Why does existential quantification appear to be predicative?

Question

When one uses the universal quantifier '∀', it is ungrammatical to not include both a variable it binds, and a formula it scopes over: ∀ x, Φ. Likewise, in natural language, "For all x, Φ is true" is meaningful.

The same is true for sentences using the existential quantifier '∃': ∃ x, Φ. However, in natural language, it sounds sensible to say, "x exists".

One reason it does not make sense in formal logic, is that variables are not "things in the domain"; when you say "x exists", the sentence isn't meaningful because x doesn't refer to anything.

c, a constant, refers to something, but it is still formally wrong to say ∃ c, because we only bind quantifiers to variable symbols, not constants.

One thing you can say, is ∃ x, x = c.

However, given the standard presentation of a signature in first-order logic, the above formula would always be unneeded in a set of axioms. If a constant symbol c exists in the signature, it is required to be mapped to an element in the domain. Thus, it is always given that in a model of a theory, for every constant c, there exists an x such that x = c. (I'll give you 1 million dollars if you can guess which value of x it is.)

There is some interesting writing here for reference, that I include as context for my question. I haven't read it yet.

This came about by trying to formalize the syntax of first-order logic, which I am told is carried out in the textbook by Enderton, which I will read soon.

When we present a signature as a set, I do not think we are declaring the existence of this set. In general, we assume the existence of a world of sets; we can only point to sets that already exist, by specifying them.

However, it feels natural to begin a formal theory by saying: "The following sets exist: a set of constant symbols, a set of variable symbols, etc."

It feels awkward to say, "x equals a particular set". But perhaps this is, after all, the 'correct' way. Which is good to know.

Answer 1

Asking why the existential quantifier appears to be a predicate is another way of asking the traditional question of whether existence should be regarded as a property. In natural language, we do often use existence as a predicate. We might say that Daniel Radcliffe exists, by way of contrasting this with Harry Potter does not exist.

In the standard way that logic is done, existence is treated as a quantifier, not a predicate. But in free logics, existence can be treated as a property that some individuals have, by contrast with those that merely possibly exist. There are many ways of handling possible or potential objects, including appealing to possible or fictional worlds, modal counterparts, or even full-on Meinongian ontology.

I'm not sure why you think specifying a set of terms within a formal logic is awkward. We don't have to employ set theory for this purpose, though it is convenient to do so.

Answer 2

OP : "We should say, ∃ x, . . . in natural language, it sounds sensible to say, "x exists".

" . . . c, a constant, refers to something, but it is still formally wrong to say ∃ c, . . ."

It seems ∃ c (c has x) is really referring to the existence of the predicate, and ∃ c on its own is without meaning.

The existence of c supposed in ∃ c where c itself exists, is not a predicate, as Kant put it in the 18th century, where c itself is unchanged by being ∃ c (in the supposed sense).

A hundred real dollars contain no more than a hundred possible dollars.

He says this in The Critique of Pure Reason, A599/B627. The form of existence in this context is apprehended by an observer, so the 'existence' happens between the observer and the observed, in judgement. It doesn't happen 'to' the object, and so is not a predicate of the object.

In Kant's terms, the judgement is the joining (copula) of the idea of $100, and sensory evidence of $100, so that the observer decides that $100 probably exists (and will continue to exist).

Being is evidently not a real predicate, that is, a conception of something which is added to the conception of some other thing. It is merely the positing of a thing, or of certain determinations in it. Logically, it is merely the copula of a judgement.

. . . there is no addition made to the conception, which expresses merely the possibility of the object, by my cogitating the object—in the expression, it is—as absolutely given or existing. Thus the real contains no more than the possible. A hundred real dollars contain no more than a hundred possible dollars. For, as the latter indicate the conception, and the former the object, on the supposition that the content of the former was greater than that of the latter, my conception would not be an expression of the whole object, and would consequently be an inadequate conception of it.

This form of subjective-objective existence of the $100 is to be distinguished from the more primordial form of existence of the observer, but that's a whole other subject (pun intended).

Answer 3

I'm not sure I understand your question, but I think you are asking about saying "x exists".

Consider the statement ∃x [x+1=2]. You aren't saying the variable x exists, you are saying there is at least one thing in the universe that is referred to by some constant c in the alphabet, which can be instantiated for the variable x, in the propositional function x+1=2, and denote a true proposition once instantiated. Reading it in natural language as "there exists an x such that x plus one equals two" hides all the information I just provided. The correct translation is:

There is at least one thing in the universe whose name can be put in place of x, in the propositional function x+1=2, and form a statement that denotes a true proposition.

