Friday, 27 March 2015

On Quine's Arguments Against QML, Part 2: The problem of "quantifying in"

The first of the two problems we look at is related to the problem of ‘quantifying in’. Versions of this argument can be found in [1,2,3]. Quine points out that modal contexts are intensional, by which he means simply that they are non-truth-functional [1, p. 122]; this is why the class of analytic truths is larger than the class of merely logical truths. Intensional contexts are opaque, and they “do not admit pronouns which refer to quantifiers anterior to the context” [1, p. 123]. To illustrate this, he gives his now-famous example of 9 and the number of planets. He says: “The identity

(3) The number of planets = 9

is a truth (so far as we know at the moment) of astronomy” [1, p. 119], [*]. Yet compare (4) “Necessarily something is greater than 7” and (5) “There is something which is necessarily greater than 7”:

(4) L¬∀x¬(x > 7)

(5) ¬∀x¬L(x > 7)

(4) “still makes sense”, according to Quine [1, p. 123], and further more it is true; take, for example, the number 9. But in contrast, (5) is “nonsense” [1, p. 124]. It is nonsense because L(9 > 7) is true, but L(The number of planets > 7) is false, even though 9 and the number of planets are the same (at least at the time he was writing). It is false because there is no analytic connection between ‘the number of planets’ and ‘> 7’.

The problem with this as an objection is that synonymy — and hence analyticity itself, since it is defined in terms of synonymy — is a contingent matter; it is accidental whether two terms are synonymous or not. In fact, the falsity of “The number of planets = 9” demonstrates the contingency of the matter; the fact that the IAU was able to redefine what it meant to be a planet, and hence change the number of planets in our solar system, shows that there is no necessary connection between the concepts ‘9’ and ‘the number of planets’. Their synonymy was only accidental.

At this point, an interesting parallel can drawn between this example and one that can be found in another area of modal logic, namely, the Aristotelian modal syllogistic. One of the long-standing difficulties commentators (ancient, medieval, and modern) have had with interpreting Aristotle’s modal syllogistic was the Two Barbaras problem: his insistence that NXN Barbara was valid while XNN Barbara was invalid. NXN Barbara is the first-figure syllogism Barbara with a necessary major, assertoric minor, and necessary conclusion, while XNN Barbara has an assertoric major and necessary minor. Commentators find common ground against Aristotle in two ways: Either they believe that neither form should be valid, or that if one is valid, there is no way to distinguish which one, and hence both should be. Here is an example in the form of NXN Barbara:

Necessarily all elms are deciduous.
All the trees in my yard are elms.
Therefore, necessarily all trees in my yard are deciduous.

One standard objection to the validity of such an argument is that the connection between being a tree in my yard and an elm tree is accidental; there is no deep underlying relation between these two concepts. This contingency in a sense “spills over” into the conclusion; it would be acceptable to draw an assertoric conclusion, but a necessary one is too strong.

Let us compare NXN Barbara with the following:

L(9 > 7)
The number of planets = 9
Therefore, L(The number of planets > 7).

In both cases, the way to rehabilitate the argument would be to necessitate the second premise; but in order to retain soundness this would require that ‘The number of planets = 9’ or ‘All the trees in my yard are elms’ be analytic (for only then would the result of prefixing them with ‘L’ be considered true, on Quine's account); but there is no reason to think that these premises are analytic.

The fact that the analogous argument is invalid, is, far from being a reason to reject quantifying into intensional contexts as incoherent, actually evidence that Quine is correctly analysing necessity-as-analyticity. This is exactly the sort of behaviour that we would want to see, since it is precisely because the identity statement is a merely accidental identity — as witnessed by the fact that while it used to be true, it is now in fact false — that we should reject the conclusion. Thus the problems that Quine sees arising from this example are not actually reasons for rejecting quantified modal logic, but rather reasons for embracing it: It is an advantage of Quine’s analytic approach to modal logic, not a disadvantage, that it makes such arguments invalid. Given that synonymy, and hence analyticity, is a matter of accident, we should not expect analytic identities to result in necessary conclusions, and if they did, we would have reason to question these conclusions on the same grounds that people question the validity of NXN Barbara.

References & Notes

• [1] Willard V. Quine. Notes on existence and necessity. Journal of Philosophy, 40(5):113–127, 1943.
• [2] Willard Van Orman Quine. Reference and modality. In From a Logical Point of View, pages 139–159. Harvard University Press, 2nd edition, 1980.
• [3] Willard Van Orman Quine. Word and Object. MIT Press, 1960.
• [*] Nowadays, of course, we know differently. It is rather amusing that two of the enduring platitudes in philosophy—that all swans are white and that there are nine planets — have both turned out to be false; Australia provided us with black swans, and the International Astronomical Union deprived us of Pluto.

Friday, 20 March 2015

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.

Thursday, 12 March 2015

A Strange Thing about the Brier Score

This post was co-written by Brian Knab and Miriam Schoenfield.

