Quantifier
In symbolic logic, quantifiers are words or symbols which indicate quantity in the sense of "all", "some", "none", or "one". The two main logical quantifiers are "For all", usually symbolized by ∀, and "There exists", symbolized by ∃. Thus, ∀x is read "For all x" and ∃x is read "There exists an x". The quantifier ∀ is called the universal quantifier because, when it is used, it indicates that every member of the "universe of discourse" has some property which will follow the symbol. The universe of discourse is just the set of all the entities of which we are making statements. So let us suppose that our universe of discourse is the set of integers; then "∀x[(x is even)(x+1 is odd)]" states that every even integer has the property that if 1 is added to it, the result is an odd integer. The quantifier ∃ is called the existential quantifier because it indicates that there is at least one member of the universe that has the property which follows it. For example, "∃x[x2=4]" states that there exists at least one integer whose square is 4. Now we know that there are, in fact, two such integers, 2 and -2, but the existential quantifier always indicates "at least one."
The truth or falsehood of a quantified statement depends upon the universe of discourse in which the statement occurs. For example, "∀x[x > -x]" is true if the universe of discourse is the set of positive integers, but it is false if the universe of discourse is the set of all integers. The statement says that x is always greater than its opposite, but if x is negative, then its opposite, -x, will be positive, so that x > -x will be false. Here is another example using the existential quantifier: ∃x[x+5=2]. This statement is true if the universe of discourse is all integers, but false if it is the positive integers. Thus it is very important to specify the universe of discourse for each statement that is made under quantification.
Suppose P represents a property and Px says that x has the property P. Since to say that something is true for all x is the same as saying that there is no x for which it is false, we can say that the statements "∀x[Px]" and "not∃x[not Px]" are equivalent. When two statements are equivalent, we say that the first statement is true if and only if the second statement is true. The logical symbol for "if and only if" is the double ended arrow, . Thus, "∀x[Px]not∃x[not Px]" expresses the equivalence of "∀x[Px]" and "not∃x[not Px]." With similar reasoning, we can write "∀x[not Px]not∃x[Px]", which essentially states that to say "Px is false for all x" is equivalent to saying "There is no x for which Px is true." Likewise, "not∀x[Px]∃x[not Px]" and "not∀x[not Px]∃x[Px]" are equivalences. Knowing such equivalences gives the logician more flexibility in proving propositions. The equivalences in this paragraph also lead to expressions for the negations of quantified statements: the negation of "∀x[Px]" is "∃x[not Px]"; the negation of "∃x[not Px]" is "∀x[Px]; the negation of "∀x[not Px]" is "∃x[Px]; and the negation of "∃x[Px]" is "∀x[not Px]. It can be seen from these statements that a general rule for negating a quantified statement is: "To negate a statement covered by one quantifier change the quantifier from universal to existential or vice versa and negate the statement which it quantifies." This rule can also be quite useful in proving theorems.
Many quantified statements simply represent what we might call common sense. For instance, let Ax represent "x is an animal." Then the common sense statement "If everything is an animal, then something is an animal" can be expressed as "∀x[Ax]∃x[Ax]". Note that this statement does not claim that everything is, in fact, an animal; it only claims that if it were the case that everything is an animal, then surely it would follow that something is an animal. We may also have statements in more than one variable with more than one quantifier. Here is an example: "There exists an x such that for all y and z, x+y+z=y+z" may be rendered in logical symbols as "∃x∀y∀z[x+y+z=y+z]." Another example from geometry: "For each pair of points x and y there is a point z such that z is between x and y." Here let Bxzy mean "z is between x and y"; then our statement may be written as "∀x∀y∃z[Bxzy].
Quantification was not always an explicit part of logic. Propositional logic uses no quantifiers and deals only with the structure of propositions. When logicians wish to speak about the content as well as the structure of propositions, they overlay propositional logic with quantification theory. The resulting system is called predicate logic or the predicate calculus because one now predicates or attributes certain qualities to all, some, or no elements in the universe of discourse. The use of quantifiers helped to clarify a number issues in logic, perhaps the most bothersome of which was the dispute over whether existence is a predicate. From Aristotle (384-322 BC), the inventor of logic, through the 19th century, existence was treated as a predicate, meaning, for example, that the statements "x works" and "x exists" were treated as having the same logical structure. There is no question that they have the same grammatical structure - both have the same subject-verb structure in the English language. The problem was that in English, "verb" and "predicate" are regarded as synonymous. Hence, it appears that "works" and "exists" are both predicates in the same sense. The great German philosopher, Immanuel Kant (1724-1804), said that while both "works" and "exists" may be considered predicates in the grammatical sense, "exists" may not be considered a predicate in the logical sense. He was essentially saying that logical predicates attribute qualities to things that are already in existence. "John works" says that the attribute of working is being claimed for John assuming that John exists. If John does not exist, then it is nonsense to attribute anything to him. So existence is a pre-condition for the attribution of qualities. It is very different from attributes that one predicates of existing objects. "John works", "John flies", "John is tall" attribute qualities to an already existing John. "John exists" may have the same grammatical structure as these, but it does not have the same logical structure when analyzed correctly. Here is where quantification can help. "John works" means "∃x(x is John)(John works)." "John exists" means "∃x[x is John]". The first proposition is a conditional; the second is a declarative statement. The first says "If John exists, then John works." The second declares "John exists." Now this might seem irrelevant to any conversation about John that the ordinary person might have, but substitute the word "God" for x in "x exists" and you have one of the classic disputes of the ages. The so-called "ontological" proof for the existence of God is valid only if existence is a logical predicate. Aristotle, St. Anselm (1033-1109), and Descartes (1596-1650) assumed that it is. Kant, Russell (1872-1970), and most modern logicians say that it is not. For more detail about this controversy, see the article entitled "Predicate."
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