On Philosophy

September 26, 2007

Possibly Necessary

Filed under: Logic — Peter @ 12:00 am

Does it make sense to claim that something is possibly necessary, or necessarily necessary? I think the idea has intuitive appeal, but that it is a logically useless construction unless we are indicating something different with each possibility quantifier, in which case we are equivocating. Of the two options the idea that we are equivocating is probably a better possibility to pursues, if we feel that we absolutely must have the ability to make such statements. But while we might be able to rescue our ability to make claims using such statements I don’t think that the claims made in this way are very interesting, certainly they don’t have the same content as the intuitive understanding of such nested quantifiers.

Let’s begin by taking a look at what necessity means. The simplest way to understand necessity is as a purely logical construct, as a way of indicating which statements must logically be true, which can be true, and which cannot be true. Thus to assert □φ is to state then that φ is a tautology, □~φ is to state that φ is a contradiction, and ◊φ is to state that φ is neither a contradiction or a tautology, and that is truth cannot be determined by appeal to the axioms alone (which perhaps coincides with the idea that φ is an a posteriori truth). If ⇒ means “proves” then this means that ∅ ⇒ φ is the same as □φ, that φ follows from the rules of logic alone, no additional axioms needed. And thus ◊φ is true if and only if there is some consistent set of axioms S such that S ⇒ φ. If we wished to understand this in terms of possible worlds each axiom set would represent all the facts that are true in a particular possible world. Thus if some axiom set proves the sentence it is true in some possible world, and if every axiom set proves the sentence it is true in every possible world.

But when we stack such quantifiers we don’t add anything. Whether ∅ proves φ is not dependant on any axioms, only the rules of logic, which we assume are constant (lest we be unable to reason at all). Which implies that all claims about necessity and possibility are themselves necessary, because they don’t depend on any axioms. But there is no need to take this on faith, we can work through it logically. Assume then that ∅ ⇒ φ. Which means that if we start from ∅ we can produce a proof of φ. But doing that in turn proves that ∅ ⇒ φ. Thus (∅ ⇒ φ) → (∅ ⇒ (∅ ⇒ φ)). To put this back in terms of necessity and possibility this means that □φ → □□φ. Similarly if we can produce a proof of ~φ from ∅ then it follows equally obviously that □~φ → □□~φ. By being unable to prove ~φ from ∅ we can deduce that ◊φ (because if it isn’t a contradiction then {φ} is a consistent set of axioms that proves φ). Which means that ∅ ⇒ ◊φ, or, in other words, that ◊φ → □◊φ. And since this exhausts all the possibilities we must conclude that once we have introduced such a quantifier we can only add further □ quantifiers. ◊□φ and ◊◊φ are thus pointless because you can always assert something stronger. Which is why I say that in strictly logical terms nested quantifiers are useless, you can’t indicate anything more with nested quantifiers than you could with just one application of them.

But this interpretation rules out a posteriori logical necessity, if something is necessary you must be able to deduce it in an a priori fashion, because if you can’t then it must not be necessary after all. Obviously there are some (Kripke) who would argue that such a posteriori necessity does in fact exist. Thus such philosophers must mean something else by necessary than logically necessary. So instead of considering what is logically possible we consider instead an arbitrary set of possible worlds, bounded only by the fact that tautologies are true in each such world and contradictions true in none of them. But other than this the members of the set of worlds are arbitrary (in the sense that we can define the set however we like as long as we abide by those restrictions). Although a fact may be logically possible there may not be a possible world in this set where it is true, or it may be true in every one. In such a framework □φ simply means that φ is true in each world belonging to this set, and ◊φ means that φ is true in at least one world in the set. While I don’t object to such a construction by itself I would object to calling □ defined in terms of it necessity and ◊ possibility. What these quantifiers really denote depends on how the set is defined; how we determine which worlds belong to it. For example, the set may contain all worlds that could result from any set of initial conditions evolved over time according the physical laws we know. And I would call this a set of physical possibilities, and so □ would mean physically necessary and ◊ would mean physically possible. But perhaps this is simply a semantic quibble that arises form the fact that I naturally think of necessity, free of any context, as meaning logically necessary. So to sidestep this debate, let us simply agree to a principle of disambiguation, where we will refrain from referring to unqualified necessity from this point forward, and in terms of quantifiers □p will mean physically necessary, □l will mean logically necessary, and so on.

Let us proceed given this understanding. Consider an arbitrary □S, where S is a set of possible worlds, and □Sφ holds if and only if φ is true in each member of S. Now consider □SSφ. Obviously this holds only if □Sφ is true in each member of S. Here we are faced with a choice. One possibility is that S is defined in the same way in each member of S. This seems quite reasonable to me, we understand S as a kind of super-logical constant and thus it doesn’t vary (for example, the definition of what is physically possible is the same in every physically possible world, the definition of what is metaphysically possible is the same in every metaphysically possible world, and so on). If this is the case then we are back to where we were with logical possibility, each □Sφ implies □SSφ and each ◊Sφ implies □SSφ, thus making such nested quantifiers as redundant as in the case of purely logical necessity.

The more complicated choice to make is to say that S is not held constant within each member of S. In this case just because □Sφ holds doesn’t mean that □Sφ is true in each member of S. Such a situation is not impossible, per se, but now by nesting our quantifier we commit an error of equivocation. Previously we established that the meaning of □S was dependant on the definition of S. Thus when considering whether □Sφ holds within a particular member of S is actually to consider a different fact then □Sφ given our initial definition of S. And so a sentence such as □SSφ is deceptive since the first and second quantifiers are not in fact the same quantifier despite the fact that they are designated with the same notation. In fact the meaning of the second □S isn’t even fixed, a fact that is not reflected at all in our notation.

Now this does not rule out our ability to talk about nested quantification when we allow the definition of the set of possible worlds itself to vary, it just means that we have to be cautious in our reasoning involving such constructions. Which brings me to the assertion that ◊□φ implies □φ, a key claim in a version of a certain famous argument. The problem with that assertion is that either ◊□φ is equivalent to □□φ and thus □φ by definition, and hence the implication demonstrates nothing (if we are working under a definition in which □φ implies □□φ), or the implication cannot be established as true or false, thinking that it is necessarily true is to commit an error of equivocation. Simply consider a set Q containing two worlds: a and b (Q = {a,b}). And let us further stipulate that in a φ is true and Q = {a} and in b φ is false and Q = {b}. ◊QQφ is thus true because □Qφ is true in a. However □Qφ is not true because φ is not true in b. Now we could add further restrictions to the way Q may vary within possible worlds contained in that set and thus force the implication to be true, but the reasoning here points out the fact that we need not be committed to such an implication even if we want to allow meaningful nested possibility quantifiers. And thus an argument that relies on such an implication cannot succeed without showing something much more complicated about the structure of possible possibility and, in addition, arguing that we should even allow such complicated structures (since I for one am inclined to the view that sets of possible worlds should not be allowed to vary within the worlds that belong to them).

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