Question

∀, ∃ what is their meanings and the differences between them. This may take a paragraph or two of some writing, which should at minimum begin with a definition of each in your words.

Answer 1

Variables behave much like pronouns. They are meaningful when they are combined with another

element in the language. In Predicate Logic, each variable combines with and is bound by a single quantifier.

Predicate Logic has two such quantifiers: ∀ (the universal quantifier) and ∃ (the existential quantifier). Since

a predicate can combine with more than one variable, it is necessary to write the variable immediately after

the quantifier to indicate which variable the quantifier interacts with. In other words, we write ∀x to indicate

that the universal quantifier ∀ is interacting with the variable x and not, say, with the variable y.

Example:

a. ∀x MAN(x) Everything is a man

b.∀x ∀y LOVE(x,y) Everything loves everything

a. ∃x MAN(x) Something is a man

b.∃x ∃y LOVE(x,y) Something loves something

When both ∀ and ∃ are used in a single expression, their order is relevant. This is illustrated below.

The paraphrases are lengthy to avoid the ambiguity present in the English sentences. The meaning of each

arrangement is different. This is also why we need to keep track of the variable that each quantifier interacts

with.

a. ∀x ∃y LOVE(x,y) For every thing (x) there is a thing (y) such that x loves y

b. ∃y ∀x LOVE(x,y) There is a thing (y) such that for every thing (x) x loves y.