Edit-I just read Bumble's response and I understood his answer. I use free logic in my personal day to day thoughts. I'm faced with existence questions all the time. "Casper is a friendly ghost", can be formulated in free logic. In first order logic you have a problem. You have the constant 'Casper', and you can existentially generalize to get a false statement: ∃ x[x is a friendly ghost]. The problem arises because you have a name that does not denote. You have a referrer without a referent. So as Bumble said, in free logic existence is a property that Casper may or may not have, but in first order logic all constants in the language denote. Thus in first order logic you have four simple rules of natural deduction.

1. Universal Instantiation
2. Existential Instantiation
3. Universal Generalization
4. Existential Generalization

In free logic the rules of inference have to be amended.

Answer 4

Keep in mind that the existential quantifier can be read as, "For some x," so it keeps that symmetry with the universal quantifier. For such a reason, some argue that it would be better to take ∃ for a "particular" quantifier. Granted, per Kant, the distinction between synthetic knowledge of existence and intuitive knowledge of particulars is rather dissolved, perhaps.

In somewhat most general terms, ∃ and ∀ are operators. They are even "functions" in a certain abstracted sense (i.e. not necessarily in terms of the ordered-pair characterization from set theory), and this even has to do with Frege's reimagining of logic as "function/argument" in style rather than "subject/predicate." A guiding principle is then nontriviality: we don't want to indiscriminately apply the concept of existence to everything, or else the concept of anything and the concept of that thing's existing would coincide, and there would not seem to be much point in differentiating between something as is, and as existing. (Anselm infamously tried to bypass this restriction by thinking of it as possibly essential to some being that it be conceived of as existing, in such a way that we could not but think that this thing exists, yet not by mere "verbal definition." But it was Anselm's judgment that Hume and Kant and then classical logic sought to permanently controvert.)

Another option is to consider how "any" and "all" are reciprocal in certain ways, and then to think that "any" and "some" also reciprocate each other in some way. One of the two senses of ∃ might seem more predicate-like, then, or "intensional," e.g. suppose we differentiated between an extensional and an intensional ∃-operator/function. Inasmuch as quantification and extensionality seem to coincide parallel to qualification and intensionality, though, we would have to think more precisely that one of the two versions of ∃ was principally a quantifier rather than a qualifier, even though either version performs both roles.

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We should also consider the question of existence as a subject. This is, roughly, the theory of subsistence, of substances (under the Aristotelian rubric "being a subject without objectively being a predicate"). (Perhaps this is what Meinong had in mind when he thought through an existence/subsistence distinction so-called.) Modulo ontological dependence, which can be understood as the quantitative version of what is qualification-theoretically known as metaphysical grounding, let us have some variable e, for existence-as-a-subject, such that (reading "⊱" as "is grounded in/by"):

• ∀x, (∃xFx) ⊱ (∃e(Ee))

... i.e., all specific existence-facts are grounded in the existence of existence-as-a-subject. e can then be variously something like prime matter, or a divine being (either Aquinean or Spinozan, say), or perhaps something else. If we don't want pure existence to ground itself, we can adjust the above to:

• ∀x, if x is not e, then (∃xFx) ⊱ (∃e(Ee))

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Finally, if existence, as actuality, can seem like a predicate, so too can possibility and necessity. But according to Kant, the other modal terms are not really descriptive or qualitative, either. So in the mainline contemporary treatment, possibility and necessity are correlated with existential and universal quantification respectively, over possible worlds. In this context, "It is actually that A," is a propositional operation, which can seem like predicating something of a proposition, but which is not typically glossed in this way. (Note that the liar-paradox sentence, "This sentence is not true," would be internally "incomplete" if "is not true" were switched from internal predicative to external operative position: "It is not true that this sentence _____.")

Answer 5

You say:

One reason it does not make sense in formal logic, is that variables are not "things in the domain"; when you say "x exists", the sentence isn't meaningful because x doesn't refer to anything.

Actually, in formal logic, a variable is a token used to express the idea that a variable is that which ranges over some domain of discourse. Thus, "x exists" is meaningful if an extension or intension is provided. A typical comprehension serves as an illustration { ∃x : x is a dog }. The act of associating a variable with a domain of discourse is called variable binding. The binding confers a semantics on the variable so that we interpret x as an unspecified instance of dog. We do the same in type theory by just listing the pair <x,dog> or programmatically x : dog.

