Tuesday, 14 March 2017

Quantification and 'Extra Constant' Semantics

(The following is a companion piece to this offsite post.)

In a fascinating new paper entitled 'Truth via Satisfaction?', N.J.J. Smith argues that the Tarskian style of semantics for first-order logic (hereafter 'FOL'), which employs the special notion of satisfaction by numbered sequences of objects, does not provide an explication of the classical notion of truth - the notion of saying it like it is - but that the second-most prominent style of semantics for FOL, which works by considering what you get if you introduce a new constant, does. I agree with him about the first claim but disagree with him about the second.

My main point in this post, however, is not to argue that Smith's preferred style of semantics for FOL fails to explicate the classical notion of truth. I will do that a bit at the end - although not in a very fundamental way - but the main point will be to draw out a moral about how we should think about the 'extra constant' semantics for FOL, and more generally, about how we need to be careful in certain philosophical contexts to distinguish mathematical relations (such as 'appearing in an ordered-pair with') from genuinely semantic ones (such as 'refers to'). The failure to do this, in fact, is what made Tarski introduce his convoluted satisfaction apparatus which others have muddle-headedly praised as some sort of great insight. (I blogged about this debacle, to this day largely unrecognized as such by the logical community, offsite in 2015.)

By way of intuitive explanation of the universal quantifier clause of his preferred semantics for FOL, Smith writes: 
Consider a name that nothing currently has—say (for the sake of example) ‘Rumpelstiltskin’. Then for ‘Everyone in the room was born in Tasmania’ to say it how it is is for ‘Rumpelstiltskin was born in Tasmania’ to say it how it is—no matter who in the room we name ‘Rumpelstiltskin’. (p. 8 in author-archived version).
But this kind of explanation is not generally correct. Get a bunch of things with no names and stick them in a room. Now, doesn’t this purported explication of what it is for quantified claims to be true run, in the case of the claim ‘Everything in this room is unnamed’, as follows: for ‘Everything in this room is unnamed’ to say it how it is is for ‘Rumpelstiltskin is unnamed’ to say it how it is--no matter what in the room we name ‘Rumpelstiltskin’? And this, I think, is very clearly false; by hypothesis, everything in the room in question is unnamed, so surely ‘Everything in this room is unnamed’ says it how it is. But if we name one of the things in the room‘Rumpelstiltskin’, then ‘Rumpelstiltskin is unnamed’ will certainly not say it how it is.

Now, as Smith pointed out to me in correspondence, this problem with unnamedness can be avoided by considering another method of singling out objects, such as attaching a red dot to them. (The worry arises that some objects are abstract and so it makes no sense to talk about attaching a red dot to them, but I won't pursue that here.) Then you can use a slightly different form of explanation, and say that for 'Everything in the room is unnamed' to say it how it is is for 'The thing with the red dot on it is unnamed' to say it how it is no matter which thing in the room has the red dot on it. Now we will of course get a counterexample involving 'red-dotlessness' but we can then just consider a different singling-out device.

But this slightly different style of explanation is also not generally viable, as becomes clear when we consider, not unnamedness, but unreferred-to-ness. Things which haven't been named but have been referred to, say by a definite description, count as unnamed but not as unreferred-to. And let's stipulate that we are talking only about singular reference - so that even if 'All the unreferred-to things' in some sense refers to the unreferred-to things, it doesn't singularly refer to them, so this wouldn't stop them from being unreferred-to in the relevant sense.

Now, applying the new style of explanation involving an arbitrary singling-out method to the case of 'Everything in this room is unreferred-to', we get:
For 'Everything in this room is unreferred-to' to say how it is is for 'The thing with the red dot on it is unreferred-to' to say how it is, no matter which thing we put the red dot on.
And this is wrong, not because the thing has a red dot on it, but because 'The thing with the red dot on it is unreferred-to' can't be true, whereas the quantified claim can be.

No analogous problem arises in the formal setting. If we specify that 'G' is to be mapped to the set of things in some room and 'F' is to be mapped to the set of unreferred-to things, and consider '(x)(Gx  Fx)', then neither Smith's preferred style of semantics for FOL nor the silly Tarskian style cause any sort of problem, since for there to exist a function which maps some constant c to an object o is compatible with o being unreferred-to. Thus we get the desired truth-value for '(x)(Gx  Fx)'.

(You might now think: OK, but what if we replace unreferred-to-ness with not-being-mapped-to-by-any-function, or whatever? Don't we then get the wrong truth-value? Well, no, because - at least on a classical conception of functions - nothing is unmapped-to-by-any-function.)

