8.4 Symbolization Techniques for Complex Sentences

Quantifiers and Scope

Nested Quantifiers and Scope

When a sentence contains more than one quantifier, the quantifiers are nested: the scope of one sits inside the scope of another. The scope of a quantifier is everything within the parentheses (or brackets) that follow it, and it determines which variables that quantifier binds.

The crucial thing to understand here is that order matters. Swapping the order of nested quantifiers can completely change what a sentence means.

• $\forall x \exists y (Fxy)$ says: "For every $x$, there exists some $y$ such that $Fxy$." The $y$ can be different for each $x$.
• $\exists y \forall x (Fxy)$ says: "There is some single $y$ such that for every $x$, $Fxy$." Now one specific $y$ must work for all $x$ values.

Think of it with a concrete predicate. Let $Fxy$ mean "$x$ admires $y$."

• $\forall x \exists y (Fxy)$: Everyone admires someone (possibly different people).
• $\exists y \forall x (Fxy)$: There's one person that everyone admires.

Those are very different claims. When you're symbolizing, always read from left to right and ask yourself which quantifier has wider scope.

Multiple Predicates in Quantified Statements

Quantified statements often involve more than one predicate, and the predicates can share variables. This is where symbolization gets interesting, because the logical connectives between the predicates shape the meaning.

A common pattern is a universally quantified conditional with an existential inside:

$\forall x (Px \rightarrow \exists y (Qy \land Rxy))$

This says: "For every $x$, if $x$ is $P$, then there exists some $y$ that is $Q$ and stands in relation $R$ to $x$."

For example, let $Px$ = "$x$ is a student," $Qy$ = "$y$ is a course," and $Rxy$ = "$x$ is enrolled in $y$." The formula then reads: "Every student is enrolled in some course."

Pay attention to which variables are bound by which quantifier. In the formula above, $x$ is bound by $\forall x$ throughout, while $y$ is bound by $\exists y$ only within the inner parentheses.

Predicates and Identity

The Identity Predicate

The identity predicate ($=$) is a special built-in predicate that asserts two terms refer to the same object. You write it between its two arguments: $a = b$ means "$a$ and $b$ are the very same thing."

Identity has three defining logical properties:

• Reflexivity: $\forall x (x = x)$. Everything is identical to itself.
• Symmetry: $\forall x \forall y (x = y \rightarrow y = x)$. If $a$ is identical to $b$, then $b$ is identical to $a$.
• Transitivity: $\forall x \forall y \forall z ((x = y \land y = z) \rightarrow x = z)$. If $a = b$ and $b = c$, then $a = c$.

You'll also frequently use the negation of identity, $x \neq y$, which asserts that two terms refer to distinct objects. This becomes essential for uniqueness claims (covered below).

Relational Predicates

A relational predicate takes two or more arguments and asserts that a specific relation holds between them. While one-place predicates describe properties of single objects (like $Px$ for "$x$ is prime"), relational predicates describe connections between objects.

Some examples:

• $Lxy$: "$x$ loves $y$" (two-place)
• $Gxy$: "$x$ is greater than $y$" (two-place)
• $Bxyz$: "$x$ is between $y$ and $z$" (three-place)

Relational predicates combine with quantifiers to express claims about how objects in a domain relate to one another. For instance:

• $\forall x \exists y (Lxy)$: "Everyone loves someone."
• $\exists x \forall y (Lxy)$: "Someone loves everyone."
• $\forall x \forall y (Lxy \rightarrow Lyx)$: "If anyone loves someone, that person loves them back." (Love is symmetric.)

The argument positions matter. $Lxy$ and $Lyx$ say different things, so be careful about which variable goes where when you symbolize.

Descriptions and Uniqueness

Uniqueness Claims

A uniqueness claim asserts that exactly one object satisfies a given condition. To express this in predicate logic, you combine the existential quantifier with the identity predicate using this standard pattern:

$\exists x (Px \land \forall y (Py \rightarrow y = x))$

Here's how to read it, step by step:

1. $\exists x (Px \ldots)$: There exists some $x$ that is $P$.
2. $\forall y (Py \rightarrow y = x)$: And for anything $y$, if $y$ is also $P$, then $y$ must be identical to that same $x$.

Together, this guarantees that at most one thing is $P$ (the second part) and at least one thing is $P$ (the first part). That gives you exactly one.

For example, "There is exactly one prime minister":

$\exists x (Mx \land \forall y (My \rightarrow y = x))$

where $Mx$ = "$x$ is a prime minister."

Definite Descriptions

A definite description is a phrase that picks out a unique individual by describing it, typically using "the": the tallest mountain, the current president, the author of Principia Mathematica.

In predicate logic, definite descriptions are symbolized using the same uniqueness formula. The idea is that "the $P$" presupposes there is exactly one $P$. So to say something about "the $P$," you assert that a unique $P$ exists and then predicate something of it.

For example, "The author of Principia Mathematica is British" gets symbolized as:

$\exists x (Ax \land \forall y (Ay \rightarrow y = x) \land Bx)$

where $Ax$ = "$x$ authored Principia Mathematica" and $Bx$ = "$x$ is British."

The steps for symbolizing a definite description:

1. Identify the descriptive condition (the property that "the" phrase specifies). That becomes your predicate $P$.
2. Write the uniqueness formula: $\exists x (Px \land \forall y (Py \rightarrow y = x) \ldots)$.
3. Add whatever is being predicated of "the $P$" as an additional conjunct inside the scope of $\exists x$.

One note: this approach follows Russell's theory of definite descriptions, which treats "the $P$" not as a simple name but as a quantified expression that asserts existence and uniqueness. This is the standard treatment in most formal logic courses.