A Note on Empty Sets and Quantifiers

We talk about many things that do not exist. Dragons, unicorns, perhaps the modern maritime megalopolis of Atlantis, and so on. More practically, we may posit and discuss theoretical scientific entities — entities whose existence we have no evidence for apart from their theoretical or explanatory virtue. For instance, we can talk of the inclement weather as a result of Zeus’s wrath. A set of non-existent entities is empty. Consider the set of all dragons and count its elements. 0. Same for unicorns and modern maritime megalopoleis. That is, they are all equivalent to the null set.

Still, we want to be able to make nontrivial assertions about non-existent entities. We should be able to say things about them, for instance, “Kronos was a titan.”  Suppose someone said “No Kronos was a titan”.  On a certain level this seems contradictory.  But on the following logical construal, we can make sense of it.

Take the proposition, (1) “all dragons are green”. Because the set of dragons is just the null set, this proposition is trivially true. For we express it as, \forall x (Dx \rightarrow Gx). Which is to say: for all x, if x is a dragon, then x is green. Because no x is a dragon, the antecedent of the conditional, “if x is a dragon”, is always false. Therefore, the whole conditional is always true; so “all dragons are green” is true.

Now take the proposition (2) “no dragon is green”. (1) and (2), prima facie, are contradictory. But (2) turns out to be true, also. We express (2) as, \neg \exists x (Dx \wedge Gx). Or: it is not the case that there exists an x which is both a dragon and green. Because there does not exist a dragon, (2) will always be true.

But the logical representation isn’t exactly faithful to the assertive content of the linguistic utterances (1) and (2).  Existence, or lackthereof, is not explicitly asserted in (1) or (2).  This becomes apparent when we translate the logical sentences back into natural language.  “\neg \exists x (Dx \wedge Gx)” translates back into, “there does not exist an object such that it is both a dragon and green.”  And “\forall x (Dx \rightarrow Gx)” translates back into, “for all objects, if that object is a dragon then it is also green.”   These seem a little bit different than their counterparts in (1) and (2).

A different approach.  Prima facie it seems that “no dragon” and “all dragons” denote the same object, namely the null set.  And the predicates are the same.  So we are supposed to admit that the sentences are saying the same thing.  But it seems contrary to intuition to say that these sentences “say the same thing” for lack of a better word.  They denote the same thing insofar as their subjects and predicates denote the same things.

 

Here’s the thought we are addressing. The set of all dragons is empty. So, prima facie, “no dragon” and “all dragons” refer to the same set (the empty set). So when we predicate “is green” to both “no dragon” or “all dragons” we are referring to the same thing, and consequently the predicate must be true of both.

But “no dragon” and “all dragons” do not denote the same set. “Dragons” denotes the (empty) set of dragons. But quantifiers do not operate on the set they are attached to (e.g. “no” [or “all”] does not qualify the subject “dragons”). Instead, quantifiers modify the denotation of the predicate; in this case, “is green”. “No” means that “is green” is not true of anything in the salient set (this case, dragons). “All” means that “is green” is true of everything in the set. So (1) says, “the predicate, ‘is green’, is true of every individual in the set of dragons”. And (2) says, “the predicate, ‘is green’, is true of no individual in the set of dragons. So the conjunction of (1) and (2) says “the predicate, ‘is green’, true of both all the elements and none of the elements in the set of dragons”. And this is more than just apparently contradictory, it is actually contradictory, for nothing, regardless of its existential status, can be both p \wedge \neg p.

“All” and “No” do not modify the set of dragons. If they did, then there is no contradiction between (1) and (2) because the same predicate, “is green”, is applying to all the same elements. But “all” and “no” modify predicates, not sets. Consequently, in the conjunction of (1) and (2) “is green” and “is not green” are applied to all the same elements, and therein lies the contradiction.

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