Vaguely True Liars

My next 4 posts explore, informally and briefly, the possibility that Liar utterances are about as true as not. Abstract:

I suggest that Liar statements – e.g. ‘this is false’ – are about as true as not. In other words, they are vaguely true, and vaguely false. And truth does seem to come in degrees (e.g. if a colour is about as blue as not, calling it ‘blue’ would be about as true as not). I also suggest that ‘this is not even vaguely true’ can be called ‘vaguely true’, even though it may then seem false, not just vaguely false. That’s because it is, more precisely, vaguely more vaguely true (similarly, a colour that’s roughly as blue-green as it is blue would usually be called ‘blue’ becaue it is a faintly greenish blue). I give similar resolutions to the paradoxes of Kurt Grelling and Keith Simmons, while I regard as of a different kind those of Bertrand Russell and Haskell Curry.

Links to the 4 posts:

......Introduction (via Grelling’s paradox)

......Liar statements are about as true as not

......My ‘revenge’ paradox (and Yablo’s paradox)

......Russell, Cantor, Curry, and Simmons’ paradox

An application of the above is a common sense refutation of the Divine Liar argument (against omniscience), see my Liars, Divine Liars, and Semantics revisited (in April’s issue of The Reasoner).

Curry's paradoxes

The Liar paradox concerns utterances such as ‘what I’m saying isn’t true,’ which is, if true, not true, and which seems true if not. Another way of saying the same thing would seem to be to say ‘if what I’m saying is true, then pigs fly.’ Yet that utterance is paradoxical in a way so different that not only has it a different name – Curry’s paradox – it’s debatable whether its resolution should even resemble that of the Liar.
......Suppose I say ‘if what I’m saying is true, then P,’ where ‘P’ stands for any proposition. For simplicity, let’s say that C is the assertion that C implies P. Suppose, just suppose, that C is true. We are thereby supposing that C implies P. So we would also have P. That much is simple enough. Given C, and that if C then P, we get P. And yet that much is too much. If it’s true that by supposing C we also get P, then C really is true, and hence P is true, even though P could be asserting anything at all (even that pigs fly).
......That seems quite unlike the Liar. E.g. there was no ‘not’ in the previous paragraph: We didn’t consider C being either true or else not, and find both possibilities inadequate; nor did we see C seeming to say that it wasn’t something that C did indeed seem not to be. Rather, just by wondering what C was – in particular, whether C might be true – we seemed to get P. And so by deriving actual being from mere possibility, Curry’s paradox seems to be more like the Ontological Argument than the Liar.
......On the other hand, if it follows from the meaning of ‘not’ that either A or not-A, where ‘A’ stands for any proposition (e.g. that it’s raining), then it seems that if A implies B (e.g. that I’m carrying an umbrella), then either B or not-A (either I’m carrying an umbrella or it’s not raining). And conversely, if it’s the case that either not-A or B, then if it’s A (if it isn’t not-A), it must be B. So in short, C could seem to be asserting that either not-C or P. And then P effectively disappears if it’s false, leaving C asserting not-C.

Simmons' paradox

The following, which is akin to the Liar paradox, is a paradox of self-reference by Keith Simmons, described on p. 231 of his ‘Reference and Paradox’ in JC Beall (ed.) Liars and Heaps (Oxford 2003):

Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.

Here is what I think about those denotations:
......The first expression, ‘pi’, referred to 3.14159..., and the second to 6. So if the third expression does denote a number, say N, then N = N + 9.14159... Given that by ‘number’ we mean finite number, it seems that the third expression can’t denote a number. So those three expressions denote only pi and six. But then the sum of the numbers denoted by those three expressions is 9.14159..., and so the third sentence does seem to denote a number after all. Or rather, it does because it doesn’t; and furthermore, it seems to denote 9.14159..., but therefore it seems to denote 18.283..., or rather, 27 and a bit, etc.
......I have, however, been implicitly assuming that reference is an all-or-nothing affair. Usually we can – and indeed, should – take it to be so, but is it so in general? Imagine, for example, a man staggering through a desert. He sees a mirage, which he takes to be a pool, and as it happens there is a pool, just where he takes one to be, but it’s obscured from his view by the mirage. As he staggers towards it, he’s constantly thinking ‘that pool looks cool’. As he nears the pool, its image gradually replaces the illusory one, without him noticing, so that the referent of ‘that pool’ gradually changes to the pool. And so at some point he may have referred only vaguely to it.
......Such a case would of course be exceptional, but so are scenarios designed to be paradoxical. And it does at least seem possible that the third expression of Simmons’ paradox referred only vaguely to 9.14159... It would also have referred, even more vaguely, to 18.283 (and so on), but while that’s even odder, it too seems possible. And if the alternative is paradoxical, vague reference may not be too odd. And such a resolution would cohere with ‘this is not true’ being vaguely true (about as true as not), and ‘is heterological’ being as heterological as not (see my previous posts this year).

God is Timeless

When people say that God is timeless, they may mean many things. They may mean that He is above and beyond the mundane world, like the truths of mathematics, for example. But they would not then be disagreeing with Open Theism:
......Under Open Theism, God is certainly above and beyond His creation, a bit like a dreamer and the dream he finds himself within. And one of the few coherent philosophies of mathematics is Open Theistic Constructivism, in which God, being omnipotent, creates the truths of mathematics—from the basic concepts of a thing and of possibility (the latter grounded in His omnipotence)—doing so endlessly because such is the nature of the former concept, according to the resolution of Cantor’s paradox that takes it to be showing that cardinal numbers are collectively indefinitely extensible (one of the few coherent resolutions).
......Of course, they may instead mean that God is not changeable. But even that isn’t incompatible with Open Theism, under which God cannot change his essential properties. Nor can you, of course. You can’t become me, for example. You might change by becoming in a manner of speaking a different person (i.e. your character might change, for better or for worse), but under Open Theism God’s character remains perfect. So the only disagreement with Open Theism could be that such philosophers are denying that God could choose to cause any real change in anything; and not only is that clearly not what most believers mean when they say that God is timeless (is rather more like an absurd denial of the reality of change), such philosophers would be denying God’s omnipotence:
......The deliberate creation of anything contingent surely requires several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change (not necessarily one that takes place within spacetime).

Only the unfit evolve

From Philosophy Now, I learn 'that people and organisations that miss their goals disastrously perform better in the long run.'

Professor Desai, who led the study, said “knowledge gained from success was often fleeting while knowledge from failure often stuck around for years.” [...] He says failure causes a company to search for solutions and it puts the executives in a more open mindset. He doesn’t recommend seeking out failure in order to learn.

...which makes me think of the merchant bankers. Will they perform better now? Only if we make them (despite our politics being dominated by the short-term), I think. You know, their performance was always chaotic, even in the good old days, as the new maths of chaos showed us in the 80's. But they took past success to be indicate a propensity for success; ironically, they were bad at applied math.
......I find that ironic because it was pointed out to me, when I was reading physics (in the 80's), that investment banking was the sort of 'good job' that a physics degree qualified one for. (That was what made physics such a good degree.) At the time, I wondered whether we really would've been a better society had more women wanted to play such games. (Margaret Thatcher, with her Chemistry degree, hardly thought of investment banking as a waste of a physics degree.)
......What can you do? The young are always with us. And we can hardly want to dilute democracy, but, why not give more experienced voters additional votes? Perhaps we shouldn't take the vote away from those who've shown themselves to be very selfish (unless we've locked them up and thrown away the key), but why not give an extra vote to all those who haven't (yet) shown themselves up; indeed, why not give extra votes to those who deserve honours? Wouldn't that be fairer? (It would certainly be safer.)

