The Timpson Lyceum · Geometry (2024)

Predicate Logic and Identity – Lecture 2 (Geometry)

Predicate Logic and Identity – Lecture 2

Overview

This lecture extends propositional logic into predicate logic, the language in which nearly all of mathematics is actually written. After a quick review of the five propositional operators and the key equivalences (negating $\land$ / $\lor$ , the converse vs. the contrapositive), we introduce predicates — proposition-valued functions — and the two ways to turn them into statements: substituting terms or attaching quantifiers. We define the universal ( $\forall$ ) and existential ( $\exists$ ) quantifiers, show that the order of quantifiers changes meaning, and develop the rules for pushing a negation through a quantifier. We then meet the special logical symbol of identity ( $=$ ) and its two governing laws, use it to prove a first theorem, and close with the idea of an axiom system and the three standard proof styles, plus the meaning of a vacuously true implication.

Key Concepts

Proposition (review). A sentence that is either true or false, never both. Two propositions $\alpha,\beta$ yield four truth-value combinations. The five basic operators are $\lnot$ (not), $\land$ (and), $\lor$ (or), $\to$ (implies), and $\leftrightarrow$ (logical equivalence). An implication $\alpha \to \beta$ is false only in the single case $\text{T} \to \text{F}$ ; a false antecedent makes the whole implication vacuously true.

Key equivalences (review).

De Morgan: $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$ and $\lnot(\alpha \lor \beta) \equiv \lnot\alpha \land \lnot\beta$ .
An implication and its converse are not equivalent: $\alpha \to \beta \not\equiv \beta \to \alpha$ .
An implication is equivalent to its contrapositive: $\alpha \to \beta \equiv \lnot\beta \to \lnot\alpha$ .
An implication as a disjunction: $\alpha \to \beta \equiv \lnot\alpha \lor \beta$ .

Predicate. A proposition-valued function: give it its term(s) and it returns a proposition (something true or false). Example: $P(x):\ x + 3 = 5$ . Then $P(2)$ is true, $P(1)$ is false. On its own $x+3=5$ is neither true nor false — it is not a statement until $x$ is supplied.

Relation. A predicate that takes exactly two terms. These are so common they get their own name and an infix notation: rather than write $L(x,y)$ for the "likes" relation, we write $x \text{ likes } y$ ; rather than $\text{LessThan}(1,2)$ we write $1 < 2$ . Predicates may take any number of terms (one, two, three, …).

Building new predicates. Because predicates return propositions, the logical operators combine them into new predicates. The defining symbol $:\!\equiv$ (or $:=$ ) means "the thing on the left is defined to be the thing on the right." Example: $x \le y \ :\!\equiv\ (x < y) \lor (x = y)$ .

Two ways to convert a predicate into a statement.

Supply its terms — e.g. $P(7)$ .
Attach a quantifier.

Quantifiers.

Universal $\forall$ — read "for all / for every / for each."
Existential $\exists$ — read "there exists … such that." The quantifier is written before the predicate. While $x > 2$ is only a predicate, both $\forall x\, (x > 2)$ and $\exists x\, (x > 2)$ are genuine propositions.

Definition of the quantifiers. A truth table is impossible when the domain is infinite, so the quantifiers are defined directly:

$\forall x\, P(x)$ is true iff $P(x)$ is true independent of the choice of $x$ ; otherwise false.
$\exists x\, P(x)\ :\!\equiv\ \lnot\,\forall x\,\lnot P(x)$ — "there is a white swan" means "it is not the case that every swan is non-white."

Order of quantifiers matters. $\forall x\,\exists y\,(x+y=0)$ is true (take $y=-x$ ), but $\exists x\,\forall y\,(x+y=0)$ is false (no single $x$ works for every $y$ ). They are different statements.

Negating a quantifier — push the $\lnot$ inward and flip the quantifier:

$\lnot\,\forall x\, P(x) \equiv \exists x\, \lnot P(x)$
$\lnot\,\exists x\, P(x) \equiv \forall x\, \lnot P(x)$

Identity. The symbol $=$ is a symbol of logic, more foundational than $+$ or $\times$ . It is governed by the law of identity, in two parts:

Everything is identical with itself: $\forall x\, (x = x)$ .
Principle of extensionality: equals may be substituted for equals.

Axiom systems and theorems. Every field of mathematics studies the logical consequences of its axioms (in geometry, "axiom" and "postulate" mean the same thing). Statements derived from the axioms are theorems. Toy system: from "all men are mortal" and "Bob is a man" you may conclude the theorem "Bob is mortal."

Three proof styles.

Direct proof — from $\alpha$ and $\alpha \to \beta$ , conclude $\beta$ . (The form $[\alpha \land (\alpha \to \beta)] \to \beta$ is a tautology.)
Proof by contrapositive — prove $\lnot\beta \to \lnot\alpha$ instead of $\alpha \to \beta$ ; from $\lnot\beta$ conclude $\lnot\alpha$ .
Proof by contradiction — if $\alpha \to \text{F}$ is true (assuming $\alpha$ leads to something impossible), then $\alpha$ is false.

Vacuous truth. "If $\alpha$ then $\beta$ " speaks about the universe as it actually is — it does not invite you to rearrange the world to make $\alpha$ hold. An implication is vacuously true whenever its antecedent cannot be satisfied. A false statement entails anything: granting $1 = 0$ lets one "prove" any absurdity, which is exactly why a false antecedent always yields a true implication.

Worked Example

Claim. If $a = b$ , then $a + c = b + c$ .

To prove an implication, assume the antecedent and derive the conclusion. We use only the law of identity: $a + c$ is identical with itself (part 1), then substitute $b$ for the equal quantity $a$ (part 2).

\begin{aligned} &\text{Assume } a = b. \\ a + c &= a + c &&\text{(law of identity: everything equals itself)} \\ a + c &= b + c &&\text{(extensionality: substitute equals for equals)} \quad\blacksquare \end{aligned}

Nothing about $+$ was used, so the same argument proves $a = b \implies a \star c = b \star c$ for any well-defined operation $\star$ . This is the logic behind the familiar rule "what you do to one side you must do to the other."

Summary

A predicate is a proposition-valued function; supplying its terms — or attaching a quantifier — turns it into a true-or-false statement. A two-term predicate is a relation, usually written infix.
The universal quantifier $\forall$ ("for all") and existential quantifier $\exists$ ("there exists") are defined directly, not by truth tables, since domains may be infinite.
The order of quantifiers matters: $\forall x\,\exists y$ and $\exists x\,\forall y$ generally say different things.
To negate a quantified statement, push the negation inward and flip the quantifier: $\lnot\forall x\,P(x) \equiv \exists x\,\lnot P(x)$ and $\lnot\exists x\,P(x) \equiv \forall x\,\lnot P(x)$ .
Identity ( $=$ ) is a logical symbol governed by two laws: $x = x$ for all $x$ , and equals may be substituted for equals.
Mathematics derives theorems from axioms via direct, contrapositive, and contradiction proofs; an implication with an unsatisfiable antecedent is vacuously true.