June 14, 2019

Pruss and Rasmussen on Modal Logic

Chapter two of Pruss and Rasmussen's Necessary Existence can be seen as preliminary to the main project of the book. The core aim of the chapter is the explanation and defense of a picture of metaphysical modality that is already (so it seems to me) standard among analytic metaphysicians. The chapter concludes with a brief demonstration of a proposition that will be a crucial lemma in many of the arguments throughout the book: if a necessary being is possible, then a necessary being is actual.

Those who are already immersed in analytic metaphysics and accept the modal system S5 could certainly skip this chapter and still be able to follow the remainder of the book. Readers not already familiar with modal logic and modal metaphysics will find this chapter heavy going (though it might be useful as an introduction to modal logic and modal metaphysics for students with strong background in symbolic logic). However, despite the fact that the central theses of the chapter are already widely held views, the chapter is full of interesting arguments. In particular, I want to mention two arguments, one for the conclusion that metaphysical modality is not equivalent to narrow logical modality, and one for the conclusion that the logic of metaphysical modality is S5. The parts of this chapter that will be of interest to people already familiar with analytic metaphysics are the most technical parts of the book, so this post will be more technical than the rest of the series.

Pruss and Rasmussen define "narrow logical necessity" as provability within a formal system with recursively specifiable axioms (p. 12). The identification of narrow logical necessity with metaphysical necessity is not feasible, according to Pruss and Rasmussen, for three reasons. The first is that "we need additional non-formal axioms about natural kinds ... in order to capture such necessities as that cats contain DNA, that water is material, or that acute thinkers aren't acute angles." This objection seems to me underdeveloped, but I can see two reasons why the addition of these non-formal axioms might be thought problematic for the proponent of the narrow logical necessity view. First, it might be thought that the introduction of additional axioms drawing connections between the predicates of the formal system (not just the logical vocabulary) undermines some of the motivation for the narrow logical necessity view. The addition of these axioms amounts to giving up on the idea that modality could be explained by or reduced to the formal features of quantifiers, logical connectives, etc. Second, it might be feared that these non-formal axioms are not recursively specifiable. As Pruss and Rasmussen note, if the narrow logical necessity view abandons the notion that the axioms are recursively specifiable, it's unclear whether it is a substantive theory any longer (see pp. 13-14).

The second reason the narrow logical necessity view won't work is that, as Pruss had previously pointed out (see section 1.2), Gödel's First Incompleteness Theorem implies that there are mathematical truths that are narrowly logically contingent, but it is highly implausible to suppose that any mathematical truths are metaphysically contingent.

The third and final reason against the narrow logical necessity view is the most original and well-developed. According to Gödel's Second Incompleteness Theorem, no sufficiently powerful consistent formal system can prove its own consistency. But (given classical logic) an inconsistent system can prove anything. Thus if you could prove that any statement was unprovable, you would thereby prove that the system was consistent. Therefore, it's not provable (within a sufficiently powerful consistent system) that any statement is unprovable. If necessity is provability within this system (as the narrow logical necessity view holds), then it will turn out that every statement is possibly necessary, even contradictions like 0=1 (p. 13).

Pruss and Rasmussen do not mention that an opponent might try to evade this argument by endorsing a non-classical logic that does not validate contradiction explosion. However, this seems like a desperate move, and it doesn't solve the other two problems.

The second argument I want to mention is the defense of S5. Essentially the structure of the argument is that if someone endorses an account of metaphysical modality that does not obey S5, that account can be used to construct a system that does obey S5, and the latter system has a better claim to be the fundamental modality than the latter. The basic idea can be seen easily in the possible worlds model theory. Formally speaking, S5 is the system in which the accessibility relation is an equivalence relation. One equivalence relation is the universal relation—every world is accessible from every other. The person who thinks modality doesn't obey S5 needs a model theory (if she uses the possible worlds model theory) that has some inaccessible worlds, but using that we can construct a system with universal accessibility which will have a better claim to be the fundamental modality than the original system.

The simplicity of the model theory of S5 has, I think, been given as a reason for endorsing it previously. However, Pruss and Rasmussen go farther by showing how to run the argument without using the model theory, and by giving some additional reasons for thinking that the system that results from this construction should be regarded as more fundamental than the system with which we started.

There's a lot more in this chapter. Like the rest of the book, it's densely packed with arguments. I don't have much to say by criticism because I, like most analytic philosophers, already endorsed the chapter's main conclusions. Nevertheless, it seems to me that the chapter makes a strong case for the picture of modality the authors favor.

Posted by Kenny at June 14, 2019 3:15 PM
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