Philosophers' Carnival #121

Welcome to Philosophers' Carnival #121. Not so much a one-to-one as a party (a rather Platonic party). The bouncer's been busy (arguably too busy, or not busy enough). And there's some of the hard stuff later on; so, enter...


......the Hallway (epistemology)

Keith DeRose imagines 'Nico at the Zoo with Zebras' (at Certain Doubts), an example of how knowledge attributions can sometimes be true even when they’re about insensitive beliefs.

John Wilkins compares Elliott Sober's 'Modus Darwin and the *real* modus darvinii' of 'affinity, explained by common ancestry' (at Evolving Thoughts), showing that the former should've been the latter.

Maryann Spikes thinks of 'Atheism and agnosticism (really, apisticism) as belief' (at Ichthus77), and also thinks that you can only be apistic if you don't claim to be.


......the Games-room (logic and language)

Ben Nelson wonders about 'Trust as a truth-maker' (at Talking Philosophy). Of course, "X trusts me" is made true by X trusting me, but Nelson takes a broader (even deeper) look at this kind of thing.

I ask 'Is 'pretty' pretty?' (It is, and it isn't:)

Matt asks 'How slippery is the slope?' (at The Consternation of Philosophy), and concludes that 'the slippery slope fallacy is a slippery beast, and is perhaps best not thought of as a fallacy at all.'


......the Dining-room (metaphysics)

Edward Feser asks 'Why are (some) physicists so bad at philosophy?' (at Edward Feser), and...

Eric Steinhart wonders 'Why Materialism is Unscientific' (at Camels with Hammers), both in response to astrophysicist Ethan Siegel asking 'Can You Get Something For Nothing?' (both answer No).

Jeremy Stangroom wonders, 'A First Unmoved Mover?' (at Talking Philosophy). He shows Copleston and Russell describing the Atomists differently; not because either was bad at the history of philosophy, but because good answers to the question Why should there be something rather than nothing? do include Because the metaphysically necessary being is perfectly good as well as the (now) obvious No reason.


......the Living-room (mind)

Kenny Pearce knows that 'Sometimes it's Rational to act Arbitrarily' (at Kenny Pearce). 'In ordinary cases it is irrational to take a certain course of action when you know there is a better one available to you,' but what if you are asked to choose any natural number of dollars? (Sobel thinks that choosing anything would still be irrational; why would he think that?)

Constantine Sandis entertains 'Enchanting Causes' (at Flickers of Freedom), and so 'tests our intuitions about what sort of desire makes an action intentional.'

Joel considers 'Killing a Vegan: Degrees of Subjectivity' (at Florida Student Philosophy Blog), arguing that chickens (as opposed to Vegans) may not feel phenomenological pain, because they don’t have the 'I' concept, or the neurological ability to do much more than react physically.


......the Kitchen (moral philosophy)

Robin Hanson asks 'What Virtue Privacy?' (at Overcoming Bias), and by discussing Thomas Nagel's 'Concealment and Exposture' argues 'that humans had huge heads to subtly evade social norms while pretending to enforce them.'

Tim Dean considers 'Morality, Health and Sam Harris' (at Ockham's Beard), arguing that Harris's moral realism makes naturalism harder to defend, and suggesting that we could just say that 'Being animals, we pursue health. And being social, we pursue morality.'

James Gray defends moral realism against Hume, by considering 'Intrinsic Values & Beliefs About Reality' (at Ethical Realism).

Antti Kauppinen explains 'How the Experience Machine Works' (at Experimental Philosophy), before objecting to Felipe de Brigard’s recent ex-phi objection to Robert Nozick’s result.

Richard looks at 'Natural Agents and Status-Quo Bias' (at Philosophy, et cetera), questioning Carolina Sartorio, who argued (via Trolleys) 'that we need stronger reasons to justify interfering in a process (e.g. deflecting a trolley) than to justify abstaining from such involvement.'

Clayton Littlejohn has 'Ethical Intuitions (II): Cosmic Coincidence' (at Think Tonk), the second in a series of posts on moral epistemology (some empirical arguments having been considered in 'Ethical Intuitions (Part I)'): A version of intuitionism on which moral properties supervene upon natural properties is defended against Matthew Bedke.

Jussi Suikkanen conjoins 'Deliberative Contractualism and the Conditional Fallacy' (at PEA Soup), arguing that the former (by Nicholas Southwood) commits the latter.

Thom Brooks announces 'Thom Brooks on "Punishment: Political, Not Moral"' (at The Brooks Blog). British Hegelians make Alan Brudner's retributivism more attractive, apparently.

Chris Bateman considers 'A Categorical Imperative for the Other' (at Only a Game), suggesting that 3 formulations of Immanuel Kant's categorical imperative are more easily seen to be equivalent if they (or something like them) are derived from Emmanuel Levinas' concept of the Other.

Anders Sandberg wonders how much 'Intolerance we ought to encourage?' (at Practical Ethics). 'At the very least we can make it a social rule that just as we frown at racist, sexist or homophobic statements we frown at pseudoscience or deceptive evidence.'


......the Backyard (other)

Paul Newall examines different views of 'Astrology and its problems: Popper, Kuhn and Feyerabend' (at The Kindly Ones), and suggests that 'the philosophical problem for astrology is not that it can always explain failures (Popper) or that it does not attempt to solve problems (Kuhn) but instead that it has stagnated (Feyerabend).'

Kieran Healy looks at 'Gender divides in Philosophy and other disciplines' (at Crooked Timber). More than 70% of US PhDs in psychology were awarded to women, and it's 60% in sociology. Still, it's only 40% in political science, and 30% in philosophy.

Brian Leiter also looks at 'Women in Philosophy in the US' (at Leiter Reports), and finds that the proportion teaching philosophy is only about 20%. That's pretty good, given our context (it's more than 5 times the proportion of female bloggers here).

Gary Williams has some 'Thoughts on Cordelia Fine's new book Delusions of Gender' (at Minds and Brains); e.g. 'Maybe 1000 years in the future there will be an equal amount of male and female physicists, philosophers, and computer scientists,' because our brains are (equally) plastic.

James Warren reveals 'Rejection letters of the ancient philosophers' (at Kenodoxia). Bitchin'


.........is there 'More on philosophy and society'? No, because the party's over (hopefully before the fighting starts). Almost all-male, and the kitchen the most popular place; what a party. But if this carnival bored (or annoyed) you, or if your entry bounced (for no good reason), the solution is to host a carnival, which you should also do if you liked this one, of course: No hosts = no carnivals.
......And whenever you find yourself reading an interesting post, of a philosophical nature, you should submit it, because no posts = no carnivals. Carnival #122 will be at Ichthus77

Is "pretty" pretty?

