The Timpson Lyceum · Discrete Math (2025)

Predicate Logic, Identity, Definitions, and Theorems – Lecture 2 (Discrete Math)

Predicate Logic, Identity, Definitions, and Theorems – Lecture 2

Overview

This lecture extends propositional logic into predicate logic, the language used to state nearly every theorem in mathematics. We meet predicates and relations, the universal and existential quantifiers, and the rules for negating quantified statements. We then add the equals sign as logic with identity, governed by two simple laws, and use them to build our very first formal proofs — including the surprising fact that $a + c = b + c$ follows from logic alone, not arithmetic. Finally we lay out the field axioms of the real numbers and use them, together with new definitions (even, odd, divisible, prime, composite), to prove results such as $2 \ne 1$ and $2 + 2 = 4$.

Key Concepts

Recap — propositions and operators. A proposition (statement) is a sentence that is either true or false, never both. The five logical operators are negation $\lnot$, conjunction $\land$ ("and", true only when both are true), disjunction $\lor$ ("or", inclusive — true when at least one is true), implication $\rightarrow$, and equivalence $\leftrightarrow$. The implication $\alpha \rightarrow \beta$ is false only when $\alpha$ is true and $\beta$ is false; whenever the antecedent $\alpha$ is false the implication is vacuously true. Two statements are logically equivalent ($\alpha \leftrightarrow \beta$) when they always share the same truth value, equivalently when $\alpha \rightarrow \beta$ and $\beta \rightarrow \alpha$.

Useful equivalences (from propositional logic).

• De Morgan: $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$ and $\lnot(\alpha \lor \beta) \equiv \lnot\alpha \land \lnot\beta$.
• Contrapositive: $\alpha \rightarrow \beta \equiv \lnot\beta \rightarrow \lnot\alpha$ ("if shoes then socks" says the same as "if no socks then no shoes").

Predicate. A predicate is a proposition-valued function: it takes one or more terms and returns true or false. By itself it cannot be called true or false (e.g. $x > 2$ is a predicate, not a statement). Plugging in a term turns a predicate into a proposition. Notation: $P(x)$ for one variable, $R(x,y)$ for two.

Relation. A predicate that takes exactly two terms is a relation. For relations it is customary to write the symbol between the terms — e.g. we write $1 < 2$ rather than ${<}(1,2)$. New predicates can be defined from old ones with logical operators; the symbol $:\equiv$ means "is defined to be," as in $(x \le y) :\equiv (x < y) \lor (x = y)$.

Quantifiers. Two new symbols convert a predicate into a proposition:

• Universal $\forall$ — read "for all / for every." $\forall x\, P(x)$ is true exactly when $P(x)$ is true regardless of the choice of $x$.
• Existential $\exists$ — read "there exists … such that." $\exists x\, P(x)$ is defined as $\lnot \forall x\, \lnot P(x)$.

Order of quantifiers matters. $\forall x\, \exists y\,(x + y = 0)$ is true (for each number, pick its negative), but $\exists x\, \forall y\,(x + y = 0)$ is false (no single number cancels every number). Swapping the quantifiers changes the meaning entirely.

Negating quantifiers. A negation flips the quantifier and moves inside the predicate:

"Not all swans are white" becomes "there exists a non-white swan." Combined with De Morgan: $\lnot\,\forall x\,\big(P(x)\land F(x)\big) \equiv \exists x\,\big(\lnot P(x)\lor \lnot F(x)\big)$.

The structure of mathematics. Every field of math is an axiomatic system with four parts: undefined terms (you cannot define everything without circularity — like learning French from a French-only dictionary), axioms (which give the undefined terms meaning), definitions, and theorems. Almost all working mathematics is creating definitions and proving theorems.

The three logics of proof. A theorem is an implication $\alpha \rightarrow \beta$ from the axioms. Three valid forms, each a tautology:

• Direct: $\big[\alpha \land (\alpha \rightarrow \beta)\big] \rightarrow \beta$ (modus ponens).
• Contrapositive: use $\lnot\beta \rightarrow \lnot\alpha$ in place of $\alpha \rightarrow \beta$.
• Contradiction: if $\alpha \rightarrow F$ (something impossible), then $\alpha$ must be false.

Logic with identity. Adding the equals sign gives two governing laws:

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

Field axioms of the real numbers (10). With undefined terms $0,\,1,\,+,\,\times$: commutativity and associativity of $+$ and $\times$; additive identity $0$ ($x+0=x$) and multiplicative identity $1$ ($1\cdot x = x$); additive inverses (for every $x$ there is $y$ with $x+y=0$) and multiplicative inverses (for every $x \ne 0$ there is $y$ with $x\cdot y = 1$); distributivity $x(y+z)=xy+xz$; and the non-triviality axiom $0 \ne 1$. All numbers are then defined: $2 := 1+1$, $3 := 2+1$, and so on.

Definitions to memorize. All are stated over the integers $\mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\}$.

• Divisible / divides: $b \mid a$ means there exists an integer $c$ with $b\cdot c = a$. (Note: $b \mid a$ is a predicate returning true/false — not the fraction $a/b$.) For example $2 \mid 8$ (take $c=4$); $3 \nmid 8$. Every integer divides $0$.
• Even: an integer $a$ is even if $2 \mid a$, i.e. $a = 2k$ for some integer $k$.
• Odd: an integer $a$ is odd if $a = 2k + 1$ for some integer $k$.
• Prime: an integer $p$ is prime if $p > 1$ and its only positive divisors are $1$ and $p$. ($1$ is not prime; $2$ is prime.)
• Composite: a positive integer $a$ is composite if there is an integer $b$ with $1 < b < a$ and $b \mid a$. ($1$ is neither prime nor composite.)

Worked Example

Claim: If $a = b$, then $a + c = b + c$ — a consequence of logic alone, using only the law of identity and the principle of extensionality. To prove an implication, assume the antecedent and derive the conclusion.

The same skeleton proves $a \circ c = b \circ c$ for any well-defined operation $\circ$. Building on it, here is the proof that $2 + 2 = 4$ using the definitions $2 := 1+1,\ 3 := 2+1,\ 4 := 3+1$ and associativity of addition:

Summary

• A predicate is a proposition-valued function; supplying its terms — or attaching a quantifier — turns it into a true/false statement. A two-term predicate is a relation, conventionally written infix (e.g. $x < y$).
• The quantifiers $\forall$ ("for all") and $\exists$ ("there exists") bind a predicate's variable; their order matters, and negation flips each quantifier while pushing the negation inside: $\lnot\forall x\,P(x) \equiv \exists x\,\lnot P(x)$.
• Every area of math is an axiomatic system of undefined terms, axioms, definitions, and theorems; a theorem is an implication $\alpha \rightarrow \beta$ proved directly, by contrapositive, or by contradiction.
• Logic with identity rests on the law of identity ($x = x$) and the principle of extensionality (equals may be substituted for equals); these alone prove facts like $a = b \Rightarrow a + c = b + c$, independent of arithmetic.
• The field axioms define $0,1,+,\times$ and let us define each number; together with $0 \ne 1$ they yield results such as $2 \ne 1$ (by contradiction) and $2 + 2 = 4$.
• Five core definitions over $\mathbb{Z}$ recur throughout the course: divisible ($b \mid a \iff \exists c,\ bc = a$), even ($a = 2k$), odd ($a = 2k+1$), prime ($p > 1$ with only divisors $1$ and $p$), and composite (positive with a divisor strictly between $1$ and itself).