You say:

c, a constant, refers to something, but it is still formally wrong to say ∃ c, because we only bind quantifiers to variable symbols, not constants

There's a notation that is often used by adding the bang to specify a single, unique element of a domain of discourse that does not vary. Thus, as a comprehension we can say {∃!c : c is a dog }. In natural language, we note the distinction with the indefinite and definite article. "The dog" means there's a specific, dog we are referring to, and "a dog" means there is some dog from a set of dogs.

You say:

When we present a signature as a set, I do not think we are declaring the existence of this set. In general, we assume the existence of a world of sets; we can only point to sets that already exist, by specifying them.

In intuitionistic type theory, this is what is done. First a collection is defined based on the intuition and is described, and then once the existential declaration is used on it, then the collection becomes a rigorously defined set. This is supposed to mimic how we go about using our intuition psychologically and then formalizing a notation. In this way, we provide the interpretation of a system, and then build the formal system.

There are a number of different formal semantics that can be built up around a natural language interpretation. FOL, ITT, and ZFC are three widely understood and used formalisms in linguistic formal semantics which is a discipline that has grown around the contributions of Richard Montague. But as a computer person (if memory serves you're a developer?), you might be interested in simple type theory. My go to for a simple type theory is the text William M. Farmer put out. In chapters 2 and 3, he discusses some of the theoretical fundamentals and defends STT as a formalism against the use of FOL, ITT, and ZFC.

It should also be noted that there are ambiguities in natural language, such as generic generalizations (SEP), that carry the challenge of vagueness. When dealing with binding, isolated form context, it might be difficult to understand which binding in formal semantics is appropriate. There's a complicated set of claims about how to best deal with this sort of binding.

One of the nice aspects of Farmer's theory is that he used the untyped lambda calculus to define six classes of binding which in the general case is Val(E,x,a) which is read as the value of E when the value x is a: set abstraction, function abstraction, universal quantification, existential quantification, definite description, and indefinite description. That is to say all six cases meet the definition of a formal binder as a triple. It would be easy to extend the six definitions to generalize quantifiers (SEP), too

Though exists can be read as a typical predicate, it is ontologically speaking generally accepted that it is not a standard, first-order property, but a second-order property; that is, it is a property of properties. This clears up some inconveniences. The SEP has a thorough article on existence (SEP) that covers it.

Answer 6

Existential quantification ("Sea urchins exist", "Mermaids do not exist", "Mermaids are not real", "Mermaids are not existing creatures") appears as a grammatical predicate only in particular natural languages. In classical Chinese for instance 'existence' is not predicated of entities as if it was a property or attribute. The implicit, intuitive model in Chinese is not one of attributing a property to an object but of locating the object in a particular realm or domain:

• (天下)有王

  (tianxia) YOU wang

  (the world) have/there-are king(s)

  There are kings (in the world).

• (天下)無敵

  (tianxia) WU di

  (the world) not-have/there-are-no enemies

  There are no enemies (in the world).

This is analogous to existential quantification over a domain ("the world" if left implicit). Because of this Anselmus ontological argument becomes pretty awkard when translated into Chinese and immediately loses most of its appeal. It's almost as if some form of (anti-Platonic) nominalism is baked in into the language.

Something that seems related to this is that early Chinese dialogic logic (as found in the Mohist canons for instance), is very much focussed on indexical terms ("this", "that") and on definite descriptors, terms which were mostly ignored in Western logic (until the advent of modern logic). Making a statement is then very "naturally" conceptualized as finding, selecting or recognizing a particular object (or class) inside a domain.

Answer 7

Oversimplifying:

Saying "there is some set [a,b,c]" is different from saying "there might be some set [a,b,c]."

Even more, "I'm positing that [a,b,c] is true," versus "I'm positing that [a,b,c] might be true, if..." is also different semantically.

The links everyone provided are excellent. To add one; David Lewis is your friend. "On the Plurality of Worlds".

Answer 8

Why does existential quantification appear to be predicative?

The expression ∃x, x² - 1 = 0 is understood to mean that x exists such that x² - 1 = 0. This translates in natural language:

Some thing exists such that its square minus one is equal to zero.

This clearly says something about some thing, but it doesn't say which thing exactly. This contrasts with any typical predicate sentence, such as:

The king of France is bald.

Here we say what it is that we are saying that it is bald, namely, the king of France.

So an existential sentence does something predicate sentences do, namely say something about some thing, but it is not really a predicate sentence, in the sense that it does not say which thing is the subject of the predicate.

This may account for the impression that it is a predicate sentence even though it is certainly not an ordinary predicate sentence.