## December 09, 2007

### Zalta's logical truths that are not metaphysically necessary

In this post, I want to summarise the argument of Zalta's paper `Logical and analytic truths that are not necessary' for non-logicians who still know a little bit about modal logic.

The basic idea is as follows: Zalta identifies certain sentences which, first, are logical truths, in that they are true under all interpretations; but, second, they are not metaphysically necessary, in that they are not true at all possible worlds under at least one interpretation.

Zalta's argument focusses on definite descriptions. I want to use a simpler language: a standard sentential language L with a single unary operator @. If p is a wff of L, then informally, @p means p is true at the actual world.

More specifically, give L countably many primitive sentential constants p1, p2, p3, ..., and build the wffs of L using the standard logical operators ^, v, -, -> and the unary operator @ on the pi. A world W is a total function {pi : i in N} -> {T,F}, ie, an assignment of truth values. An interpretation A is an ordered pair , where D is a set of worlds and W0 is in D. Informally, W0 is the actual world under A. `p is true at W under A' is defined in the usual recursive way for the standard logical operators. `@p is true at W under A' is true if p is true at W0 under A. For purposes of order of operations, @ is stipulated to be of the lowest order, ie, it is evaluated after all other operators when there is ambiguity. `p is true under A' is true if p is true at W0 under A. `p is necessary under A' is true if, for all W in D, p is true at W.

p is logically true if, for all interpretations A, p is true under A. p is necessarily true if, for all interpretations A, p is necessary under A. Zalta goes to great pains to show that his semantics are entirely standard for languages which involve rigid designators; the only difference here is that the language is much simpler than Zalta's primary example.

For some q, let Q be the sentence

@q -> q

Informally, Q says that, if q is actually true, then q is true.

1. Q is logically true.

Q is logically true iff for all interpretations A, Q is true under A. Choose an interpretation Q. Q is true under A iff Q is true at W0 under A. Q is true at W0 iff either @q is false at W0 or q is true at W0. @q is false at W0 iff q is false at W0. So Q is logically true iff q is false at W0 or q is true at W0. So Q is logically true iff either q is false or q is true at W0. Since W0 is total, either q is false or q is true at W0. Hence Q is logically true.

2. Q is not necessary.

Q is necessary iff for all interpretations A, Q is necessary under A. Q is necessary under A iff, for all W in D, Q is true at W. Q is true at W iff either @q is false at W or q is true at W. @q is false at W iff q is false at W0. Hence Q is necessary iff for all interpretations A and all worlds W in D either q is false at W0 or q is true at W.

Hence Q is not necessary iff there is some interpretation A with some W in D such that q is true at W0 and q is false at W.

Let A be an interpretation such that q is true at W0 and false at all other W in D and let W != W0. Then q is true at W0 and q is false at W. Hence Q is not necessary.

3. There are logical truths which are not necessary.

The `counterintuitive' gap arises because of a difference in scope between necessity and logical truth. Logical truths must be true at all actual worlds under all interpretations. It does not look at any worlds besides the actual world in each interpretation. Hence the actuality operator @ is inert with respect to logical truth: @p is logically true iff p is logically true.

It is with respect to necessity that @ is doing some work. Necessity does not just consider the actual world of each interpretation. It considers the full `multiverse' of each interpretation. Necessity is actually much broader than logical truth, in the sense that it takes more into consideration. Indeed, necessity implies logical truthhood, but as 3 shows, the converse is not true. In my experience, metaphysicians think exactly the opposite -- that logical truthhood implies necessity, but not vice-versa.

This version of Zalta's example show why metaphysicians should want necessity to be the broader notion in a very clear way. Q said that, if q is actually true, then q is true. If this were necessary and q were actually true, then q would itself be necessary. That is, if logical truth implies necessity, then rigid designation makes every true sentence necessary. Modality collapses, and every `multiverse' consists of a single possible world.

If modal metaphysicians wish to retain the equipment of rigid designation or actualism and any robust notion of possibility, then they must deny the inference from logical truth to necessity, at least with respect to statements that might involve rigid designation or reference to the actual world.

Yet, as Zalta points out, this creates a serious problem for metaphysical methodology. That a modal statement is self-contradictory -- is logically false -- does not imply that it is a necessary falsehood -- necessarily not the case. It implies only that it is not necessarily the case.

Zalta's paper is `Logical and analytic truths that are not necessary', J Phil (Feb. 1998: 85.2), pp 57-74. JSTOR link