Are any words pretty? Maybe not (outside of calligraphy or song). But I’m reluctant to say that “pretty” isn’t pretty because it is not too odd-looking (as words go) and it does make us think of prettiness. Still, I am reluctant to say that it is pretty, and so it seems to me that “pretty” is about as pretty as not. Perhaps you think that “pretty” is pretty. Or perhaps you think that “pretty” is not pretty. But if about as many people think it is as think it is not, and lots of people have no strong opinion either way, then a good case could be made for “pretty” being about as pretty as not.
...... Descriptive accuracy is, in general, a matter of degree. E.g. “is long” is not a long English predicate, while “is so far from being short that, not only is it not short, it is rather long for a predicate written in ordinary English” is quite a long English predicate, and so it seems fairly plausible that there might be some predicate that means the same as “is long” and which is about as long as not. For just one more example, “is boring” seems a little interesting, now that I come to think about it, although it does not hold my interest for long.
...... Words are heterological if they do not describe themselves, and so it seems to me that “pretty” is about as heterological as not. And if words are heterological insofar as they don’t describe themselves very well, then the fact that if “heterological” is heterological then it is not heterological, and the fact that if it is not then it is, those facts show that “heterological” is about as heterological as not. Those two facts are called the Grelling-Nelson paradox, and they are a version of Russell’s paradox.

The most significant version of Russell’s paradox concerned collections. When we refer to some things collectively, we are referring to their collection. E.g. “all the words in this post” refers us to all of the words in this post (including those words). Some collections include themselves (e.g. the collection of all the collections would include itself), but most do not (e.g. the collection of all the words in this post is not itself a word in this post). Russell’s paradox is that the collection of all the collections that do not include themselves would include itself if it did not include itself, and would not include itself if it did. Mathematicians tend to think that Russell’s paradox shows that we should only use well-defined kinds of collections. But if we could not talk about things that we were finding it hard to talk about, we would never learn anything.

A similar resolution of the Grelling-Nelson paradox would be that it shows that the word heterological is not a well-defined word. Perhaps Grelling and Nelson (who were mathematicians) should not have introduced that word. But what is wrong with saying that “heterological” is about as heterological as not?

Classical Logic: how is it correct?

According to Stewart Shapiro, Classical Logic is first-order (and hence formal) predicate logic, which he describes in some detail (in that SEP entry), having first noted that:

Formal languages, deductive systems, and model-theoretic semantics are mathematical objects and, as such, the logician is interested in their mathematical properties and relations. Soundness, completeness, and most of the other results reported below [in that SEP entry] are typical examples. Philosophically, logic is the study of correct reasoning. Reasoning is an epistemic, mental activity. This raises questions concerning the philosophical relevance of the mathematical aspects of logic. How do deducibility and validity, as properties of formal languages--sets of strings on a fixed alphabet--relate to correct reasoning? What do the mathematical results reported below [in that SEP entry] have to do with the original philosophical issue?

Shapiro goes on to list some possibilities; e.g. perhaps "the components of a logic provide the underlying deep structure of correct reasoning."
......Another possibility is that "because natural languages are vague and ambiguous, they should be replaced by formal languages," or rather (since formal languages are all defined using natural languages) "regimented, cleaned up for serious scientific and metaphysical work." Now, scientists often do define their own scientific terms; but how could that process apply to logic? Informal logic must be good enough for us to work out the correct formal language to use, if there is one (otherwise our justifications would become circular). "Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects." Such a view makes sense, in view of the many logics studied by logicians; but what, then, can we say about logic in the sense of correct reasoning? We use such a logic when we use any mathematical model scientifically; we must reason correctly about the model. Is classical logic a good model of informal logic?
......Despite the various logics that logicians work on, most arguments are presented in a classical logical style (even those about other logics). So presumably we do think that such components cut our reasoning pretty close to its joints, so to speak. Is classical logic correct? Some philosophers say so (e.g. Alexander Pruss recently gave that as a reason for rejecting open future views, in comments on this post of his), but I wonder how it is. It's hard for me to specify my worries without having already resolved them; but let's look at some simple logical arguments, such as might be used to introduce classical logic, and see how they may fail to be examples of classical logic (without going too far into the related problems of metaphor and vagueness).
......Grass is green, and all flesh is grass, so, is all flesh green? That's clearly invalid, the second premise being metaphorical; but did we use classical logic to work that out? And suppose I was holding something that wasn't green; could I deduce that it wasn't grass? Again no, because it's not true that all grass is always green. But what if all Blurps were always green; could I deduce that I wasn't holding a Blurp? I don't see why not. And surely I could put that argument in classical logical form. And yet if "is green" is a classical predicate, then not only what I'm holding, but anything and everything is either green or else not green (LEM); whereas, in reality, there's clearly a shading from green, through greenish and vaguely greenish colours, to those that aren't green. Is it that classical logic is a good model, but that real (informal, or natural-linguistic logic) must be slightly different? If so, what's the point of all that maths that Shapiro introduces (rather well)? I'm not saying there's no point to it. (I'm hoping it's not turning philosophy into a pseudo-science.) I'm rather asking whoever's reading this post, what do you think about classical logic?

Reasoning badly from Yablo’s paradox

Paradoxes can be hard to resolve, so it can be hard to reason well from them. A nice example is a recent argument that the past is finite, by Laureano Luna (2011: ‘Reasoning from paradox’, The Reasoner 5(2), 22–23). I shall vary the details. Suppose there’s a place where, once a year, every year, someone says “No previous utterance here was true,” with nothing else ever having been said there. Details can be varied, so long as we would, were the past infinite, have an infinite sequence of similar utterances. Indeed, it’s because we can vary the details that the following contradiction seems to follow from supposing the past to be infinite (rather than from, say, supposing that language need not begin with evident truths).
......Each utterance in that place concerned the past, so it seems that each utterance should be either true—were none of the previous utterances there true—or else false—were at least one of them true. But none of them can be either, on pain of Yablo’s paradox: Were no utterance there true then, via what each said, each would be true (and not true); but were any of them true, then since none of the earlier ones would have been true, its immediate predecessor would also, via what it said, have been true (and not true).
......But even if such sequences of utterances are impossible, the past might be infinite. One possibility is that simply infinite sequences, e.g. the natural numbers, are indefinitely extensible (are Potential Infinite) in the sense that while there’s always a next element, e.g. a bigger number, there’s no complete collection of them all. Standard mathematics assumes that such isn’t the case, but we’ve yet to discover that it isn’t. And if it is, then although we naturally think of past years as stretching back in time forever, the past couldn’t be the whole of such an infinite sequence, and so our infinite sequence of utterances would’ve been impossible too. And yet the past might, even so, be infinite. E.g. there might have been, before the Big Bang, some infinitely slow process, which took an infinite time to complete (and before which there might have been something else, possibly with no beginning); such a process has an infinite duration in the sense that we might go any natural number of years back into it and not reach its beginning, and also in the sense that were it the unit of time, all the time since the Big Bang would be relatively infinitesimal.
......Another possibility is that truth, or descriptive accuracy, is essentially a matter of degree. We might take an utterance of “No previous utterance here was true” to be asserting that none of those previous utterances described the past well enough for it to be classed as true. Yablo’s paradox would then be ruling out every possibility except the possibility that all those utterances described the past only vaguely, that they were all vaguely true. (Does it seem that they would then have been failing to describe the past well enough to be classed as true? If so, note that the suggestion is that either classification—true or not—would be less accurate than that of vaguely true.) So, our contradiction may well have been due to our having used, in effect, a rather artificial language. So we seem to have shown only that either the language of those utterances doesn’t allow such sequences of sentences, or something else (e.g. maybe the past must be finite, or maybe the natural numbers are indefinitely extensible).