So, quantified propositions are not correctly explicated by talking about the truth-values of propositions you get by naming things. Nor are they correctly explicated by adopting a non-semantic singling-out device and then considering propositions which talk about 'The thing' singled out. This in itself shouldn't really be news, but also noteworthy is that, despite such explications being incorrect, the style of semantics for FOL which works via consideration of an extra constant gives no undesired results, and is arguably better than the Tarski-style semantics, which is needlessly complicated and is born of philosophical confusion. (Still, it does create a danger that students of it will wrongly think that you can explain quantified propositions in the way shown here to be incorrect.)

What does this mean for Smith's claim that 'extra constant' style semantics for FOL explicates the classical conception of truth, the conception of saying it like it is? Well, I think that's an interestingly wrong idea anyway, and probably deeper things should be said about it, but: Smith's incorrect informal gloss of the formal quantification clause - which gloss, as we have seen, cannot be corrected by moving to an arbitrary singling-out device and talking about 'The thing' singled out - certainly seems to be doing important argumentative work in his paper. His main claim, bereft of the spurious support of the informal gloss, is as far as I can see completely without support.

Many thanks to N.J.J. Smith for discussion.

Wednesday, 15 February 2017

The Resurgence of Metaphysics as a Notational Convenience

Reading Jessica Wilson's interesting new SEP entry on Determinables and Determinates, the following speculation occurred to me: the oft-remarked-upon resurgence of metaphysics heralded by the work of David Lewis, D.M. Armstrong and others was driven in part by cognitive resource limitations and practicalities of notation; putting things metaphysically often lightens our cognitive loads and makes thinking and writing more efficient in many philosophical situations.

Wilson's piece is dripping with metaphysical turns of phrase, but much of what she says could be re-expressed in a conceptual or linguistic key. I think this goes for a good deal of contemporary metaphysics. However, converting metaphysically-expressed ideas and claims into a conceptual or linguistic key may make them a bit fiddlier to think and express. And if you're doing hard philosophy and need to think and express a lot of things, this extra cost is going to pile up. Sometimes, having things in a conceptual or linguistic register may make things clearer, and sometimes it may be essential. But for many purposes the metaphysical register does fine, and often has the benefit of being less resource-hungry.

Yes, some metaphysics may not be capturable in conceptual or linguistic terms, and perhaps even in favourable cases the capturing will not be complete or perfect. And there are doubtless other important things going on behind the sociological phenomenon of the resurgence of metaphysics. But maybe this is part of the story.

UPDATE: Brandon Watson (at the end of a post on Fitch's paradox) links to this post, writing: 'I'm very interested, of course, in accounts of how philosophical scenes get transformed, how ideas transmogrify, and the like. This hypothesis for the rise of analytic metaphysics makes considerable amount of sense, and is probably true.' This is encouraging!

Friday, 20 January 2017

'Close Enough' Closer to the Truth About Counterfactuals

Lewis would have liked to be able to say that a counterfactual A > C is true iff the corresponding material conditional is true at all closest worlds. But his example of the inch long line seemed to show that sometimes there are no closest worlds - you can get closer and closer without limit to being one inch long while still not becoming one inch long. Wanting to avoid the Limit Assumption - the assumption that you do hit a limit as you get closer to actuality, after which you cannot get closer except by reaching actuality - he plumped for a clever, but complicated and costly solution; requiring that no A & ~C world is closer to actuality than any A & C world. (In his (1981) he admitted that this is costly in terms of simplicity and intuitiveness.)

I think Lewis here was too hung up on the idea of minimal change to the actual world. Proposing instead that a counterfactual A > C is true iff the corresponding material conditional is true at all close enough, or relevantly similar, worlds is a better way to avoid the Limit Assumption. (This theory might work for indicatives, too, but that's an especially vexed issue.) Why is this better?

(1) It lets you have a simpler, more intuitive form of account, with a set of worlds which are relevant.

(2) This also lets you have nice things like these results about when it's OK to use certain inference patterns.

(3) It better handles what might be called 'categorical' or 'no matter what' conditionals like 'If you had seen a cat then you would have seen an animal', where this is intended in such a way that you could add 'definitely' or 'no matter what' after the 'then' without changing the truth-condition, and is generally a more flexible and hence powerful account.

