Two Propositions or One (or none)?

At the beginning of this month, in Divine Liars: the devil's in the details, I wrote about sentences like ‘this sentence is not true,’ where the phrase ‘this sentence’ is referring to that very sentence. That sentence seems to be saying that what it is saying is not true, in which case it would also be saying that it is not true that what it is saying is not true, so it would also be saying that what it is saying is true. Does that mean that that sentence expresses two propositions?

Propositions are the logical meanings of sentences. Sentences in different languages (and in particular, different formal languages) can express the same proposition. And in particular, the logical meaning of the self-referential ‘this sentence is not true’ seems to be that that sentence is not expressing a true proposition. Now, that certainly seems to be a different proposition to the proposition that that sentence is expressing a true proposition, because were it the case that that sentence expressed a true proposition, the first of those two propositions would seem to be false, and the second true.

That self-referential sentence seems to be saying, of itself, that it is not true. But if that sentence is saying of itself that it is not true, then it is not standing outside itself when it says that it is not true. It is not like when I say that it is not true. And when it says that what it is saying is not true, so that it is also saying that it is not true that what it is saying is not true, the latter is just part of what that sentence says when it says that it is not true. Because the former includes the latter, there is a sense in which there is only one proposition here, albeit one with a very strange part. Now, there is a sense in which we are talking about two propositions here, because the latter is only part of the former (and it is a part that seems to contradict the former). But the latter is part of the former. And this self-referential sentence is a very simple sentence, so we can see that it is not equivocating. I therefore think that the former proposition with that very strange part is all that that self-referential sentence would be expressing, were it expressing a proposition. Perhaps that strangeness is so extreme that no proposition is being expressed, but the main thing here is that:

When we think of the proposition that that sentence is not expressing a true proposition, we might be thinking of the proposition that would be true were that sentence not expressing a true proposition, or we might be thinking of the very strange proposition that that sentence would be expressing were it expressing something propositional.

Because those are two propositions, not one, talk of propositions seems to have brought with it an increased risk of equivocation, not increased clarity. With other kinds of sentences, talk of propositions may well introduce clarity, but for these self-referential sentences, we might be better off sticking with talk of sentences and their meanings, logical and other (after all, formal languages are usually designed to avoid self-referential sentences and other complications).

A Linguistic Puzzle

Where, in ‘my tabletop is flat,’ is the meaning of that string of letters? Any meaning that it has must be read into it (it is not a string of magical sigils). You read those words, which are about the flatness of the table at which I am sitting, and perhaps you think of something like the tables that you know. You might think of such a thing being flat, and it being my table (whoever I really am). You get that string of letters, and you put your meanings into it. But the meaning of that phrase is, I think, the thought that I expressed with it, my thought that my tabletop is flat. And that is not the only mysterious thing about reading, of course. When we read something truly meaningful—e.g. a great novel—we do not seem to be getting out of it some rearrangement of what we already know. We seem to learn something about the world around us. Are we fooling ourselves?

Our Lewisian paradise

Modern philosophy (which began with Bacon and Descartes) looks antiquated until 1973, the year it acquired the now-standard way of analyzing counterfactuals: David Lewis’s possible worlds analysis (PW). If a one-eyed man is king in the land of the blind, Lewis seems to me to be like a blind man in the land of the one-eyed, successfully selling them a white stick:

The blind man (Lewis) tells them that they can use the white stick to find out whether things are near enough to them to be a problem. I think that they should have told him that it is just a stick, and the wrong colour for their outfits.

Lowe’s description of Lewis’s analysis is better than mine:

A counterfactual of the form ‘If it were the case that p, then it would be the case that q’ is said to be true if and only if, in the closest possible world in which p is the case, q is also the case – where the ‘closest’ possible world in question is the one in which p is the case but otherwise differs minimally from the actual world.

Suppose that I’m trying to tell you something, and I know that you’ll find it hard to understand what I’m saying; I might say ‘If you knew what I was trying to tell you, you’d know how difficult this is.’ Now, if you did know, it would be trivially easy to tell you about it, so presumably I meant that after I’d told you, and you’ve understood me, you’d agree that it was difficult to tell you. But suppose it’s so hard to tell you that you never do get it. Does my meaning really depend upon which of the possible worlds in which I’ve told you is most like the actual world, in which you didn’t get it?
......What if the difference is just a few neurons that you were born with, for example, but that those neurons also make it hard for you understand why it was difficult to tell you (since you then find such things so obvious)? What if lots of things; and so basically, how could all that really affect the meaning—and hence the truth—of what I’m saying? After all, we do seem to have got bogged down in an awful lot of fallacious arguments and counter-arguments since 1973; which may be ideal for professional philosophers in a stupid economy, but less so for those applying logic to the real world.