Philosophers' Carnivals: Now & Next

Philosophers' Carnivals "showcase the best philosophical posts from a wide range of weblogs," as it says on the carnival's homepage. From today, carnival #120 is at nicomachus.net. And carnival #121 will be here in 3 weeks time, so if you find yourself reading something nicely philosophical, posted between now and then, please consider submitting it, via the online submission form, even if you wrote it yourself: "Don't be shy, we want to hear from you, that's the whole point of this project! Your post doesn't need to be anything earth-shattering - it just needs to be something that other philosophically-minded people might enjoy reading."
......As for what you can submit, there are No Rules, except: "No self-help, mysticism, marketing spam, etc." Of course, marketing spammers are unlikely to have bothered reading as far as this, so telling them not to bother submitting seems pointless. And I wouldn't rule out what some academic philosophers like to call 'mysticism', e.g. Mathematical Platonism, Substance Dualism, Open Theism and so forth (since such is just realistic metaphysics). Nor shall I reject whatever formalized craziness such academics work on instead, of course (since I should be unbiased in my hosting). Indeed, since the number of the carnival will be 121 (which sounds like "one-to-one") there's even some hope for self-helpers (and Continental Philosophers) whose positive thinking has carried them thus far, because insofar as their posts describe how the ideal of the Socratic Dialogue relates to their brand of self-help (or Derrida) I shall look upon them kindly.
......Here's a cautionary tale about rule-following: Many years ago, a port on the east coast was industrializing. To its north and south were two large estates, the country houses of two progressive squires, who built factories and docks in the port, and cheap housing for their workers there. Peasants to the west of the port flocked there, to earn more and to be free from their old-fashioned and relatively oppressive squire. As his peasants deserted his lands, that squire soon found himself with cashflow problems, and eventually he was reduced to opening his mansion to the public. He even built an inhumane zoo in its overgrown grounds; but things got no better. He got more and more depressed. One day he became quite deranged, and smashed up his zoo. Then he climbed onto the back of a huge hippopotamus and rode it towards the port. Now, the two rich squires heard of him crashing through their workers' slums, but they were unable to stop him because he had the law on his side, the law which states that the squire on the hippopotamus is evil to the slums of the squires on the other two sides.

Liars, Divine Liars and Semantics revisited

Divine Liar arguments aim to show that there’s no omniscient being—that no one knows all that’s true—in the following way. Suppose I say “No omniscient being knows that what I’m now saying is true.” If (as I believe) no one is omniscient, then no omniscient being exists, to know anything. So in that case, what I said was true. What I said was therefore an assertion, whether it was true or not. And if it wasn’t true—if it’s not the case that no omniscient being knows that what I said was true—then some omniscient being knows that what I said was true, despite it not being true, which is impossible (knowledge being of truths). So I asserted a truth; and so either that was a truth that some omniscient being doesn’t know, which is also impossible, or else there’s no such being.
 ......However, resolutions of the Liar Paradox might show that such arguments are invalid, e.g. according to Daniel J. Hill (2007: ‘The Divine Liar Resurfaces’, The Reasoner 1(5), 11–12) and my earlier article (2008: ‘Liars, Divine Liars and Semantics’, The Reasoner 2(12), 4–5). So, suppose I say “What I’m now saying isn’t true.” If what I said was true then, as I said, what I said wasn’t true. Does it follow that what I said wasn’t true? The paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. The resolution defended earlier by me (2008) takes my utterance to have been meaningless, so that I didn’t really say anything. But we may then wonder how it was that it seemed so clear what my utterance would have meant had it been true; and my Divine Liar utterance was even more obviously meaningful. Another popular resolution would regard my Liar utterance as equivocal, with the word ‘true’ naming many different predicates in Hill’s (2007) Tarskian hierarchy. But formal languages can only be defined via natural language; and my informal Divine Liar utterance wasn’t obviously that equivocal.
 ......Questions of truth are essentially questions of how well our words are describing the world. So insofar as my Liar utterance wasn’t meaningless, it was asserting that it wasn’t describing itself very well, not well enough for it to have been true. And since it was nothing if not self-contradictory, it certainly wasn’t describing itself very well. But therefore, in view of what it was asserting, it seems to have been describing itself quite well after all. Was it describing itself well enough for it to count as true? I’m reluctant to call it ‘true’ as follows. If it was true because it wasn’t, then it was true and not true, but surely something’s only not some way if it’s not the case that it is. Nor do I want to say that it was neither true nor not true, as that’s just to say that it was not true and also true. Nevertheless, my utterance wasn’t describing itself very well, and was therefore describing itself quite well; so perhaps it was only partially true. If so then calling it either ‘true’ or ‘not true’ would both be inaccurate, would both be only partially true.
 ......We naturally focus upon whatever truth we can find in what people say, or upon an obvious untruth. And things are usually described accurately enough for some obvious purpose, or not accurately enough. But would it be unrealistic to think of truth (descriptive accuracy) as a matter of degree? The classic example is that of Vann McGee (1991: Truth, Vagueness, and Paradox, Hackett, 217): If “Harry is bald” is true insofar as Harry is bald, ‘true’ should be at least as vague as ‘bald’. And quite generally, why should we believe that our words are much better defined than our purposes have required them to be? Maybe natural language has a ubiquitous—since usually unobtrusive—vagueness. (That would explain why the discovery of a contradiction so naturally triggers an attempt to clarify our terminology.) And in particular, the Liar Paradox might be revealing this ordinarily obscure vagueness of ‘true’. That’s because if my Liar utterance was only partially true, then it would follow from what I said only that it was also partially not true, which clearly coheres with it being only partially true. There’s no inconsistency—no more paradox—and it seems that much the same could be said of any Liar sentences.
 ......And if that is how the Liar Paradox should be resolved, then my Divine Liar utterance would have been only partially true if there is an omniscient being. My Divine Liar argument was therefore fallacious, because arguments should have premises that are unequivocally true enough to count as true under all relevant hypotheses. But if you asked an omniscient being whether my Divine Liar utterance was true, she might say that it contained an element of truth. That might be a more informative—more true and less misleading—answer than a simple ‘yes’ or ‘no’.
 ......Similarly, the best answer to the question “Is this colour blue or not?” could be to say that it’s vaguely bluish. Ordinary objects are almost always either blue or not, but colours don’t really divide into those that are blue and those that aren’t. On the two sides of any such line, between the blue and the other colours of some spectrum, would be colours that were indistinguishable. So there’s no such division; and so there’s some colour of which, rather than saying that it’s blue, or that it isn’t, we ought to say that it’s bluish. Note that such a colour might look blue against a background of colours that weren’t blue, or even if you just wondered whether it belonged to that class of colours, and so postulated it amongst them (cf. what we find paradoxical about the Liar Paradox).
 ......Incidentally, some formal work on ‘true’ as a vague predicate is well described as Fuzzy Logic.