(4) It lets you straightforwardly explain why 'If this person had been taller, they would have been only a tiny bit taller' and the like are not true. Lewis can do this at the cost of saying that here close similarity in height just isn't important, but this is a little awkward given his case against the Limit Assumption.

(Donald Nute long ago proposed a 'close enough' account as better than Lewis's (see references below), but it seems few people listened. Also, some of his reasons can be diffused by being clever and flexible about what matters for similarity, and he didn't have (2) above to offer, and maybe not (3) either.)

Why wouldn't you go this way? One reason I can think of is that it may seem like a regrettable move to a less definite, less informative form of account. After all, if I am told 'the tallest people will be given a prize' this seems more informative than 'everyone tall enough will be given a prize'. But in the present context, this is illusory. You need to build in so much contextual flexibility into Lewis's account to make it at all plausible that the indefiniteness there swallows up the apparent difference in informativeness. Either that, or you keep the edge in definiteness but at the cost of implausible truth-value verdicts. Minimal change, I suspect, is a good way of thinking about lots of counterfactuals, and maybe those were the ones on Lewis's mind - but I see no reason why the change would ever have to be so minimal that you need to abandon having a set of relevant worlds and move to Lewis's official 'no A & ~C world closer than any A & C world' account. For other counterfactuals, minimality seems not to the point at all. So it's good to have a more flexible 'close enough' style account for your general theory of counterfactuals (or conditionals in general).


Lewis, David (1981). Ordering semantics and premise semantics for counterfactuals. Journal of Philosophical Logic 10 (2):217-234.

Nute, Donald (1975). Counterfactuals and the similarity of words. Journal of Philosophy 72 (21):773-778. (Title [sic]. As far as I know it's just a very unfortunate typo and it should be 'worlds'.)

Saturday, 24 December 2016

What Are My Problems Now?

This is a follow-up to What Was My Problem?.

1. The Basis of Puzzlement about Modality

One line of investigation I would like to pursue now is into what might be called the basis of the puzzlement about modality. And as suggested by my experience of vaguely wrestling with a bunch of problems, before realizing that my strongest leading ideas for my thesis were really about some of these problems rather than others, I think this line of investigation may itself call for the distinguishing of various problems.

One locus of puzzlement about modality is the notion of metaphysical or subjunctive necessity as it applies to propositions. And one question about this notion is whether, and how, meaning comes into the picture. Also, just the question of how this notion relates to other notions, and the extent to which it can be analyzed (not necessarily in non-modal terms). Those problems are addressed, properly I hope, by the account in my thesis. But lots of what I was wrestling with at the beginning of my research remains, and does not attach specifically to the notion of subjunctive necessity de dicto; there is a lot that is puzzling about modality that my thesis does not address.

One puzzling thing which borders directly on my thesis work, and does have to do with the notion of subjunctive necessity de dicto, is the question of how this relates to de re modal constructions and quantification into modal contexts. But I have been very frustrated in my research here, and to be honest I have come to feel like it is a bit of a minor, abstruse issue compared to some of the more fundamental problems about modality (although I have no doubt that very interesting work could be done on that issue, and have a couple of ideas).

A more fundamental area I would like to work on is indicated by the question: Why is modality puzzling at all? But here too there are probably several puzzling things to distinguish. One thing I am not primarily thinking of, although it may end up becoming relevant to the problems I am grasping at, are questions about modality in an extremely general sense. For instance, the question of what unifies all uses of expressions which we call modal, or which we say are about possibility or whatever - including 'You can come in if you like', 'It could be that John is on his way', 'It is impossible for two colours to be in the same place', '"Hesperus is Phosphorus" is necessarily true', 'I can lift this weight', 'This apparatus has four possible configurations'. Also, questions about what generalizations can be made covering all or at least a great diversity of such uses, for instance about logical implication relations between them.

Rather, I am interested in trying to get at the basis of our puzzlement about what may be called objective modality. What does 'objective modality' mean? Well, one clear thing it does is exclude epistemic modals, like 'It could be that John is on his way' in natural uses. These are to be put to one side - at least initially - in the line of investigation I want to pursue. Likewise with uses of modal language having to do with permission. Within the puzzlement attaching primarily to objective modality as opposed to these set-aside kinds, important distinctions may have to be made. For instance, there may be a need to distinguish between more down-to-earth uses of modal language, for instance 'I can lift this weight', from what may be called more metaphysical uses - but not 'metaphysical' in the sense often used in modal philosophy, to mean either something like 'objective' or something a bit more specific, like picking out what I pick out with 'subjunctive'. Rather, by 'more metaphysical uses' I mean uses which are so to speak puzzling from the start. That is, where there isn't as much non-problematic, clearly useful use as in the case of 'I can lift this weight' and the like. E.g. 'The world could have been otherwise', 'Aristotle is essentially human'.