In the literature on epistemic utility theory, the Brier Score is offered as a paradigmatically reasonable measure of epistemic utility, or epistemic accuracy. We offer a case meant to put pressure on the claim that the Brier score in fact reasonably captures epistemic utility or epistemic accuracy.

1. A Simple Case

Consider two people contemplating the origin of the universe.

The simple deist is confident that a being exists that designed the universe. She is aware that cosmologists have developed non-design theories about the origins of the universe. However, she's confident that the non-design thesis is false.

So, according to the simple deist: deism is true, and the non-design thesis (which we’ll call “adeism”) is false. Deism and adeism form a partition of her possibility space.

The simple adeist, on the other hand, is confident that deism is false. She's confident that the universe came about without any help from a designer at all, and that the non-design thesis is true.

So, according to the simple adeist : deism is false, and the non-design hypothesis is true. Deism, and adeism form a partition of her possibility space.

It turns out: There is a deisgner!  (So deism is true, adeism is false). Who is more accurate?  The deist, obviously.

The Brier Score straightforwardly confirms this -- the simple deist is more accurate, according to the Brier Score, than the simple adeist.

2. A Problem Case

Now, consider again two people contemplating the origin of the universe. Both of them are admittedly somewhat uncertain about the existence of a designer. Both of them are aware of a large number of non-design theories of the origin of the universe.

The sophisticated deist  is more confident in deism than adeism. She has, moreover, also carefully considered all of the available non-design hypotheses, and has concluded that only one of them could possibly be true.  The rest, she thinks, are non-starters.

The sophisticated adeist  is, on the other hand, more confident in adeism than deism. She has also carefully considered all of the available non-design hypotheses, and although she thinks it’s likely that one of them is true, she has no opinions concerning which is the true one.  In her estimation, the non-design hypotheses are all equally likely.

Now, suppose it turns out: Deism is true (and so every non-design hypothesis is false). Who is more accurate?

We think: the sophisticated deist!  After all the sophisticated deist has the following two advantages over the sophisticated adeist: she has a higher credence in the truth than the sophisticated adeist does, and she has less credence invested in falsehoods than the sophisticated adeist does. So what advantage does the sophisticated adeist have over the sophisticated deist?  The only remaining difference between them is the way in which they distribute their confidence among the false hypotheses.  But why should the way in which the adeist distributes her confidence among the various false hypotheses make her more accurate in a world in which deism is true?

From the first personal side of things: if I want to have an accurate attitude about the origin of the universe and my choices are between being a sophisticated deist or a sophisticated adeist, I’d prefer to be the sophisticated deist, in the world in which deism is true.

But, in certain situations, and given enough non-design theories, the Brier score delivers the opposite verdict. For example, let D be the design hypothesis, and suppose there are 58 non-design theories, T1, T2, ... T58. Thus our partition is

By the description of the case, the ideal credences, across this partition, are (1,0,0,0,0...0)

Suppose the sophisticated deist’s credences are

Then her Brier Score is

Suppose the sophisticated adeist’s credences are

Then her Brier Score is

That’s a small victory for the adeist, admittedly, but the point is a structural one.  The adeist  -- in spite of the fact that she is a good deal less confident in the truth and, overall, a good deal more confident in the false -- is more accurate than the deist, according to the Brier. (For a related point -- one which trades on this same structural feature of the Brier Score --  see Knab, “In Defense of Absolute Value.”)

3. Discussion

That, we think, is enough of a puzzle to put some pressure on the Brier understanding of epistemic accuracy. More generally, the Brier Score fails to satisfy what looks like a plausible desideratum:

Falsity Distributions Don’t Matter: For any partition of theories: T1...Tn, a probabilistic agent’s accuracy with respect to this partition at world w should be determined solely by the amount of credence she invests in the true theory at w, and the amount of credence she invests in false theories at w.  The way she distributes her credences amongst the false theories at w shouldn’t affect her accuracy.

Tuesday, 3 March 2015

Second MCMP Summer School on Mathematical Philosophy for Female Students

After the huge success of last year's event, the Munich Center for Mathematical Philosophy will be hosting the second installment of its Summer School for female students, from July 26th to August 1st 2015. From the website:
The summer school is open to excellent female students who want to specialize in mathematical philosophy. Since women are significantly underrepresented in philosophy generally and in formal philosophy in particular, this summer school is aimed at encouraging women to engage with mathematical methods and apply them to philosophical problems. The summer school will provide an infrastructure for developing expertise in some of the main formal approaches used in mathematical philosophy, including theories of individual and collective decision-making, agent-based modeling, and epistemic logic. Furthermore, it offers study in an informal setting, lively debate, and a chance to strengthen mathematical self-confidence and independence for female students. Finally, being located at the MCMP, the summer school will also provide a stimulating and interdisciplinary environment for meeting like-minded philosophers.
Instructions on how to apply can be found here. This is a fantastic opportunity for all female students interested in the more technical, mathematical areas of philosophy to strengthen their skills and become more familiar with work on the area. Although I missed last year's event (and will sadly miss this one too), I am told that the general atmosphere was very friendly and encouraging, and so extremely conducive to the stated aims of the Summer School. So time to get going with those applications!