Liars Are Fairly True

Suppose I say “what I’m saying isn’t true.” If what I said was true, then as I said, what I said wasn’t true. Does it follow that my words weren’t true? The famous paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. A fairly popular resolution takes my words to have been meaningless, so that I didn’t say anything. But if my words had been meaningless, you could hardly have known what they would have meant had they been true. Is our ordinary conception of truth shown by such Liar-style sentences to be deficient? Let’s see why not.
......To begin with, such sentences are in some ways like Truth-teller-style sentences. If I said “what I’m saying is true,” for example, what would I be saying? Not much. Questions of truth are essentially questions of how well our words describe the world, and “this is a good description” isn’t much of a description. Still, it might not be too bad a self-description, precisely because there isn’t much to describe. If someone saying “what I’m saying is true” intended to be speaking the truth, should we deny that she was telling the truth? It may be hard to say, but therefore it might be that such sentences are not so much vacuous as vague. Since “what I’m saying isn’t true” also addresses nothing but its own descriptive power, might it also be, in its own way, rather vague? Consider the following analogy.
......If I said of some colour, “I wouldn’t say that it’s blue,” I might not be saying that it wasn’t blue, because colours don’t divide into those that are blue and those that aren’t. To see that, consider a spectrum: On the two sides of any such line, between the blue and the other colours, there would be colours that were indistinguishable. So there’s no such division; so there’s some colour of which, rather than saying it was blue, or that it wasn’t, I’d prefer to say, more precisely, that it was bluish but not very blue. (Since the perception of colour is subjective, you might say it was blue, or that it wasn’t.) Our perception of colour is also context-sensitive, e.g. it’s affected by surrounding colours, and by our preconceptions. So if I wondered if our colour really was blue, I might thereby see it as not blue, while if I then wondered if it was therefore not blue, it might seem pretty blue (even to me).
......And similarly, it’s when “what I’m saying isn’t true” has been thought of as definitely not true that it seems most clearly to be true. More precisely, while those words aren’t giving us a very good description of their own meaning—they’re self-contradictory—we therefore have a description that isn’t too bad, insofar as it’s saying that it’s not a very good description. In short, they’re rather nonsensical (and false), but therefore fairly true (and false). And that’s basically how Liar-style sentences are compatible with our ordinary conception of truth. We need a bit more clarification, but it should soon become clear that while we can always be more precise, there’s no threat to truth here.
......What is truth, if not a sufficiently accurate description? Usually we describe things accurately enough for some obvious purpose, or else we don’t, so we tend to assume that truth is black-or-white. But it’s really a matter of degree, in a context-sensitive way. E.g. the table at which I’m writing this is flat enough for that purpose, so “this table is flat” is true enough, but might be false were I writing about geometry. And in general, our words tend not to be much better defined than our purposes have required them to be. So natural language has a ubiquitous—since ordinarily unobtrusive—vagueness (whence the way to resolve paradoxes, and uncover other fallacies, usually involves clarifying some terms). Of course, the words of “what I’m saying isn’t true” have clear enough meanings, so there’s no simple equivocation to discover. But it should help us to resolve the paradox if we don’t demand anything too unrealistic. (Similarly, we shouldn’t demand that colours be either blue or else not blue.)
......Liar-style sentences present themselves as misrepresenting themselves, so their meaning is self-undermining. And they can be read (or heard) in two basic ways—each a necessary part of the other’s context—because their meaning self-undermines in a loopy sort of way. Insofar as Liar-style sentences are true they’re also false, and they need concern nothing but their own truth, so they can certainly be read as nonsensical. But they’re not just senseless, and hence not at all true, because insofar as they’re not true they’re easily read as true. So they also have that sense. But they can’t be nothing but partly true and hence partly false, because that would leave nothing for them to be true or false about.
......This resolution—that Liar-style sentences are fairly true, in that loopy way (they’re fairly true because they’re rather nonsensical, and they’re rather nonsensical because insofar as they’re true they’re also false)—is a strengthened version of the resolution that takes them to be nonsensical. So for those who believe that an omniscient being is logically possible, it allows a similar reply to Divine-Liar-style sentences. E.g. the problem with “no omniscient being knows this” is that it can’t be true if there’s an omniscient being, but if it isn’t true then, since no one could then know it, it would seem to be true. My new reply is that if it’s only fairly true (in this loopy way) then no epistemically perfect being would have to know it, except to know it for what it is. And note that “no omniscient being knows any of this” is simply false, e.g. such a being would know those words. (Similarly, “what I’m saying isn’t at all true” is fairly false.)

Omniscience Again cont.

This is the last of 17 posts, which are collectively Eternity, etc.
......There is something counter-intuitive about the suggestion of the previous post, of course (even on the modern view of arithmetic). If B is the biggest Beth that has been constructed, then my suggestion denies that 2-to-the-power-of-B exists, where 2-to-the-power-of-B is the cardinality of P(X) when X has cardinality B. Were there no such B, my suggestion would deny that the union of the existing sets has an actual cardinality, on the modern view of the natural numbers (on the older view, it would deny the existence of M + 1, where M is the biggest natural number divinely constructed). Either way, my suggestion is effectively that there are true statements that God did not know but which were bound to be true and which, if we could come to know them, we would most naturally say had always been true. Intuitively, that seems to fall short of divine omniscience.
......Nevertheless, we know from section III that what can seem, with hindsight, to have been timeless truths may not have been. And according to section VII it is logically, not just physiologically, impossible for anyone to say or know all such things. Such statements therefore belong to an indefinitely extensible totality. Would it therefore be more accurate to talk of possible statements here? Maybe not [i], but it’s certainly logically possible that our intuited shortfall is due to our being dependent creatures. For us, even physics is immutable, but God is certainly the ground of metaphysical possibility. And He may well be the ground of all meaning and value. So the counter-intuitiveness of divinely created mathematics may prove, upon reflection, to be no more conclusive than the counter-intuitiveness of Divine Command Metaethics [ii]. After all, my suggestion does not deny that, in the time we took to think of 2-to-the-power-of-B, God had already constructed it [iii].
......So, to recap, God’s omnipotence conflicts with His timelessness, according to section VII, unless we deny arithmetical Realism, or deviate further than Presentism does from standard logic. And under Presentism, even such an omnipotent God could be necessarily omniscient. So what follows from God being necessarily omniscient—and our freedom being libertarian—is primarily disjunctive. Either God has timeless knowledge of the future—if that does cohere with our freedom being libertarian—but Realist arithmetic is paradoxical, or Realist arithmetic is divinely constructed and time is Presentist, or some other option. So even if they regard God as necessarily omniscient, Perfect Being Theists who take a libertarian view of free will—and regard the future as (partially) real—should not reject Open Theism.
......Notes:
......[i] Statements are basically possible assertions (see note ii of Eternity), but a possible statement is not necessarily just a statement. Similarly, one might be unable to say something in French, and yet be able to learn (more) French, so that one would be able to be able to say it. It is of course hard to tell how apposite that analogy is, for the language (so to speak) of God’s thoughts.
......[ii] Lois Malcolm, “Divine Commands,” in Gilbert Meilaender & William Werpehowski (eds.), The Oxford Handbook of Theological Ethics (Oxford Univ. Press, 2005), pp. 112–29.
......[iii] Suppose (see note v of Possible Worlds) that a God who could change had made our 4-dimensional world in an instant. Then some biggest Beth, say B, would be known by Him at all (of our) times. But then we might use “2-to-the-power-of-B” as a definite description of a Beth that He does not know, at any (such) time, which hardly coheres with His being the greatest conceivable being. By contrast, a Presentist God would most plausibly be learning arithmetic too quickly for us to describe any such number.