One way forward in this line of investigation would be to look critically and closely at philosophers' attempts to give a sense of the puzzlement about (objective) modality, often as a preliminary to some account or a survey of accounts. For instance, Sider's remarks on the subject in 'Reductive Theories of Modality'. But I think it will also be important to look within, so to speak, and keep seriously asking myself 'What is it that puzzles me about this?'.

2. Propositions and Meaning, Language Systems, and Our Expectations

Another line of investigation I would like to pursue has to do with the account of propositions and meaning sketched in chapter 6 of my thesis. That account appeals to a notion of an expression's internal meaning, cashed out in terms of the expression's role in the language system to which it belongs. This may raise questions about the nature of the system, and how we should think of it and describe it. In my thesis, I tried to remain quite open about this, emphasizing that I was offering a sketch, and that different fillings in of the detail here may be possible.

It was hard to avoid striking a false note here. For I do not think this is the whole story about my sketch, and the middle-Wittgenstein idea about role-in-system which it takes over; it may not be quite right to just think about it as a sketch of a theory, where some aspects are not filled in. For the very idea of what needs filling in, and how, should I think be scrutinized. It is not that I am advocating quietism, or defeatism, about questions about the 'language system' I appeal to. But I think that some of our expectations here may be in need of examination.

A curious thing happens in this territory - it is easy to become disoriented, and wonder what the problem was and what is needed now. Maybe sometimes in philosophy, as we solve problems, they slip from our grasp. Sometimes there is a strange feeling where we wonder something like: how could there be a solution here which is given in mere words? How could that ever do? We feel we still need to be taught something, or shown something. Could it be something practical, so to speak? I.e. something we could get through practice?

In the new year, I intend to use this blog to try to make some inroads into these and related problems.

Wednesday, 21 December 2016

On Why Crude Analyses are Often the Most Enlightening

The crudest conceptual analyses in philosophy are often the most enlightening - in metaethics, for example. It would certainly be wrong to thing that these enlightening analyses are only incidentally crude and false as analyses. Rather, their crudity and literal falsity is essential to their being illuminating. We are noticing an illuminating analogy - and for an analogy, you need two different things, so the analysis which shows clearly such an analogy is bound to be crude and false qua analysis.

Saturday, 17 December 2016

What Was My Problem?

From notes made early in my PhD research:
The central flaw in much of what I have written is that it has the unintended appearance of giving a kind of unexpected grounding for modal discourse. Perhaps it also might look as though what I'm saying might change our evaluative criteria, cast doubt on earlier beliefs about what's possible, etc. All of this has to be avoided. But it isn't satisfactory to simply insist 'No no no, that's not what I'm doing at all' - this has to be made evident. And the way to do that is to be better (more effective) at what I'm really doing. Very often, my problem is that I don't know what my problem is. I have to take it easy about that, and just look for frontiers where they arise.
I think I ended up avoiding this flaw. It gradually became clear that the problem to which I had a solution in inchoate form (now it's hopefully in a clear form) was not about the grounds of modality in general, but rather about the particular, partly-themselves-modal grounds, of the modal notion of subjunctive or metaphysical necessity as it applies to propositions. More specifically, about whether and how semantics comes into the picture. (A proposition is necessary iff it is or is implied by a proposition which is both inherently counterfactually invariant and true, and it makes sense to think of inherent counterfactual invariance as a broadly semantic property.)

Monday, 14 November 2016

On Carnap's 'Empiricism, Semantics, and Ontology' - Towards a More Nuanced View

Below are some notes on the first two sections Carnap's classic paper 'Empiricism, Semantics, and Ontology'. (Carnap's ideas in this paper have been very influential, and there has been a recent flurry of interest in them, as reflected in the 2016 publication of a volume entitled Ontology After Carnap. Thomas Hofweber, whose ontological project I have criticized, is a contemporary philosopher who has been very influenced by these ideas of Carnap.)

The notes below end up suggesting a more nuanced view of what is going wrong in metaphysical debates about the existence of ordinary things and numbers, according to which Carnap has correctly diagnosed that a kind of impossible jumping out of a framework is being attempted. But on the more nuanced view, this kind of jumping out is possible in some cases. (Carnap's view is that it never is, except as a potentially misleading way of switching to talk about the practical question of whether to adopt a linguistic framework.) That it is possible in some cases better explains why we so much as attempt it in the case of ordinary things and numbers.

