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.

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