Cartesian dualism, ii

I’ve yet to find a good philosophical argument against such substantial dualisms as (for the commonest) that our psychology results from the interaction of spiritual souls with the physical brains in which (so to speak) they’re incarnated. The two commonest arguments are (i) asserting the closure of the physical and (ii) failing to see how the spiritual could interact with the physical. Both are clearly fallacious as I’ve stated them, but I’ve yet to find a substantially fuller, non-fallacious expression of either. Now, I’ve blogged on (i) already, and have little to say about either anyway, but I’ve just been reading Lowe (Erkenntnis 65, 5–23), who put (ii) as follows (2006: 7, 11):

[...] according to Descartes, whereas the mind has beliefs, desires, and volitions, but no shape, size, or velocity, the body has shape, size, and velocity, but no beliefs, desires, or volitions. [...] it is often complained that it is completely mysterious how an unextended, non-physical substance could have any causal impact upon the body – the presumption being, perhaps, that any cause of a physical event must either be located where that event is, or at least be related to it by a chain of events connecting the location of the cause to the location of the effect.

As put, the problem seems to be one of mere conceptual possibility, which is easily answered. By typing into your keyboard you can make virtual beings move about in cyberspace. Clearly you don’t have to be where they are, in cyberspace, to be able to move them about. So it isn’t so very mysterious how such things are possible. And even if it were, why presume that would be a problem for dualism, rather than a personal failing?
......As Lowe notes, people said that Newtonian action-at-a-distance was completely mysterious, and maybe it was, and is, but there was hardly any argument there against Newtonian physics (except in the minds of some philosophers). The truth turned out to be far weirder again, and it was to be had by working through Newtonian physics. There is that other problem, of how exactly the interaction works, but the way towards answering that is the relatively hard way of science, and why should it not go through Cartesian dualism?
......My analogy only worked because of the causal link between your fingers moving on the keyboard and the consequent virtual motion (as expressed in actual space on the screen), which goes via continuous paths in space (if we include force-fields in our ontology), but still, it did work. It suggests that a possible Cartesian response is to give the body, not only a spatial location, but also another, non-spatial location, at which the soul acts. How plausible is that? In the natural theistic context of Cartesian dualism, it’s very plausible, since God created space, and is himself located elsewhere.
......And suppose that Cartesian dualism is false. Then there’s some other true theory of mind. Somehow the physical brain, which changes its form and its atomic constituents continually, is associated with a subjective unit (the mind, which we know directly), which is continuously the same person. So if there could be a non-Cartesian theory, then there’s some way of associating with the physical brain a unique continuant of some sort. It is only to that that the Cartesian theory has to associate a soul. And a very simple and natural (in the Cartesian context) way to do that would be by divine stipulation, God associating each such brain-correlate with a unique soul.
......In many ways that’s far simpler and more natural than the sort of Humean regularity approach to scientific laws that philosophers are often led to by considering how mysterious are nomological necessities (a consideration that most scientists rightly ignore). If souls are possible, then they would have individual existences, in some logical space (say heaven), and would interact in some way (say via spiritual bodies). And if so then matter would’ve been created to be such as could be used in such ways (for some reason). The details are for scientific discovery, but the mere possibility is not really so mysterious.

What's a mathematician?