* * *

'If someone wishes to speak in his language about a new kind of entities, he has to introduce a system of new ways of speaking, subject to new rules; we shall call this procedure the construction of a linguistic framework for the new entities in question.' - What is the status of this proposition? Is it meant to be a tautology? What does it take to be a 'new kind' of entity in the relevant sense? For in some reasonable sense it seems clear that we can begin to talk about a kind of entity which we have not previously been talking about without introducing a new linguistic framework. The framework can have, so to to speak, advance provisions for that in some cases.

'To recognize something as a real thing or event means to succeed in incorporating it into the system of things at a particular space-time position so that it fits together with the other things as real, according to the rules of the framework.' - Shades of pragmatism, or coherentism. Is 'recognize as F' a success verb here? Can't we have an idea of something which does fit well with our other current ideas but which is nevertheless the idea of something which doesn't actually exist?

Also, why a particular position? That seems false. Surely we can come to recognize the existence of something without knowing where it is. Also, the language Carnap uses here is quite unclear because while he is actually talking about changes on our end, i.e. in our linguistic representations and thoughts, he makes it sound like we are operating with the things themselves, incorporating them into a system.

- 'From these questions we must distinguish the external question of the reality of the thing world itself.' - This is a strong case for Carnap (who is talking here about the world of ordinary, observable physical objects), but one might wonder if it fails to generalize. Isn't there something special about our talk about the thing world? Carnap himself admits that the members of the thing world are 'the simplest kind of entities dealt with in the everyday language: the spatio-temporally ordered system of observable things and events'.

Couldn't it be this - the special foundational role played by talk of ordinary things - that makes it nonsense to ask whether they exist? (Or makes the question unsettlable?) Couldn't you have other cases where you've set up a framework which seems to licence certain "internal" existence statements, but where you can quite intelligibly and productively ask whether the things posited really exist at all? (And where this is not plausibly construed, as Carnap would want to construe it, as a practical question about whether to accept certain linguistic forms?)

It may be instructive to attempt to construct a clear, if artificial, example of this. (Here is a first thought, though there may well be much better examples available: a legal linguistic framework may treat the existence of a court as a basic assumption, without which the framework could not be applied. This doesn't mean we can't drop the legalese - step outside the legal linguistic framework - and ask about the existence of the court.)

Remember, linguistic frameworks can be embedded in larger linguistic frameworks. And in that way, we may be able to call the existence of the entities posited in some framework into question outside that framework, by remaining in a larger containing framework which allows us to treat the question in what Carnap would allow is a 'scientific, non-metaphysical' way. So that in the larger framework, the existence of the kind of entity in question is a question which may be answered empirically or a priori, in a non-trivial and non-metaphysical way, while in the embedded framework, the existence of the kind of entity in question is a basic assumption. I.e. something without which we can't really get off the ground with the embedded framework at all.

'The acceptance of the thing language leads on the basis of observations made, also to the acceptance, belief, and assertion of certain statements. But the thesis of the reality of the thing world cannot be among these statements, because it cannot be formulated in the thing language or, it seems, in any other theoretical language.' - Again, it seems like the thing language is a special case here, which makes Carnap overgeneralize. In the cases of less basic, less foundational frameworks, the latter disjunct 'or, it seems, in any other theoretical language' may be weak indeed.

Now Carnap turns to numbers, and again his case there is strong.

So, what I am saying is no threat to the core of Carnap's way of understanding what is wrong with metaphysical questions about the reality of ordinary things or the reality of numbers. Rather, it may lead to a nuancing of this and greater plausibility for it.

The problem isn't that you can never get outside a framework and ask about the reality of the things posited in the framework. On the contrary, you often can. And that helps explain why the attempt in the case of things and numbers is made at all.

So this more nuanced view has greater explanatory power. On Carnap's simpler view, according to which there is no such thing, ever, as getting outside a framework to ask about existence (literally, not as a disguised practical question), it is less clear why we would ever try.

Furthermore, the very idea of this going outside a framework, i.e. the very idea of what Carnap would call an 'external existence question', becomes clearer on the view I am suggesting. Rather than this mysterious thing which cannot in any possible case be done, it becomes something which can happen, and which we have examples of. On that basis, we may then argue that certain cases which bother philosophers are such that there is no properly analogous going-outside-the-framework to be done.