Omniscience Again

This is the sixteenth of 17 posts, which are collectively Eternity, etc.
......You may be wondering how, if the Open God is forever acquiring arithmetical knowledge (see previous post), He could ever be omniscient (or how His understanding of His options could ever be perfect). It would not even help us here to think of omniscience in Swinburne’s terms, because however much arithmetic God knows it is logically—indeed, metaphysically—possible for Him to know more (according to section VII); and nor could He know all the interesting Beths (that could ever exist) [i], because the smallest Beth that He did not know would be of some objective interest.
......Nevertheless, such Perfect Being Theists as Augustine and Duns Scotus took the Platonic Forms to be divinely created (in view of God’s omnipotence) [ii], and similarly, Open Theists might take arithmetic to be divinely constructed [iii]. Suppose that arithmetical statements are true or false only when divinely proved or refuted (respectively). That process could not always be instantaneous, according to section VII, and of course, not yet knowing something that is not yet true would not obstruct omniscience. And while most of us think of arithmetic as timeless, by “arithmetic” we ordinarily mean finite arithmetic, and on the modern view such a God could have always known all of that (instantaneously constructed in His primal state). Indeed, He could have always known all the Beths that are not, for us, unimaginably large.
......My suggestion is therefore that God, being epistemically omnipotent, constructs each and every modal consequence of the concept of a thing (which He understands perfectly), and in particular the cardinalities of possible creations (doing so endlessly because such is the nature of that concept). He knows all the Beths that exist. Before He constructs a Beth, it has only a potential existence. And eventually (and arbitrarily quickly) He knows any Beth that could ever exist. And His understanding of such possible Beths is perfect (cf. how we could understand the essence of an arbitrary natural number, even on the older view of arithmetic).
......Notes:
......[i] For a similar suggestion, see Menzel, “God and Mathematical Objects,” pp. 93–4 n. 42.
......[ii] For Augustine, see Sorabji, Time, Creation and the Continuum, p. 252. For Duns Scotus, see Gunton, The Triune Creator, pp. 118–9. For more details, see Copan & Craig, Creation out of Nothing, pp. 173–80.
......[iii] Menzel, “God and Mathematical Objects,” defends such a view, called “theistic constructivism” by Copan & Craig, Creation out of Nothing, p. 191.

Cantor’s Paradox again

This is the fifteenth of 17 posts, which are collectively Eternity, etc.
......Because of all those unions (see previous post), our collection is a nested hierarchy of sets, whose cardinalities the Beths are defined to be. And so because our collection is not quasi-spatial, nor are the Beths, collectively. So even on the modern view, cardinal arithmetic is indefinitely extensible [i]. And while that result is of a kind with Cantor’s Paradox [ii], it is the belief that cardinal arithmetic is timeless that makes it paradoxical. The atemporalist faces some tough choices, because the truths known by a timeless God would be collectively quasi-spatial, rather than variable.
......But an everlasting God could acquire arithmetical knowledge endlessly. And Presentist time is merely our natural reification of the possibility of change, not a real dimension. And under Presentist Open Theism, that possibility originates with the greatest conceivable being’s power to change. So such a God would have enough time to know each arithmetical truth, and to know it arbitrarily quickly. Time would then be indefinitely extensible, and absolutely continuous [iii], in the sense that for any duration, and for any Actual Infinite cardinal number (that could ever exist), that duration has more than that many instants (possible instantaneous changes).
......It seems, then, that only a God with the power to change is, for every Actual Infinite cardinal number, able to know all about a possible world of so many things, and hence able to create such a world perfectly freely (with a perfect understanding of His options). So the argument at the end of section VI becomes an argument that God is, since omnipotent, not timeless. And note that the informality of this rather mathematical section does not make that a weak argument. Formal proofs can only prove theorems within axiomatic systems, and since the justification of such axioms is necessarily informal, so informality also suits a more direct argument about metaphysically possible creations.
......Notes:
......[i] For more details, see W. D. Hart, “The Potential Infinite,” Proceedings of the Aristotelian Society 76 (1976): 247–64; Alvin Plantinga & Patrick Grim, “Truth, Omniscience, and Cantorian Arguments: An exchange,” Philosophical Studies 71 (1993): 267–306; Stewart Shapiro & Crispin Wright, “All Things Indefinitely Extensible,” in Agustin Rayo & Gabriel Uzquiano (eds.), Absolute Generality (Clarendon Press, 2006), pp. 255–304; Nicholas Rescher & Patrick Grim, “Plenum Theory,” Noûs 42 (2008): 422–39.
......[ii] For Georg Cantor, sets were consistent Actual Infinite collections. But he thought that all Potential Infinite collections presuppose Actual Infinite collections, much as mathematical variables range over fixed domains. So he thought of collections like that of all the sets (Cantor’s Paradox) or all the cardinal numbers as Actual Infinite but inconsistent. For more details, see Michael Hallett, Cantorian set theory and limitation of size (Clarendon Press, 1984), pp. 24–48. Of course, taking inconsistency on the chin like that is a high a price to pay for Realism (whence the foundation of mainstream mathematics is now an axiomatic set theory). But even if Potential Infinite collections do depend upon something being Actual Infinite, that might be a power (see note iv in Cantor's Paradox) or a length (see following note) rather than a collection.
......[iii] For such continua, see my “To Continue with Continuity,” Metaphysica 6 (2005): 91–109; Philip Ehrlich, “The Absolute Arithmetic Continuum and its Peircean Counterpart,” in Matthew Moore (ed.), New Essays on Peirce’s Mathematical Philosophy (Open Court, forthcoming).

Cantor’s Paradox cont.

This is the fourteenth of 17 posts, which are collectively Eternity, etc.
......You may be familiar with N (see previous post) from school mathematics. Such informal sets are basically collections that are quasi-spatial, in the sense that their members coexist (insofar as they do exist) altogether. Given any spatial collection—e.g. some ordinary objects in a room—any sub-collection of them is clearly also spatial; and similarly, a definitive property of our informal sets is that every conceivable sub-collection of such a set is itself quasi-spatial, is a subset [i].
......Surprisingly, the modern view (of arithmetic as timeless) offers little support to atemporalism, as follows. To say that two collections have the same cardinality—the same cardinal number of members—is to say that the members of each collection could all be paired off, one-to-one, with those of the other [ii]. And for any set, S, if the collection of all its subsets is also quasi-spatial, then that collection—including (for simplicity) the so-called improper subsets, S and the empty set—is the powerset of S, say P(S). And according to Cantor’s Diagonal Argument [iii], P(S) always has a greater cardinality than S.
......In particular, the cardinality of P(N)—which Peirce called “Beth-1”—is greater than the cardinality of N, which is Beth-0. And the cardinality of P(P(N)) = P-squared(N) is Beth-2, which is greater than Beth-1. And so on; for each natural number n, P-to-the-nth-power(N) has Beth-n members. And the union of N and all those P-to-the-nth-power(N) is the collection of all their members. For each n it contains at least Beth-(n + 1) members. So its cardinality, say Beth-omega [iv], is greater than Beth-n for every n. And if that union is also a set, say U, then by Cantor’s Diagonal Argument, P(U) has an even greater cardinality, Beth-(omega + 1) [v],
......We might expect that union to be a set, because Beth-0 being an Actual Infinite number means that all those Beth-0 sets coexist quasi-spatially (like a row of houses, whose contents therefore coexist similarly). So the next union might be of U and all the P-to-the-nth-power(U). But by continuing in that way, taking powersets and unions as far as is logically possible [vi], we cannot end up with a set because from any set we could have continued further in that way. We have, then, a collection that is not quasi-spatial, being generated by a process that cannot be completed (as a matter of logical necessity). Continued...
......Notes:
......[i] By contrast, if the natural numbers are forever growing, according to the rule of add 1 repeatedly, then only those sub-collections that are similarly specified by a finite rule exist in the same kind of way.
......[ii] The natural numbers are finite cardinal numbers. And N has the same infinite cardinality as the subset of just the even numbers because n can be paired with 2n for all natural numbers n. There are, in an obvious sense, more natural numbers than even numbers, but cardinality is fundamental to our number concept; Shapiro, Thinking about Mathematics, pp. 133–8.
......[iii] If S and P(S) had the same cardinality, there would be one-to-one mappings from S onto all of P(S). Let M be one such mapping, and let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. That is, D is contradictory, and so there is no such M, which means that S and P(S) do not have the same cardinality. But for each member of S, say m, P(S) contains {m}, and so the cardinality of P(S) is greater than that of S.
......[iv] Omega is the first ordinal number after the natural numbers. Ordinal numbers generalize counting numbers as such beyond the natural numbers (whence their use indexing the Beths).
......[v] Such ordinal addition corresponds to a rearrangement of the natural numbers, e.g. from their natural order (to which omega corresponds) to 2, 3, ..., 1.
......[vi] We could also take unions of Beth-1 sets, since Beth-1 is Actual Infinite; and similarly, Beth-2 sets, etc.