A mathematician is a device for turning coffee into theorems, according to Renyi's famous joke. But since most mathematicians don't prove theorems nowadays, I tend to think of the natural mathematicians as those finding More or Less as gripping as an Agatha Christie mystery (whether or not they like set theory).
......On the news this morning, there was a story about researchers at Durham have discovered that coffee makes us hear voices. They conceded that maybe those who hear voices tend to be more stressed, and so drink more coffee, but that seems like an odd concession to me, as people who're stressed usually turn to, if not alcoholic beverages, then chocolate or cigarettes, or cannabis.
......Scientists have also claimed that cannabis can trigger psychosis. That seems more plausible, as cannabis is the famous hippy drug, but I wonder even about that. Many of the most obvious direct tests of that hypothesis would be rather unethical, and the indirect tests (e.g. statistical correlations) would be vulnerable to selection biases. It's not just that schizophrenics might be more likely to disobey the law. There is apparently a part of the brain that is involved in religious experiences, e.g. Richard Dawkins had his stimulated and experienced nothing, apparently.
......If some people are more disposed to such things (again, whether they're born that way, or whether the brain adapts to their chosen way of life, is less obvious) then that would be a relatively obscure but effective source of such a bias. Incidentally, such studies need worry religious people surprisingly little. The traditional view of God has him timelessly creating us, and so faces similar problems, e.g. from our choices to turn to him etc. And Open-theistic views must face the facts of life anyway, e.g. that some of us are born richer, or better looking, or with better brains in other ways. If they can do that, they'll be able to live with similar facts.
......Anyway (oh how tangents attract the active mind), it occurs to me that those with such a religiously inclined brain might be more inclined (statistically, not each of them of course) to hear voices and also more inclined (similarly) to drink coffee, whether because they don't drink alcohol for religious reasons, or because they like to be awake to creation, or whatever. They may also be a little more inclined (statistically) to smoke cannabis, insofar as that's associated with the mystical side of hippies, or the religious side of Rastafarians, etc. If anything, we'd expect a greater corrolation with coffee, of course (and it does seem more plausible that coffee is not actually causing us to hear voices).
......Anyway, that's one possible explanation: a common cause leading to interlinked tendencies. Another explanation is that coffee in large quantities makes some of us irritable and tense, and perhaps over-sensitive, and so maybe more likely to hear voices insofar as we're already slightly inclined towards that (although I'm not ruling out the possibility that people take coffee because they're feeling stressed), but even in that possibility there's room for biases of the former kind—a common partial cause—e.g. a weak willed person might be more likely to over indulge in coffee, and less able to resist hallucinating. Similarly, they might drink more and get into fights, for such a reason. Or they might (also) react to the drink by getting more aggressive themselves. Note that that would be no reason for those who drink to relax and socialise to drink less (you may have guessed that I drink a lot of coffee:)

Faith, a definition

Personal faith is not assent to evidence which is so strong as to be beyond reasonable doubt. It is assent to a discernment of God which is personally overwhelming but not objectively testable. This is not discernment of a historical God, timeless and unchanging. It is discernment of an active, loving God, making himself known in personal lives at specific points which become the matrix of a communal response to his will.

Keith Ward, Divine Action (London: Flame, 1990), 238.

How mysterious is Platonism?

Arithmetical Platonism is supposed to be prima facie suspect because how, it’s asked, could we have arithmetical knowledge if the objects of that knowledge are in a world apart from us, a timeless world of Platonic objects, with which we cannot interact causally? I reply by wondering, how strange are abstract objects? You have just been reading this, for the obvious example. What have you been reading? You have been reading sentences. You look at the physical instances of these words, but you see the words, you read the words, and as you do so you are (hopefully) thinking about the thoughts expressed by means of them. So, there are, in the physical world around you, those physical instances of the shapes of (written modern English) words, and there are in your mind those thoughts; so, where are the sentences? What are the sentences? Sentences are made of words, and words are parts of a language (i.e. modern British English). They can be spoken or written, and can sometimes be spelt in different ways. Furthermore they have a meaning, a sense, and they have it essentially. The mere shape of a word is not a word, no more than meaningless strings of letters are words. Words, it seems, are abstract objects (I’m not entirely sure about that, or about what abstract objects are, so I’d welcome corrections) and you’ve just been reading some words of mine (and I’ll add that words can be true, insofar as they describe the world sufficiently accurately, or not, in case anyone wants to argue that thoughts and not words are truth-bearers:)

Who's the Philosopher?

If good mathematics tells us that there are sets of some size, or functions of some type, or if good science tells us that there are waves or particles or forces or fields, then who is the philosopher to pipe up otherwise? As David Lewis forcefully argued, the philosopher taking any such line is apt only to make himself look foolish. Considering the case of sets in mathematics, Lewis wrote:

Mathematics is an established, going concern. Philosophy is as shaky as can be. To reject mathematics for philosophical reasons would be absurd [...] I’m moved to laughter at the thought of how presumptuous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways, and abjure countless errors now that philosophy has discovered that there are no classes?

