Remember that the sentence “\(x\) is an American,” is not a proposition. This is because the truth of the statement changes based on different values for \(x\text{.}\) Such a sentence is called an open sentence or predicate. However, it is quite useful to include variables in our statements, so our logic should be able to accommodate that. To this end we introduce quantifiers. A quantifier is a symbol which states how many instances of the variable satisfy the sentence.
For an open setence \(P(x)\text{,}\) we have the propositions
The existential statement \((\exists x) P(x)\) may be read as:
The symbol \(\exists\) was chosen as a backwards E for “exists.”
Similarly, the universal statement \((\forall x) P(x)\) may be read as:
The symbol \(\forall\) was chosen as an inverted A for “all.”
Perhaps you think quantifiers are now obvious, and if so, what is the truth value of the proposition \((\exists x) (x^2 = 2)\text{?}\) The answer is, “it depends!” In particular, it depends on the universe of discourse; that is, what is the set of \(x\)'s over which we are quantifying? This universe needs to be specified beforehand in order to make sense of the proposition and determine its truth value. For example, the existential statement above is false if the universe is the set of natural numbers, \(\mathbb{N}\text{,}\) but it is true if the universe is the set of real number, \(\mathbb{R}\text{.}\)
If the context does not make clear the universe over which we are quantifying, we may specify it explicitly by using the set membership symbol \(\in\text{,}\) which may be read “is an element of.” For example, we read \(x \in \mathbb{R}\) as any of:
When we are feeling lazy, we omit “the set of.” Now, we may rewrite our statement with respect to different universes as
\begin{gather*} (\exists x \in \mathbb{N}) (x^2 = 2),\\ (\exists x \in \mathbb{R}) (x^2 = 2). \end{gather*}Now, the first statement is false, and the second statement is true, so the universe of dicourse matters.
Does it surprise you that we only have two quantifiers? At first it might seem strange but what are the other possibilities? In general, it is not very useful to embed the idea “there are 5 \(x\) satisfying \(P(x)\text{,}\)” directly into our logic. This is for a variety of reasons including
In addition, it can be difficult to make statements like “most sheep are white,” precise. However, there is another quantifier that we find useful.
For an open sentence \(P(x)\text{,}\) the proposition \((\exists ! x) P(x)\) is true when there is exactly one \(x\) making \(P(x)\) true. We read this proposition in English as “there exists a unique \(x\) such that \(P(x)\text{.}\)” The symbol \(\exists !\) is called the unique existential quantifier.
The proposition \((\exists ! x) P(x)\) is just an abbreviation for
\begin{gather*} \Big((\exists x) P(x)\Big) \wedge \Big((\forall y)(\forall z) (P(y) \wedge P(z)) \implies (y = z)\Big). \end{gather*}You should interpret this as saying, “there is some \(x\) satisfying \(P(x)\text{,}\) and whenever any two elements \(y,z\) both satisfy this open sentence, then they are actually the same element.”
The next theorem explains why we chose the universal \(\forall\) and existential \(\exists\) quantifiers as the de facto standard of quantification: they play nice with negation.
Negation interacts with the existential and universal quantifiers in the following ways.
\begin{gather*} \neg ((\exists x) P(x)) \iff (\forall x) (\neg P(x));\\ \neg ((\forall x) P(x)) \iff (\exists x) (\neg P(x)). \end{gather*}Translate the following quantified sentences into English.
\begin{gather*} (\forall x \in \mathbb{N}) (\exists y \in \mathbb{N}) (x \lt y),\\ (\exists y \in \mathbb{N}) (\forall x \in \mathbb{N}) (x \lt y). \end{gather*}What is the difference between these two sentences? Do they have the same truth value, or different truth values?
The first statement may be translated as “for every natural number \(x\text{,}\) there exists a natural number \(y\) such that \(x\) is less than \(y\text{.}\)” The second statement may be traslated as “there exists a natural number \(y\) such that for every natural number \(x\text{,}\) \(x\) is less than \(y\text{.}\)”
The key difference between these two sentences is the following: in the first sentence, \(y\) can depend on \(x\) because \(x\) came first; in the second sentence, \(y\) must be independent of \(x\text{.}\) From here, we can recast our translations as: “for every natural number, there is some other natural number bigger than it,” and “there is some natural number which is simultaneously bigger than every natural number (including itself!).”
Now, it should be clear that the first statement is true (if \(x \in \mathbb{N}\text{,}\) choose \(y = x+1 \in \mathbb{N}\)), but the second statement is false (because we've just shown there is no biggest natural number).
This issue with the order of quantifiers only comes into play among different quantifiers. In particular, the order of consecutive universal quantifiers may be changed at will and yield an equivalent proposition. Similarly, the order of consecutive existential quantifiers may be changed at will and yield an equivalent proposition.