Cantor's Paradox

This is the thirteenth of 17 posts, which are collectively Eternity, etc.
......This section is rather mathematical, but we can—indeed, should—begin with the simplest numbers, 1, 2, 3, etc. Mainstream mathematics has axiomatic set theory for a foundation, for such reasons as Cantor’s Paradox [i], and pure mathematicians are certainly free to explore any interesting formal possibilities. But we are primarily interested in possibilities insofar as they are (or might be) grounded in the God that is.
......The natural (or counting) numbers are the products of endlessly reiterating the addition of 1, starting with 1. They are clearly instantiated, because you and I are 2 people. Arithmetic is prima facie the science of such elementary metaphysical possibilities as the possibility of two individuals. Such arithmetical Realism may be difficult to justify atheistically [ii], but we may think of the natural numbers as existing amongst God’s thoughts, arising via His epistemic omnipotence from His perfect grasp of the concept of a thing, whence our informal 1, together with a concept associated with His omnipotence, such as possibility, whence informal addition and its endless reiteration. Note that Realist arithmetic can be discovered a priori under Theism because we instantiate the concept of a thing and were created in God’s image (and we might also be divinely inspired) [iii].
......The endless reiteration of the addition of 1 means that the natural numbers are (collectively) infinite. Many mathematicians have taken them to be Potential Infinite, as Aristotle put it [iv], or as J. S. Mill put it, indefinitely extensible. The addition of 1 is a definite process, but the natural numbers would have no fixed extension if the endless reiteration of the addition of 1 led to growth that could not even in principle be completed. But most of us think of arithmetic as timeless, and the modern view of the natural numbers is that they are (collectively) Actual Infinite, existing as an immutably complete collection, N = {1, 2, 3, …}. Continued...
......Notes:
......[i] Within an axiomatic set theory, Cantor’s Theorem says only that there is no such set of all such sets. For Cantor’s Paradox, see note iii of Divine Attributes.
......[ii] For some well-known problems, see Stewart Shapiro, Thinking about Mathematics: The philosophy of mathematics (Oxford Univ. Press, 2000), pp. 107–289; George Lakoff & Rafael E. Núñez, Where Mathematics Comes From: How the embodied mind brings mathematics into being (Basic Books, 2000), pp. 342–3.
......[iii] For more details, see Christopher Menzel, “God and Mathematical Objects,” in Russell W. Howell & W. James Bradley (eds.), Mathematics in a Postmodern Age: A Christian perspective (Eerdmans, 2001), pp. 65–97 (especially pp. 92–6).
......[iv] To see what Aristotle meant, consider an everlasting fruit-tree. The tree’s endless production of fruit is the ever-incomplete expression of its power to fruit. The total amount of fruit produced is always finite, but always increasing; it is unlimited—is Potential Infinite—because the tree’s power to fruit remains infinite. For more details, see Copan & Craig, Creation out of Nothing, pp. 200–10; Peter Fletcher, “Infinity,” in Dale Jacquette (ed.), Philosophy of Logic (Elsevier, 2007), pp. 523–85.

English Numbers

Hartley Slater in The Reasoner 4(12), 175–6, tried to show, from the fact that the number of elements in the empty set is zero, that zero is not, as a matter of English grammar, the empty set—and in general, that numbers are not sets—because we don’t say that the number of elements in the empty set is the empty set. But things aren’t quite that simple.
......To begin with, Slater’s example of zero—which is often defined to be the empty set in pure mathematics—was an unfortunate choice, because mathematicians introduced zero relatively recently. Consequently English remains rather ambivalent about its status. There being no elephants in this room, for example, it’s false that there are a number of elephants here. So from it being true that there are zero elephants here, surface grammar might seem to indicate that zero isn’t even a number (a cardinal number). But zero is of course a number (the number of elephants in this room, the number of elements in the empty set).
......For another example, the numbers one, two, three etc. correspond to the positions first, second, third and so forth. And since no sequence has an element before the first one—that’s what ‘first’ means—so, in that ordinary sense, there’s no zeroth element, and so again, surface grammar seems to indicate that zero isn’t a number (an ordinal number). Nonetheless, there’s a more mathematical sense in which whenever an element is indexed by 0, it’s a zeroth element.
......In many mathematics textbooks there’s a Chapter 0, for example, containing the set-theoretic basics. Of course, such chapters don’t amount to much evidence that mathematicians take numbers to be nothing more than sets. Mathematicians make the standard identification of numbers with sets in order to prove theorems from set-theoretic axioms. They are thereby following in the footsteps of those who did geometry by proving theorems from Euclid’s axioms. And surely few if any geometers thought that there was nothing more to space than Euclid’s axioms. Space was rather the obvious space around us, of which Euclid’s axioms were taken to be true (and obvious enough to be the premises of proofs).
......Now, even if the space that we see around us is Euclidean—having been constructed as such by our brains from our sensory input—it’s surely not unlikely that what Aristotle meant by ‘space’ is non-Euclidean. So, similarly, even if our concept of zero comes (for example) from reifying the definitive property of an absence, it doesn’t follow that it’s impossible that Euler’s ‘0’ referred to an empty set. Indeed, the standard empty set can be an urelement—can be anything that has no members (where membership is an axiomatic primitive)—because its job within standard set theory is simply to have no members, and so in that sense (at least) zero can be an empty set.
......But more to the point, Slater’s argument may beg the question. That’s because if ‘the empty set’ was a definite description of zero then we could say that the number of elements in the empty set is the empty set (for all that we wouldn’t usually). After all, we can say that the number of ones in zero is zero. In general, for natural numbers n, the number of ones in n is n. Perhaps it would be more natural for us to say that two twos are four (for example), and hence that the number of twos in four is two. But such equations all follow from the natural numbers—most obviously those greater than 1—being essentially sums of ones, which seem to be some sort of collection, perhaps not unlike sets of points in that, while their elements are in obvious ways identical, they are distinguished in ways that derive from their origins (as positions in space, in the case of points).
......Two twos are four because any two things plus another two things are four things. And in English, there being a number of things of some kind is just there being some things of that kind. So again, surface grammar indicates that numbers—most obviously those greater than 1—are some sort of collection. And we might expect mathematicians to be the experts on what exactly numbers are. So, since mathematicians prove theorems about numbers from set-theoretic axioms, we’ve some evidence that numbers are sets.
......Still, such evidence is compatible with numbers being axiomatic sets only in a rather abstract way (cf. how the integers with addition are an abelian group). Slater’s argument was based on surface grammar, so it was presumably that numbers are not sets in some more obvious sense. So note that collections in the usual, informal sense can be variable, like a stamp collection, or non-variable, like a chess set. A fundamental question in this area is therefore whether mathematicians have discovered that numbers behave like sets—at least to the extent that the natural numbers are, collectively, non-variable—or whether they’ve just tended to assume that (even though we can’t so easily assume that cardinal or ordinal numbers are non-variable, in light of the famous set-theoretic paradoxes).
......Mathematicians don’t prove the standard Axiom of Infinity—which asserts the existence of a set containing one element for each natural number (amongst other axioms giving such sets the properties one would expect of non-variable collections)—but rather prove theorems from that axiom (along with the others), or work from some other foundation. Philosophical arguments are therefore needed, to assess whether the standard axioms are giving us a scientifically adequate description of the natural numbers or not. But arguments based on surface grammar are unlikely to be of much help in this area. After all, they can’t even show zero to be a number. (For a more apposite sort of argument, see my 2003: ‘Infinite Sequences: Finitist Consequence,’ The British Journal for the Philosophy of Science 54, 591–9, and my 2010: ‘Two Envelopes, two paradoxes,’ The Reasoner 4(5), 74–5.)