Philosophy simply has not got the track record of certainty, or utility, or progress, or unanimity, to mount any such high horse. If it is a question of philosophy versus physics, or philosophy versus maths, everyone knows which side to back.

That was Blackburn on Lewis on Philosophy, in Moore and Scott (eds.) Realism and Religion: Philosophical and Theological Perspectives (2007: 49, where he was trying to draw a line between the expertise of scientists and that of theologians).

    Mathematical classes necessarily go beyond standard set theory, as a result of the famous set-theoretic paradoxes of a hundred years ago. For the most part, their proper study is regarded by mathematicians and philosophers as the province of logicians and metaphysicians. And even the sets that mathematicians study are formal, axiomatic entities entirely suited to the algebraic ways of the mathematicians. Some mathematicians even work on constructive maths or category theory, for example, and insofar as they do so formally they are regarded as doing maths. In short, the pure mathematicians have passed the metaphysical buck to those applying the formal structures studied by them, while few applied mathematicians regard themselves as doing set theory.

    Lewis doesn’t accept that there are waves or particles or forces or fields, but only spaciotemporal points with arbitrary properties that either happen to fit a pattern or don’t. He’d let us keep whatever physicists say about such things, but would change their meanings (such being his idea of what philosophers can do). But those spaciotemporal points have, he presumes, a structure isomorphic to some set-theoretical structure. He might even allow that they could have any structure, but he needs some such points, and what if such structures happen to be metaphysically impossible? Standard mathematicians actually pass the buck on such applications.

    Lewis was talking about standard set theory, not mathematics (the two are often confused by American academics and those—most of the world’s leading academics—who’d like to become one), and sets have never got over the set-theoretical paradoxes. Or rather, the mathematicians—for the most part (Lewis conveniently ignores those mathematicians who’re constructivists or category theorists)—got round the problem by doing axiomatic set theory in an algebraic way, and thereby left the problem of interpretation up to those applying their maths. If there turn out to be no infinite sets really—no such metaphysically possible spatiotemporal points as Lewis presumes—then there would need to be some applicable mathematics done by someone, and whoever did it would be a mathematician. Lewis ought to be as sceptical about set theory as he is about particles and natural laws.

    Lewis may be moved to laughter at the very idea—a common response to his own hopelessly unrealistic work on metaphysics—but what was he thinking of? Brouwer and Heyting were professional mathematicians, and those philosophers (e.g. Wittgenstein and Dummett) who agreed with them were agreeing with mathematicians. So what was he thinking of, the maths that he was taught at school in America? If philosophers have something to contribute then they should contribute it, and if not then they’re philosophers in name only. Was Descartes a philosopher or a mathematician? He was clearly both, and we surely need more people like Descartes, not fewer. Scientists tend to say that to say nothing of God isn’t to say that God is nothing, and those who think otherwise do so for philosophical reasons.

    Mathematicians have as much right as anyone to think philosophically, and perhaps more right when it’s the philosophy of maths (e.g. Hamming and Fletcher). And what’s especially interesting nowadays is the increasing popularity of category theory, within maths. For the most part, that increasing popularity is not due to metaphysical concerns, but to concerns more internal to maths. But the underlying logic of category theory is intuitionistic. One can even envision a day when the professional mathematicians choose category theory as their standard foundation (one need only think of how popular Lewis is amongst professional philosophers) just because it provides the most interesting line of immediate research (cf. the dominance of string theory in physics) et cetera ad nauseum.

    Note that there’s some difference between philosophy the professional job (cf. Lewis’s concern with having the job of saying what some of his colleagues have said), which does seem shaky, and philosophy the pursuit of truth.

