# Does the existential quantifier express existence?

Does the existential quantifier express existence?

The existential quantifier is a symbol of symbolic logic which expresses that the statements within its scope are true for at least one instance of something.

In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

But some Pegasuses have flown: surely there does not exist any Pegasus at all.

• In the standard logic, yes, it expresses existence. "Pegasus" is interpreted as a predicate, and "Pegasus flies" is translated as "if x is a Pegasus then it flies", so its existence is not asserted. This is called Russell's paraphrase. There are alternative logics that divorce existence from the quantifier, and introduce an existence predicate instead. Aug 14, 2019 at 22:29
• again, thanks, esp for for the last sentence (for which a link would be amazing) @Conifold
– user38026
Aug 15, 2019 at 1:34
• As well as free logics, which others have mentioned, there is also the substitutional interpretation of quantification, under which the existential quantifier is understood not as expressing the existence of a thing within a domain of things, but as expressing that there is a true substitution instance of a sentence in which some linguistic expression replaces the variable... Aug 15, 2019 at 4:14
• There is a brief description of this in plato.stanford.edu/entries/quantification/#SubQua A more definitive but much longer and more technical account is Saul Kripke “Is There a Problem About Substitutional Quantification?”, in Truth and meaning, Evans and McDowell, Oxford: OUP, pp. 324–419, (1976). Aug 15, 2019 at 4:14

Yes, the existential quantifier expresses existence.

If you assert that

Some pegasus are flying

then you do assert that pegasuses exist, at least by the classical logical treatment of the existential quantifier, and I would claim also by the intuitive understanding of the sentence. If there are some pegasuses which are flying, then well, there are some pegasuses. If you argue that there exists nothing which is a pegasus, then stating that some of them are flying can not be a true statement.

Of course you can talk about fictional domains, and the universe you are talking about may not coincide with the actual world. Since predicate logical formulas are interpreted relative to a structure (that is, a pair consisting of a domain (= a set of objects that is being talked about) and an interpretation of all the predicates, names and function symbols), whether we can draw the conclusion that pegasuses exist in our objective physical universe depends on whether the structure you are evaluating the formula in reflects that actual physical universe we live in. A predicate logic statement is never just true or false -- truth is only defined relative to a structure, so before you can evaluate such a statement in the first place, you need to fix which universe you are talking about. If you are talking about a fictional universe, e.g. the characters in some fantasy book, then you are not making any claims about the existence of objects in the real world. But within the structure you are evaluating the statement in, that is, in the universe of objects that you talk about with your logical language, anything which is quantified by "some" or "there is" exists. That's just what the existential quantifier means.

I was deliberately turning your past perfect "have flown" into a present tense "are flying" because the past tense adds additional complication -- intuitively, it seems plausible that the existence of objects can change over time; pegasuses might once have been flying around but then went extinct, so at the present time, "Pegasuses have flown (at some point in time earlier)" could be a true statement while at the same time pegasuses presently do not exist. Tense is not systematically accounted for in standard predicate logic; we must assume that all formulas are evaluated at the same point in time, and that any object to which some predication (as "being a pegasus") applies exists as an actual object in our structure. Extensions of predicate logic which do capture time relativity and changing domains are modal logics and time logics (and combinations thereof).

If you assert that

Pegasuses have flown

then the question is how to best translate this into predicate logic, because there is no overt quantifier present in the natural language sentence -- it's just the raw predicate "pegasuses".

In this particular context, the most natural interpretation for me seams to be that this sentence is more or less synonymous to "Some pegasuses have flown", with the interpretation explained above.

In other contexts, like

Pegasuses have wings

an interpretation with a universal quantifier seems more reasonable: This statement is probably best to be read as "All pegasuses have wings" -- or, paraphrased more closely along the lines of its formalization, "For any individual it holds that if it is a pegasus, then it has wings". In classical logic, unlike the existential one, the "all" quantifier does not have existential import: A statement of the form "All A B" will -- maybe a bit counterintuitively -- be vacuously true, rather than false, if there are no objects which have property A at all -- here are some reasons why. So when interpreting the sentence "Pegasuses have wings" as "All pegasuses have wings", we do not assert that pegasuses necessarily exist. But if we translate the sentence as "Some pegasuses have wings", then there must be pegasuses.

Of course, the unquantified English sentence one could have all kinds of more fine-graned meanings -- like "Most pegasuses have wings", or "A prototypical pegasus has wings" -- that the two standard quantifiers "for all" and "exists" can not capture, but this is already out of the scope of this questin.

• so "existence" can also mean fictional existence. that's great, thanks. fwiw, i have a hunch you believe i'm more confused about basic philosophy than i am :)
– user38026
Aug 15, 2019 at 1:42
• I think the confusion here is not about basic philosophy -- I don't doubt you can imagine fictional universes exist -- but about how predicate logic works, formally. That truth is only defined relative to structures, and evaluating a proposition like "Some pegagsuses are flying" without explicitly specifying which domain and interpretation is being talked about is senseless because that's simply not defined, is the crucial point here. Aug 15, 2019 at 10:50

According to the authors of forallx quantifiers range over a domain. Furthermore, the domains for classic, first-order order logic are not empty: (page 174)

A domain must have at least one member.

Logic that permits empty domains are called "free logics". As John Nolt describes:

Classical logic requires each singular term to denote an object in the domain of quantification—which is usually understood as the set of “existing” objects. Free logic does not.

Regarding Pegasuses, either one would not consider using a first-order logic to describe them or one might construct a domain of fictional characters. In any case, the domain has to be non-empty unless it is a free logic.

Nolt, John, "Free Logic", The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2018/entries/logic-free/.

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf

• I don't see how this answers the question. The problem is not whether the domain in the universe being talked about is allowed to be empty or not, but whether asserting "Some pegasuses x" implies the existence of individuals to which the predicte "pegasus" applies. Aug 14, 2019 at 23:01
• @lemontree Whatever is in the domain exists in the sense that it is in the domain. Both quantifiers range over the domain including the existential quantifier. Aug 15, 2019 at 1:04
• As an aside, whether "first-order logic" is used to refer to the version allowing empty domains or the version requiring nonempty domains depends on the subject and/or author.
– user6559
Aug 15, 2019 at 3:10

On some understandings of this question, the answer is no. In the SEP article on ontological commitment, for example, they discuss Meinongian quantifiers, prefacing the discussion thus:

According to the third objection to sufficiency, the quantifiers of first-order logic, properly understood, do not carry existential commitment; they are not “existentially loaded”. Indeed, calling ‘∃x’ the “existential quantifier” is a misnomer; it would be better to call it the “particular quantifier” in contrast with the “universal quantifier”. Ordinary language, on its face, supports the view that quantification need not be existentially loaded (see §4).

To illustrate: it is seemingly true that, "There does exist not..." = "There does not exist..." but, "Not for some..." ≠ "For some not..." So thus far, and on the level of ordinary (English) language, the quantifier for "some" modulo "all" has an at least slightly different significance than "there exists." Now, I wouldn't recommend pushing this subtlety too far, however; the word "existence" is ambiguous, and is used modulo "some"-quantification faithfully (see e.g. Kant's argument about the weakness of the difference between one hundred actual and one hundred possible ðalers).

And yet even "for some" is ambiguous enough to sustain, sometimes, the indicated symmetry, for we might mean to say, "Not for some..." so as to refer to a few that are not X (in whatever intended sense), rather than to say that there are none that are X.