recall," and "The Matrix" are excellent examples of scenarios that raise the theoretical possibility of false belief that is (at least for those who don't escape the Matrix!) invulnerable to refutation.

Are statements about resemblances objectively true/false, or are they merely

Are statements about resemblances objectively true/false, or are they merely statements about the way things seem to us, hence subjective? Is it "objectively" true that pentagons are more like hexagons than circles? Is it objectively true that the paintings of Monet are more like those of Renoir than those of Picasso?

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 context) with some indication of the resemblance-respects which we care about when we ask. But once those indications have been provided, it can (as our examples show) be a straightforwardly objective matter what the answer is.

Similarly, the question "are the paintings of Monet more like those of Renoir than those of Picasso?" again invites the reply "like in what respect(s)?". Where they were painted? Average market value? Average size? Well, it's perfectly objective what the right answer is, if it turns out that it's resemblance in one of those respects which is question. But of course, we might very well be more interested in some more purely aesthetic features; and you might suppose that now things do indeed become more subjective. But the issues don't become more subjective because they are issues about resemblances but because they are issues about the aesthetic. (Another possibility is that the questioner just isn't clear what kinds of resemblance are being asked about: but then we don't have a determinate question to answer.)

In sum: at least some questions about resemblances -- once it is made clear what resemblances are in question -- can have just as objective answers as other questions about a thing's properties. There isn't anything instrinsically subjective about issues of resemblance per se: it depends what kind of resemblances you are interested in.

John Carey has written a book called "What good are the arts?" His central idea

John Carey has written a book called "What good are the arts?" His central idea is that our evaluation of the visual arts and music is completely subjective and relativistic. Art and art creation are seen as an important part of being human but no one can make a case for a work being of higher value because this is just opinion. Fine. However, he goes on to argue the case for the higher value of literature. Predicting the obvious objections one might have after his previous relativist argument he says: "let me emphasize that all the judgements made in this part including the judgment of what 'literature' is are inevitably subjective". Here and in live debates he has stated this as a means of getting himself off his own hook. So my question is, surely there is some contradiction involved in arguing a position while at the same time stating that it is just subjective? Aren't we trying to lay claim to some objective truth as soon as we begin arguing?

One can create axioms that make statements like "all bachelors are married" true

One can create axioms that make statements like "all bachelors are married" true. What is wrong with calling these truths analytic as a shorthand for the type of truth it is based on the type of axiom it is derived from, much in the way we use the adjectives arithmetic, set-theoretic, or logical to denote those types of formal truths? I feel like one could decide whether a truth is analytic by seeing which (kinds of) axioms need to involved in making it true.

There is nothing stopping you from defining an analytic theorem of a formal system to be one whose derivation requires appeal to at least one member of a designated subset of axioms. But on what basis are you deciding to single out that particular subset of axioms? If you say you're being guided by the fact that those particular axioms express truths about meanings, whereas other axioms express substantive truths about the world, then you owe an explanation of what that distinction amounts to -- and arguably, that will be no easier to give than an outright analysis of "analytic". (You might also look at W.V. Quine's discussion of Semantic Postulates in his paper "Two Dogmas of Empiricism.")

How could we distinguish facts and interpretations of facts? Some say that facts

How could we distinguish facts and interpretations of facts? Some say that facts are given, others say that they are constructed by theories. Could we still say that facts are independent or previous to theories?

The tricky thing about this issue is to decide what the issue is. Some people seem to want to say that all facts are constructed, but I've never really understood what this is supposed to mean. Let me yank at a few threads and see if any of them are connected to the worry.

Some facts depend on our conventions, institutions and so on. A well-worn example: I have a shiny round bit of metal in front of me. As a matter of fact, it's a quarter; it's worth $.25. That really is a fact, but it wouldn't be a fact if we didn't have certain practices, institutions and so on. In at least some sense of "constructed," it's a constructed fact.

We also classify things in various ways. Some of those classifications grow out of our interests, beliefs and so on. Classifying music according to genre is relatively benign; classifying people according to the racial categories of apartheid-era South Africa or the antebellum American South is anything but benign. Sometimes we take our classifications to mark deep distinctions in nature when all they really reflect our our own shallow points of view. But it may be, all the same, that some ways of classifying things "cut nature at its joints," as they say. Protons and electrons are arguably real kinds, and among the basic things from which nature is built. Charge and mass may be basic, perfectly natural properties.

Some philosophers -- nominalists of various sorts -- reject the very idea that some ways of classifying things are more true to the world than others. Others -- David Lewis is an important recent example -- would say that we can't get around presupposing that there really are natural properties, even though there's room to fight over just what sort of beast a property is. If this sort of view is right, then there are facts independent of any theories.

Interpretation is a slippery concept. Sometimes my interpretation of a situation is just plain wrong. I may have seen you through the window and interpreted the expression on your face as deep sorrow. In fact, you may have been laughing hysterically. My interpretation was wrong. On the other hand, when what we're trying to interpret is an artifact, "better" and "worse" are sometimes more useful ways of judging interpretations than "right" and "wrong." (Is Hamlet a story about a man with an Oedipus complex? That interpretation got a good deal of mileage, but it's not clear whether there's any hard fact of the matter.)

Here's a view. (It's a crude version of what David Lewis believed.) At bottom, there are perfectly natural properties, distributed in space-time in some particular way. There are facts about all that, whether we know them or not. And everything else is fixed by those facts; there couldn't be any differences at the level of ships, shoes, food fights and French literary theory without differences at that basic level. If Lewis was broadly right (and I've never seen any good reason to think that he was wildly wrong), then the idea that it's interpretive mush all the way down is a mistake.

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