Divine Liars: the devil's in the details

Hartley Slater falsely supposed that I think there’s no essential difference between the sentential and the propositional formulations of (strengthened) Divine Liars, at the start of his ‘Supposed Liars, Divine Liars and Semantics’ (The Reasoner 3(1), 3–4). My ‘Liars, Divine Liars and Semantics’ (The Reasoner 2(12), 4–5) had merely questioned a presumption of Patrick Grim’s sententially formulated (4) = ‘God doesn’t believe that (4) is true.’ Grim had presumed without comment, let alone justification, that those two instances of ‘(4)’ could be the same proper name. And although such naming practices are fairly standard nowadays, that’s why the idea that their legitimacy is challenged by the paradoxicality of their application to Liar-style sentences could be of some interest.
......Propositional reformulations, such as my tentative (4*) = ‘God doesn’t believe that (4*) ever expresses a true proposition,’ raise different and more difficult considerations, which I’d simply ignored in order to state my question as clearly as I could within a thousand words. An obvious difference is that although a semantic problem with sentences is prima facie a problem with propositions, and vice versa, hardly anyone argues over whether sentences exist. Less obviously, although when considering (4)—which can be so-named even if I’m right about the illegitimacy of that naming practice, if the occurrence of ‘(4)’ within (4) isn’t also as (4)’s name—it had seemed acceptable to implicitly presuppose some ordinary linguistic notion of literal truth, perhaps I should make that notion more explicit when properly reformulating (4) propositionally. So instead of (4*) consider (4**) = ‘God doesn’t believe that the modern English sentence (4**) could ever express any true proposition literally.’
......That’s still pretty tentative, e.g. the modality might be difficult to explicate, and even literal truth is notoriously difficult to define. ‘Snow is white,’ for example, literally means that snow is white, and is true if indeed snow is white, but it’s hard to say what such repetitions mean in other, more general words. Still, there’s presumably some such semantic rule, say (T), that’s both true enough and applicable to such sentences as (L) = ‘This sentence is not (now) true’ (that parenthetical ‘now’ is there to reduce the risk of equivocation).
......Suppose, if only for the sake of reductio, that (L) was expressing something literally. What (L) would then be expressing would, via (T), at least include that (L) isn’t true, which is (in view of the self-reference) that it isn’t true that (L) isn’t true, which is (via double-negation elimination) that (L) is true. Now, what I’ve just shown is, I think, that the last two italicised expressions would be expressing the same proposition if (L) was being used to express any proposition literally. And maybe I’ve effectively shown that (L) expresses no proposition literally, i.e. that it’s nonsense. But I’ve certainly not thereby allowed that sentences may express more than one proposition. Recall that Slater said (ibid, 4):

Cooke says, with regard to ‘Liar sentences’, that ‘they do seem to be saying, not only that they are not true, but also, if less obviously, that they are (therefore) true’. So sentences, he allows, may express more than one proposition, even if they may express one proposition more obviously than another. But if so then one cannot immediately derive, with respect to the previous case that the (one and only) proposition that (4*) expresses is (the obvious one) that God doesn’t believe that (4*) ever expresses a true proposition.

A sentence may of course express different propositions, e.g. literally and analogically, or by being equivocal, or when it’s expressed by different people, or at different times or places, or because the language in which it exists changes, etc. But Slater will, I suspect, have difficulty indicating what other proposition could have been expressed by (4*)—or better, (4**)—literally. If he has to use different words to those of (4*), then is it really expressed by (4*)? And if he doesn’t, then why wasn’t it expressed when he used those same words to express the ‘obvious’ proposition?
......Furthermore even if one may, in such a way as Slater’s (above), point unambiguously to one proposition, something like the original problem with (4) would arise. Note that if (4**)—or (4*)—is, as I believe it is, nonsense, then it doesn’t express any proposition literally, and so ideally one wouldn’t be able to derive that it does. But if, counterfactually, (4**) was expressing anything literally, then presumably by (T) that would be the proposition that God doesn’t believe that the modern English sentence (4**) could ever express any true proposition literally. And that does seem like a proposition to me, if only because I’d assert it, in so many words, on the grounds that (4**) is literal nonsense and that God would, were he real, be wise enough to know that. So, whether or not any other propositions are expressible by (4**), literally or otherwise, we’ve something like the problem that (4) presented to theists. That is, if (4**) could be used to express a true thought literally then God doesn’t believe—via the truth of that (last italicised) proposition—that it could be so used, whereas if (4**) couldn’t be so used then that proposition is true, whence any completely omniscient being would believe it.
......Something like it; but as I say, it may be that a reformulation more precise than (4**) is required to yield a paradox sufficiently close to that of (4). In fact, I happen to be agnostic about whether that’s even possible, about whether there’s at least that essential difference between the two sorts of formulation. But the point is that the traditional resolution of Liar-style sentences—that you can’t, by talking nonsense, say that you aren’t telling the truth—applies to any legitimate formulation, and would in particular apply to sentential formulations, were they allowed.