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.

An Analysis of Davidson's Slingshot Argument

There is a peculiar kind of logical fallacy which, ironically, is only committed by people who have an acquaintance with formal logical theory. Fallacies of this kind arise when principles of inference from formal logic are applied inappropriately to arguments carried out in a natural language.

Here I make a case-study of Donald Davidson's famous version of the Slingshot argument against facts. The argument, in its dialectical context, is meant to show that if true statements correspond to facts, then every true statement corresponds to every fact. Davidson tries to demonstrate this conditional in order to motivate us to give up its antecedent (that true statements correspond to facts). Here is the argument:

The confirming argument is this. Let ‘p’ abbreviate some true sentence. Then surely the statement that p corresponds to the fact that p. But we may substitute for the second ‘p’ the logically equivalent “(the x such that x is identical with Diogenes and p) is identical with (the x such that x is identical with Diogenes)”. Applying the principle that we may substitute coextensive singular terms, we can substitute ‘q’ for ‘p’ in the last quoted sentence, provided ‘q’ is true. Finally, reversing the first step we conclude that the statement that p corresponds to the fact that q, where ‘p’ and ‘q’ are any true sentences. (Davidson 1969, p. 753.)

Let us go through it bit by bit.

The first apparent inference in the argument is curious. 'Then surely' suggests that reasoning is taking place here, but from what? Apparently:

Let ‘p’ abbreviate some true sentence.

But that is an instruction, not something we can infer from at all. This shows that what Davidson has supplied is not an argument, so much as a recipe for making one. And since this first step is not an inference, it can't be a fallacious inference. Still, in its slightly confusing use of a technique from logic (in this case, schematization) it gives us a small taste of things to come.

Now, following Davidson's recipe, we shall let 'p' abbreviate 'snow is white'. For perspicuity, we shall not use these abbreviations in our writings-out of the steps of the argument. (Surely this could not affect validity.) Thus our first real premise is:

The statement that snow is white corresponds to the fact that snow is white.

Now we are told we may make a substitution, yielding:

The statement that snow is white corresponds to the fact that (the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes).

The first thing to note about the above is that it doesn't obviously mean anything. This should make us suspicious. After all, we are not supposed to be merely calculating with signs here. This is supposed to be an argument - a reasoned chain of statements leading to a conclusion. How did we get to the above sentence, then? There are two things Davidson needs us to accept if we are to go along with this inference:

(1) That it is valid when arguing in English to substitute, for a sentence, a logically equivalent sentence - even when this sentence is embedded in a larger one.

(2) That 'snow is white' is logically equivalent to '(the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes)'.

In trying to assess these claims, we face a stumbling block: the lack of a clear, agreed upon notion of logical equivalence as a relation between sentences of natural languages. Some would say that 'snow is white' is logically equivalent to 'snow is white and Socrates is either mortal or not mortal'. Others would deny this, on the grounds that Socrates' existence is not implied by the original sentence. Some would say that 'John is a bachelor' is logically equivalent to 'John is an unmarried man', by the logics of bachelorhood, gender and marriage. Others would say these are perhaps analytically, but not logically, equivalent, because the equivalence does not turn on the use of "logical vocabulary".

Having made due note of this difficulty, let us observe that Davidson has no problem bringing in, out of the blue, mention of Diogenes. This gives us some handle on Davidson's intended notion of logical equivalence - enough, I think, to justify us in sweeping the difficulty under the carpet so that we may proceed to ask if (1) might be true.

That the answer is 'no' can be seen from these invalid instances:

(i) It is obvious that snow is white. Therefore, it is obvious that snow is white and [some elaborate and opaque tautology].

(ii) If you assume that the square root of two is rational, it is easy to derive a contradiction. Therefore, if you assume that [some elaborate and opaque logical equivalent to 'the square root of two is rational'], it is easy to derive a contradiction.

(iii) The statement that snow is white involves no semantic concepts. Therefore, the statement that snow is white and "grass" either refers to grass or does not refer to grass, involves no semantic concepts.

