An odd quantifier shift fallacy

I was flipping through the course packet for my Intro class this semester, and the professor included a chapter from some contemporary textbook. The chapter includes this paraphrase of Anselm's second ontological argument:

Premise 1: A being that cannot be thought of as not existing is greater than a being that can be thought of as not existing.

Therefore, if God can be thought of as not existing, then a greater being that cannot be thought of as not existing can be thought of.

Premise 2: God is the being than which nothing greater can be thought of.

Conclusion: God cannot be thought of as not existing.

Which is a little weird. But it's the ontological argument, so I guess weird shouldn't be a surprise.

Let Gxy abbreviate `x is greater than y' and Tx abbreviate `x can be thought of as not existing', and g name God.

(1) (x)(y)([-Tx & Ty] -> Gxy)
Therefore, Tg -> (Ex)(-Tx & Gxg)
(2) (x)(-Gxg)
Conclusion: -Tg

Even if we include both premisses in the inference to the `therefore', it's invalid. The hypothesis that God can be thought of as not existing does not imply that there is something which cannot be thought of as not existing. That is, a model in which only g exists, and g does T, is entirely consistent with (1).

One might try to read Anselm as instead arguing from

(1') (x)(Tx -> (Ey)(-Ty & Gyx))

Then the `therefore' follows by just instantiating the universal quantifier. Notice that (1') follows from the conjunction of (1) and

(1'') (x)(Tx -> (Ey)-Ty)

That is, `if something can be thought of as not existing, then something else cannot be thought of as not existing'. But what does the first clause have to do with the second? Maybe (just maybe) you can argue from the contingent-yet-actual existence of my chair to the existence of a necessary, first cause sort of being -- something had to cause my chair to be, and something had to cause that thing to be, and so on. But that won't work if the something is, say, my pet unicorn. Logically speaking, there's no reason to rule out the model with the empty domain -- in possible world terms, there's no reason given at this stage of the argument to rule out the possibility that nothing (not even God) exists.

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

An epistemological argument against metaphysics

Infamously -- at least, as infamously as I can make it -- I am radically indifferent to the concerns, problems, and techniques of metaphysics. I am thoroughgoing here: whether it's Shapiro, Quine, Heidegger, van Inwagen, Aristotle, Leibniz, or Aquinas, I think metaphysics is less interesting and important than the number of hairs on my head. This post is an attempt to justify that radical and thoroughgoing indifference to any metaphysicians that might happen to read it.

If the job of the sciences is to describe the way the world is, then, or so I've been told, the job of metaphysics is to describe the way the world must be. That is (well, this isn't exactly the same thing, but probably what's intended), metaphysics is in the business of discovering (metaphysically) necessary truths, truths of the form []p. This discovery often involves considering (metaphysically) possible truths, truths of the form <>p. I have problems with the idea that we can know such truths, if any such truths there be. That is, I don't think that beliefs that []p or beliefs that <>p are ever justified.

Let's consider an arbitrary necessary truth []p. How could we be justified in believing that []p? I want to consider this by asking what sort of argument could be given for []p. What sort of argument? Unless the metaphysician is prepared to say that we can come to know that []p by generalising empirical truths, presumably the argument should be deductive. More specifically, it should be logically valid and sound.

But what logic? The presence of the modal operator [] suggests that a modal logic. S5 is the most widely accepted, or so I've been told. Perhaps the reader prefers another modal logic. Or perhaps the reader thinks she can do her metaphysics with classical first-order logic. In any case, she will have a set of inferences rules for her logic, which can be stated as axioms, and these axioms will have to be necessary truths themselves. If they are not necessary, then her inferences will not apply across all possible worlds -- her logic will not be metaphysically sound. Furthermore, in order for our belief in the conclusions of the arguments built using these inferences to be justified, our belief in the necessity of these axioms must be justified itself.

In short, in order for metaphysical beliefs to be justified, our beliefs that logical axioms are necessarily true must be justified.

There, I think, only three meta-doxastic attitudes one can take towards axioms. First, on the Aristotelean or Classical attitude, axioms are contentual and must be self-evident. Alternatively, on the Hilbertian or Mathematical attitude, axioms are formal and simply stipulative. (Note that Hilbert himself was not a Hilbertian in this sense about all branches of mathematics -- he was an Aristotelean about finite arithmetic.) On the first attitude, axioms are the most basic and fundamental truths -- and probably this `fundamental' carries both metaphysical and epistemological tones. On the second attitude, axioms are simply laid down arbitrarily, and need not be true of anything. The first attitude has been prominent in Western thought since Antiquity. The second was a late nineteenth-century development out of non-Euclidean geometry and abstract algebra. The third or Contingentist attitude is to take axioms to be assumptions we make contingently, either implicitly or explicitly. Perhaps the axioms express reliable-but-not-necessary and contingent features of our cognition (a certain sort of neo-Humean might think this), or perhaps they express the theory of logic that best exemplifies the features we want a theory of logic to have (Quine thought this, Michael Friedman's `contingent a priori' is somewhat related, and my own view lies in the vicinity of both of these).

Illustrate these three with an example: modus tollens.

(((p -> -q) ^ q) -> -p)

The Aristotelean says that this is self-evident and fundamental in both metaphysical and epistemological senses. Bivalence and non-contradiction are deep features of reality and our knowledge of it. The Hilbertian says that this is just a certain rule for manipulating arrays of strings of symbols -- when you have a subproof that terminates in -q and a separate proof of q, then you can write down the negation of the premiss of the subproof. And the Contingentist says, perhaps, that this is a pragmatically useful posit -- it lets us create proofs by contradiction, which are very powerful indeed, and is only objectionable if you have a very strict understanding of the nature of mathematical reasoning or think truth means justified assertion.

