Existence part II. Quantifiers

The conception of non-existent objects creates more problems than it solves. But there is an alternative approach. We can insist that all objects are existent and still solve the semantic problem of negative existential statements. One such solution is based on the observation that "existence" is a second-order notion, i.e. it applies not to things but to concepts. The sentence "Vulcan does not exist" is in fact about the concept of "Vulcan", and it states that the concept is empty (and not that it does not exist - obviously we have to assume that concepts exist).


Another solution is as follows. When we state that Vulcan does not exist, we in fact talk not about (non-existent) Vulcan, but about all objects in our universe - and we say that there is no Vulcan among them. Hence the sentence can be understood as being equivalent to "No object is Vulcan", which clearly does not carry the presupposition that Vulcan is an object. Using the definition of the term "Vulcan" we can express the same thought in an even more straightforward fashion as "No planet is closer to the Sun than Mercury", which leaves no trace of suggestion that it is some non-existent object that we talk about.

According to the quantifier conception of existence, all existential statements have to be reformulated in order to reveal their genuine logical structure, different from their surface grammar. The sentence "Elephants exist" does not attribute the property of existence to elephants, but rather speaks in a general fashion about all objects in the universe, claiming that some of them are elephants. Thus the literal form of the sentence should be "There is an x such that x is an elephant", where expression "there is" is known as the existential quantifier, "x" is a variable that can take any value from the domain of all objects, and "is an elephant" is a predicate. The formalization of the statement "Vulcan does not exist" leads to either "It is not the case that there is an x such that x is Vulcan" or, equivalently "For all x, x is not Vulcan" ("for all" is another quantifier, called universal).

It has to be stressed that the expression "x is Vulcan" is to be understood as consisting of the predicate "is Vulcan", and not as being equivalent to the identity statement x = Vulcan. In logic you can apply the identity symbol only to variables and terms called "proper names", of which it is assumed that they represent exactly one object. "Vulcan" cannot be interpreted as a proper name, since there is no object that is represented by that name. Generally, no empty name can be interpreted as a proper name, but only as a predicate.

One consequence of the quantifier view of existence is that certain expressions in natural language become "ontologically committing" (this notion was introduced by Willard V.O. Quine). For instance, the use of the word "some" indicates that we have to accept the existence of certain objects. If I say "Some thoughts of philosopher A are hard to understand", my utterance should be literally interpreted as "There is an x such that x is a thought of philosopher A and x is hard to understand", which implies that there are objects which are thoughts. In order to avoid unwanted ontological commitments, we can try to paraphrase our statements so that they no longer imply the existence of dubious entities.

Some critics point out that the quantifier view of existence commits us to the truth of the statement "Everything exist", which is clearly absurd. While it is true that this is a consequence of the quantifier view, it is by no means absurd. "Everything exists" means "For all x, x exists", and if we accept the quantifier interpretation of existence this statement has to be interpreted as "For all x there is a y such that x = y", which states a trivial truth that everything is identical with something. Instead of saying that not everything exists (which is self-contradictory on any interpretation that eliminates non-existent objects) we should better express our intuition in the statement "Some concepts are empty (non-referring)".

Further reading:

B. Garrett, "What is existence?" in What is this thing called metaphysics?