(In classical formal logic, the range of possibilities for sentential embedding is far narrower than in natural languages, and therefore no analogous counterexamples arise.)

How about (2)? For a start, can we even understand '(the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes)'? The use of the variables and brackets is, in itself, not a deal-breaker, since we can understand '(the x such that x is identical with grass) is green'. But now: on this understanding, what is the role of that which comes after 'such that' in the bracketed construction? Intuitively, the construction as a whole is a referring term, and after the 'such that' ought to go conditions relating to the variable which are met by exactly one of its possible values, thus determining a unique referent.

But then what happens if, as well as conditions involving 'x', we insert closed sentences like 'snow is white'? Well, on the intuitive idea behind the bracketed construction, this just doesn't make sense. Nevertheless, "appropriate" reference-conditions come to mind: a bracketed 'the' construction refers iff the conditions relating to the variable are met by exactly one object and all constituent closed sentences are true. To complete the semantics, we can stipulate that if such a construction refers, it refers (of course) to the condition-meeting value of the variable.

Thus we can define a new kind of referring construction, albeit a strange one. Also, it does appear that our complicated identity sentence, in light of this definition, is logically equivalent (in some sense) to 'snow is white'. Of course, this is of no use to us, since the principle whose application we wanted the equivalence for is invalid.

Before we move on: the addition of this new referring construction to our language may render previously valid principles invalid, so we must now be extra careful. (If, earlier, we had decided that (1) was true - that the unrestricted substitution of logical equivalents was valid - we would now have to go back and reconsider.)

Now, despite the fact that things aren't going very well for our argument, let us press on. We have gotten as far as:

The statement that snow is white corresponds to the fact that (the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes).

And now, citing the principle that we may substitute coextensive singular terms, Davidson has us substitute some true sentence - let us pick 'grass is green' - for 'snow is white'. (This then yields a new 'singular term', '(the x such that x is identical with Diogenes and grass is green)'.) Thus we get:

The statement that snow is white corresponds to the fact that (the x such that x is identical with Diogenes and grass is green) is identical with (the x such that x is identical with Diogenes).

And now we must ask: does the principle of substitution of coextensive singular terms hold in natural language? Notoriously, and as anyone familiar with twentieth-century philosophy of language will know, it (very arguably) does not; there are numerous contexts where such substitutions (strongly seem to) fail. (Witness the existence of intensional logics.) Here is an example of one kind of invalid instance:

Lois Lane knows that Clark Kent is Clark Kent. Therefore, Lois Lane knows that Clark Kent is Superman.

There are also well-known problems with substitution into modal contexts. Furthermore, and closer to our current context: 'the fact that Clark Kent is Clark Kent' does not obviously have the same reference as 'the fact that Clark Kent is Superman', even though the differing embedded singular terms are coextensive. And certainly the statement that Clark Kent is Clark Kent is not identical to the statement that Clark Kent is Superman. For all these reasons, we can not accept an unrestricted principle of substitution of co-extensive singular terms. Thus our last inference was invalid.

Since the final inference is a reversal of the first substitution, that concludes our step-by-step evaluation.

If there be any residual doubt about the invalidity of Davidson's argument (recipe): note that no special properties of the sentence 'The statement that snow is white corresponds to the fact that snow is white', beyond its embedding 'snow is white', are drawn upon in the derivation of 'The statement that snow is white corresponds to the fact that grass is green'. If this were really a valid way of arguing, we would also have to accept the following:

Suppose there is a chameleon, Euclid, who lives in a field of grass. Suppose further that Euclid is green because grass is green. Using Davidson's form of argument. we can infer from this supposition first:

Euclid is green because (the x such that x is identical with Diogenes and grass is green) is identical with (the x such that x is identical with Diogenes).

Then:

Euclid is green because (the x such that x is identical with Diogenes and Davidson is the author of 'True to the facts') is identical with (the x such that x is identical with Diogenes).

And finally:

Euclid is green because Davidson is the author of 'True to the facts'.

Tristan Haze

Reference

Donald Davidson. True to the facts. The Journal of Philosophy, 66(21):748–64, November 1969.