Why $\forall$ is not a predicate

Question

There is a reason why existence can not be a predicate, namely:

1. Let's prove that unicorns exist.
2. It is sufficient to prove that there is an existing unicorn.
3. There are two possibilities: either an existing unicorn exists or it does not.
4. The second possibility is a contradiction: how could an existing unicorn not exist? That's the same as saying that a blue ball is not blue.
5. So, unicorns exist.

This argument probably dates back to Kant and the ontological argument of God's existence. It shows that $\exists$ is not a "property" and should be treated in a special way.

My question is why we need $\forall$ as a special symbol. Are there any arguments like the one (for existence) that I mentioned?

I suspect that that text about unicorns could be rewritten using negations and $\forall$, but I don't exactly see how.

Answer 1

In first-order logic, $\forall$ cannot be considered a predicate for the fundamental reason that its syntax behaves completely differently from predicates.

A predicate is something that combines one or more terms into a formula.

In contrast $\forall$ combines a variable name and a formula into a new formula.

Because the syntax is not parallel, one cannot even write down rules or principles that apply in the same way for $\forall$ and predicates, so nothing would be gained by considering it one.

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This holds equally for $\exists$, of course. In your argument you seem to be confusing "such-and-such particular thing (whose identity we somehow already agree on?) exists" with "there is something that has such-and-such properties". Formal logic's $\exists$ models the latter concept, not the former.

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In higher-order logic we can phrase things such that $\forall$ becomes a predicate on predicates, which is true if the predicate we apply it to is the always-true predicate. Making this work probably requires having something like lambda abstractions in the syntax such that we can write down the predicates we apply $\forall$ to. There is impeccable historical precedent for this, but since the result is not a direct generalization of the usual syntax of first-order logic, it seems to be somewhat uncommon in "pure logic" theoretical treatments. (And computer proof systems such as HOL seem to prefer to go to a much richer type system where $\forall$ as well as its higher-order relatives are still primitive constructs).

Answer 2

The quantifiers $\forall$ and $\exists$ are fundamental symbols - originating towards the end of 19th Century with C.S. Peirce and G.Frege - that allow us to express in a formal manner basic sentences of natural language, like e.g.

"Every Man is Mortal" : $\forall x ( \text {Man}(x) \to \text {Mortal}(x))$

and mathematics :

"$0$ is not the successor of any natural number" : $\forall x \lnot (0=s(x))$.

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In modern formulation of predicate logic, quantifiers are "special" symbols with specific syntactical rules governing them (and specific semantical rules to interpret them).

We use them with terms ("names" for objects) and predicates (denoting properties of objects) and logical connectives to form meaningful sentences (aka : well-formed formulas).

No one of the above "categories" of symbols is sufficient by itself to express sentences.

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If you are interested, you can see a "special" type of logic : Free Logic, where quantifiers (especially the existential one) are treated difefrently and you can have a "special" predicate $\text E !(x)$ for "actual" existence.

Answer 3

You turn an $\exists$ into a $\forall$ by means of double negation.

$$\exists\,p:P(p)\iff\lnot\forall p:\lnot P(p).$$

It is a lie that all unicorns are inexistent.

Answer 4

Here I understand " predicate" as a term that is applied to an individual ( a first level, concrete, individual, a " thing" ) and that attributes a property to this individual.

When you say " there is an x such that x has run a 100m race in less than 10 sec", you are not talking about a person, you are talking about a formula ( an open formula : " x has rum a 100m race .... " ) and say that this formula is true for at least 1 value of the variable x. Or, if you prefer, you are talking about a set ( the set of all x such that x has run...) and say about this set ( not about a person) that this set is not empty.

From this follows that existence is not a predicate you attribute to things themselves ( existence is " being satisfied" for a formula or " non-emptiness" for a set).

In the same manner, when you use the universal quantifier, you do not talk about things, you talk about formulas or ( equivalently) about sets.

Using " for all x " ( belonging to some domain D) you say that an open formula with x as variable is satisfied for all values of the domain.

The following reasoning would count as a sophism:

(1) Musicians are all artists. (2) Therfore Mozart is " all artist", Wagner is " all artist", etc.

or, even worse,

(1) Musicians are all artists. (2) Therefore, Mozart is all and Mozart is an arrtist, Wagner is all and Wagner is an artist, etc.

The word " all" does not refer to individuals as a predicate, it refers to the set of musicians and means that the whole set of musicians is included in the set of artists.