On Quine's Arguments Against QML, Part 1: Modality and Analyticity

When teaching philosophical logic to undergraduates, I feel I have two responsibilities: (a) To teach them logic and (b) To teach them something of the historical development of the field. (Alas, given constraints arising from not enough time, (b) generally means saying something about 20th C developments, rather than what I'd really like to tell them about, namely, 13th and 14th C developments!) This means that when the part of the module where I teach quantified modal logic (QML) came around, I felt honor-bound to introduce them to Quine's arguments against it, and, further, to say something about how I view this arguments. This post and its successors arose from that project.

Philosophers often appeal to Quine's conclusions that QML is "meaningless" [1, p. 124] or has "serious obstacles" [2, p. 43] to justify why they do not consider QML. This, I think, does a great disservice, not only to QML, but also to other philosophers (particularly undergraduates) because it merely parrots his conclusions without engaging in them. Since I fall firmly on the side of thinking that QML is a worthwhile area of research which can be done coherently, the responsibility falls to me to explain where I think Quine's arguments against QML have gone wrong.

I have found that explanation rather easy: I don't think his arguments are wrong. I think where he has gone wrong is taking the phenomena that they demonstrate to be problematic, rather than recognizing that they are the natural consequences of his definition of necessity, in terms of analyticity. In the following posts, I will look at two of his arguments and show that what he is picking out by them are exactly what you would expect to happen in QML if necessity is defined as analyticity. In this, I will first look at what he says concerning the relationship between necessity and analyticity.

Because he wishes to define necessity in terms of analyticity, Quine first looks at the notion of analyticity in non-modal contexts. In such contexts, it is possible to identify a notion of logical truth which can be used as a touchstone against which to measure the concept of analytic truth. In a non-modal context, every logical truth, he says, is "deducible by the logic of truth-functions and quantification from true statements containing only logical signs" [2, p. 43], such as ∀x(x = x). [3] The class of analytic statements is "broader than that of logical truths" [2, p. 44], because it contains statements such as the following:

(1) No bachelor is married.

The truth of this statement is warranted on the basis of the relation of synonymity, or sameness in meaning (or intension, cf. [2, p. 44]), between ‘bachelor’ and ‘unmarried man’, and in fact synonymy proves to be the crucial concept in defining what it means for a sentence to be an analytic truth:

Definition A statement is analytic if by putting synonyms for synonyms (e.g., ‘man not married’ for ‘bachelor’, it can be turned into a logical truth [2, p. 44].

In order for this definition to prove fruitful, it must be spelled out precisely what is meant by ‘sameness of meaning’; this, however, is a complicated task, and one that many have struggled with to date without achieving full success. It is not necessary, thankfully, to have a complete answer here: If we suppose, as Quine does, that "there is an eventually formulable criterion of synonymy in some reasonable sense of the term" [2, p. 44], then we can appeal to this criterion even if we don’t yet know what it is.

That (1) is an analytic truth on this definition is clear by seeing that

(2) No man not married is married.

is a logical truth.

It is important for Quine that he provide a suitable definition of what counts as analytic because of the close relationship that he sees existing between analyticity and modality. He asserts that there is an analogy between necessity and analyticity in exactly the same way that there is between negation and falsity [2, p. 45]:

The contrast between ‘necessarily’ and ‘is analytic’ is exactly analogous to the contrast between ‘¬’ and ‘is false’. To write the denial sign before the statement itself. . . means the same as to write the words ‘is false’ after the name of the statement [1, p. 122].

When it comes to modality and analyticity, this close relationship is expressed in the following way:

Lemma The result of prefixing ‘L’ to any statement is true if and only if the statement is analytic [2, p. 45].

Given the usual connection between necessity and possibility, it follows that the result of prefixing ‘M’ to any statement S is true if and only if ¬S is not analytic.

References & Notes

• [1] Willard V. Quine. Notes on existence and necessity. Journal of Philosophy, 40(5):113–127, 1943.
• [2] W. V. Quine. The problem of interpreting modal logic. Journal of Symbolic Logic, 12(2):43–48, 1947.
• [3] Whether = is, strictly speaking, a logical sign he does not discuss; and for our purposes it does not matter if we grant to him that it is.

---

© 2015 Sara L. Uckelman