Possible Worlds

This is the twelfth of 17 posts, which are collectively Eternity, etc.
......As well as all those (correct) statements (see previous post), there are also a lot of truths about other metaphysically possible worlds [i]. Statements of the former kind naturally seem more important than truths about merely possible worlds; and I have not shown that not knowing the former would not make God liable to make mistakes. But balancing the possibility that they do is the possibility that such ignorance is required for our genuine freedom [ii]. And in any case, would it follow from God being maximally knowledgeable—as well as maximally powerful—that He is timeless even if He could only be completely knowledgeable about the future if He was timeless?
......Since the answer is no [iii], let us consider all metaphysically possible worlds, not just this one, under each candidate conception of God. And since God’s omniscience may be less certain than His omnipotence (see section II), let us consider His power as well as His knowledge. And in view of section IV, let us compare libertarian atemporalism with Presentist Open Theism, taking both those conceptions to be prima facie logical possibilities [iv].
......God being possibly timeless means that a world like ours could conceivably be the 4-dimensional creation of a God who transcends its temporal dimension. But it is therefore conceivable that a Presentist God could instantaneously make a 4-dimensional world that is similarly like ours—as it has been so far, and happening also to be that way in the future—but with a fourth dimension substantially unlike Presentist time [v]. So a world as ours would be were God timeless could conceivably be made by Him whether He is timeless or not. And He would be completely knowledgeable about it whether He made it or not.
......Similarly, for any possible spatiotemporal world that a timeless God could make, a Presentist God could conceivably make—and so would know all about—an isomorphic world. But Presentist Open Theism being possible means that this world might have a future that is open, in the sense that there are statements that are neither true nor false but which will be either true or false. And a timeless God could hardly make such a world, because for Him the future has to be completely real.
......Now, atemporalists may not regard that inability as detracting from His omnipotence, whether or not the creator of such a world would be liable to bodge things up in it. But there is also an argument that there are many more metaphysically possible creations if God is able to change (see section VIII), which uses what is shown in the next three posts, that there is no immutably complete totality of all the metaphysically possible whole numbers.
......Notes:
......[i] That is clearly so under Open Theism; and although the deliberate creation of something contingent seems to require several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change, creation is also taken to be contingent by Mawson, Belief in God, p. 71.
......[ii] Boethius famously argued that our freedom would not be limited by God’s knowledge of what we will be doing were that, not so much foreknowledge, as timeless knowledge (the analogy was with someone knowing what we do, not before we do it, but as we do it). (E.g. see Mawson, “Divine eternity,” pp. 38–40; Sorabji, Time, Creation and the Continuum, pp. 254–6.) Nevertheless, it would still have been true in the past that God knows (timelessly) all about the future. And to see why that might be a problem for libertarian atemporalism, consider a timeless God revealing truths about our future free actions to some of His saints in the remote past (and perhaps even on a distant planet). Is it obvious that our now being responsible for our actions could depend upon Him having done no such thing? (For more details, see Helm, Eternal God, p. 101 ff.)
......[iii] To see why not, consider someone choosing between Open Theism and the hypothesis that she was made by a transcendent computer, which has a complete database on (and complete control over) its creatures’ relatively virtual lives. Only the computer could be completely and infallibly knowledgeable about her whole life. But it would clearly be less knowledgeable (and powerful) than those who might have built an isomorphic computer, and who might have been created by an even more knowledgeable (and powerful) Open God.
......[iv] Cf. Mawson, “Divine eternity,” p. 46 n. 10.
......[v] If this world had been made like that, we should think of Presentism as false, because the analogical—as we should then see it—instant at which God was fully present would then include the past and future. Such a God would be neither everlasting nor timeless (see note vii of Divine Attributes cont.), but is also relatively implausible (see note iii of Omniscience Again cont.).

Bodging Up cont.

This is the eleventh of 17 posts, which are collectively Eternity, etc.
......How closely would such interventions as we might expect under Open Theism have to resemble Mawson’s scenario (see previous post)? Why, to begin with, should the world’s aggregate happiness have decreased? The immediate consequence of Adolf’s birth was a little more joy in the Hitler household. And surely God would have intended to intervene further, as necessary to ensure that aggregate’s continued increase, if that had been His motivation. (Making such interventions would hardly cause further problems, as simply making evermore planets or heavens of inherently happy animals or angels might suffice.)
......But in any case, a more plausible motivation for an Open divine intervention would be that aggregate’s eventual perfection [i], e.g. by our becoming a communion of saints. Rather than the Open God intervening to ensure a baby’s birth [ii], He would more plausibly have answered Mrs. Hitler’s prayers in order to help her to relate to Him more fully [iii]. And had He done that, then the satisfaction of His desire would hardly have depended upon how her child grew up. It would have depended upon her free choices—to some extent (He would definitely have so helped her)—but that amounting to luck is generally rejected by libertarians [iv]. Furthermore, even if Mrs. Hitler did not respond by becoming a saint, the Open God could surely try again, and again. And His attempts might become irresistible, as Mrs. Hitler wised up, or perhaps she might become very undeserving.
......So in short, Mawson’s scenario did not show that the satisfaction of the Open God’s most beneficial desires—perhaps that everyone (who is not too undeserving) should end up somewhere heavenly forever—could not be inevitable. So we are left with no reason why we should think of the Open God (of any variety) as a “well-intentioned buffoon”[v], rather than as Jesus [vi], and hence no reason why God should know all about the future (cf. end of section IV). So although there are statements about the future that would not be known by the Presentist Open God despite them being in a sense correct, that sense has not been shown to be significant enough to obstruct divine omniscience.
......Notes:
......[i] Keith Ward, Divine Action (Flame, 1990), pp. 134–9.
......[ii] Swinburne, Is There a God? pp. 114–5.
......[iii] Robert M. Adams, “Theodicy and Divine Intervention,” in Thomas F. Tracy (ed.), The God Who Acts: Philosophical and theological explorations (Penn. State Univ. Press, 1994), pp. 31–40.
......[iv] Timothy O’Connor, “Is It All Just a Matter of Luck?” Philosophical Explorations 10 (2007): 157–61.
......[v] Mawson, “Divine eternity,” p. 48.
......[vi] Richard Swinburne, Was Jesus God? (Oxford Univ. Press, 2008). If Jesus is, like us, a continuant, then it’s hard to see how he could be, not just the signature, but the identity of a timeless God. But clearly an everlasting God could incarnate as fully human, e.g. if we are essentially spirits that produce human minds because we animate human brains.