If we take one of these attitudes explicitly, what sort of attitudes towards the axioms is justified? Are we justified in believing that they are necessarily true? For the Contingentist, we are at most justified in believing that axioms are true (that is, true of the actual world). Similarly, as there is no need on the Hilbertian attitude for the axioms to be true, we are at most justified in believing that axioms are true. To infer that axioms are necessarily true is to make a ridiculously hasty generalisation. We are justified, on these two attitudes, at best in believing them to be contingent truths. (Note that even this may not be justified: on certain sorts of Contingentism, such as the neo-Humean, we are not even justified in believing that they are true!) Finally, for the Aristotelean, if self-evidence is indeed justification, then it seems that we are justified in believing that the axioms are true. But self-evidence doesn't imply necessity. Indeed, for classical epistemological foundationalists, sensory beliefs are self-evident, but are assumed not to be necessary at all. If self-evidence is not justification, then it doesn't seem that we are even justified in believing that the axioms are true.

The problem is just this: Any story we tell about being justified in believing that the axioms of logic are true is consistent with them not being necessarily true. Indeed, any story we tell is consistent with the axioms of logic only being true of the actual world, and false of every other possible world.

Something entirely parallel happens if we prefer an externalist account of knowledge, replacing justification with warrant. Every non-ad hoc story the externalist epistemologist-metaphysician tells us about how our belief that the axioms of logic are true is warranted is consistent with them not being necessarily true, and indeed with them only being true of the actual world. The only way out that I can see is to propose that belief that []p is warranted just in the case that []p is true, or something equivalent, which is clearly ad hoc.

To summarise: In order to be justified in our belief that []p, a paradigm example of an important claim of metaphysics, we must first be justified in believing that the axioms of logic are necessarily true. But on no reasonable account of the status of axioms are they necessarily true. Hence our belief that the axioms are necessarily true is not justified. Hence we are justified in believing that []p.

The indispensability of identity conditions

Sometimes metaphysicians can only be dealt with by doing some metaphysics. So, as much as it pains me, this is a post on metaphysics.

Philosophers who fancy themselves `Quinean meta-ontologists' believe in a simple condition for existence claims: something of type K exists if and only if statements of the form [there is something such that it is a K] are true. That is, if and only if

(Ex)(Kx)

For example, to say that cats exist, represent [x is a cat] as Cx, and then assert that

(Ex)(Cx)

It's not immediately obvious how to translate `something exists' into the formal idiom. A typical strategy is to use self-identity, so that `something exists' is taken to be formalised as

(Ex)(x=x)

(NB Such a strategy is, I think, incompatible with the claim that existence is captured entirely by the existential quantifier. If that were so, then one wouldn't need identity to say that something exists. But that's not the main argument I'm making in this post.)

All cats are self-identical. Or, at least, the identity of cats (ie, whether one cat is or is not the same as `another' cat) is not considered deeply controversial. How do we say that all cats are self-identical?

(x)(Cx -> x=x)

Note that, from the claims that (1) cats exist and (2) all cats are self-identical, one easily concludes that something exists. Here's the derivation in all its glory, if you're curious:

(Ex)(Cx)
(x)(Cx -> x=x)
Ca
Ca -> a=a
a=a
---------------
(Ex)(x=x)

Some `Quinean meta-ontologists' believe in propositions. That is, they believe that propositions exist. If [x is a proposition] is symbolised as Px, then they believe that

(Ex)(Px)

Now, this is a departure from Quine. Quine very firmly believed that propositions did not exist. His argument was that there were no clear identity conditions for propositions -- it is generally impossible to tell whether one propositions was the same as `another'. Self-identity of propositions is, therefore, controversial.

The `Quinean meta-ontologists' who depart from Quine in this way claim that they do not need identity conditions. Some have claimed that they do not need identity conditions at all, for any type of thing, while others seem to commit to the far more modest claim that they do not need identity conditions for propositions. I take this to mean that they do not assert or assume that

(x)(Px -> x=x)

I claim (or at least I suggest as likely to be true, pending proof) that, from these two assumptions (or, more precisely, one assumption and one lack of assumption), one cannot conclude that something exists. That is, from (Ex)(Px) and the ordinary rules of first-order logic *without identity*, and without using (x)(Px -> x=x), one cannot derive (Ex)(x=x). In short, from the assumption `there is something such that it is a proposition', one cannot conclude that `there is something'.

It may be objected that I have crippled the `Quinean meta-ontologist' by forbidding the use of the logic of the identity relation =. But the `Quinean meta-ontologist' has done it to herself, by abjuring identity conditions for propositions. If there are no identity conditions for propositions, then neither of these two claims, nor any equivalent to them, can be assumed:

(x)(x=x)
(x)(Px -> x=x)

The first of these captures, as an axiom, the standard rule of =-introduction used when identity is added to first-order logic. The second is equivalent to it under the assumption (Ex)(Px). There are therefore both unacceptable.

And I claim (or at least suggest as likely to be true, pending proof) that at least one of these, or something equivalent to them, must be used to derive the desired conclusion. I thereby conclude that identity conditions for propositions (and, indeed, every other type of entity one would countenance) are indispensable to make good on the programme of `Quinean meta-ontology'.

The challenge to the `Quinean meta-ontologist' who believes in propositions is to give the required proof.