The Timpson Lyceum · First-Order Predicate Logic (2024)

Identity and Theorems of Logic - Lecture 10 (Logic)

Overview

This lecture completes first-order predicate logic by adding identity — the equality symbol = — as a built-in symbol of the logic. Doing so requires only one new rule (Rule I), yet that single addition lets us prove foundational facts that every branch of mathematics quietly relies on. We prove that identity is an equivalence relation, derive the quantifier-negation laws as theorems rather than definitions, and establish that you may "do the same thing to both sides" of an equation. The lecture closes with a worked proof that uses identity essentially, and then introduces group theory as the first axiomatic system built on top of this completed logic.

Key Concepts

Identity in language. Identity is encoded by a "to be" verb (is, was, are, were, being, been) used in the sense of is identical with. "Charles III is the king of England" asserts identity; "snow is white" does not — it ascribes a property. Test: if the "to be" verb can be replaced by is identical with without changing the meaning, it encodes identity.

Logic with identity. Adding = puts us in propositional and first-order predicate logic with identity — the complete system underlying the foundations of mathematics. Equality is now a logical symbol, not an ordinary predicate.

The two principles encoded by identity:

• Law of Identity — $\forall x\,(x = x)$. Everything is identical with itself.
• Principle of Extensionality — if $x = y$, then anything true of $x$ is true of $y$ and vice versa. Mnemonic: equals may be substituted for equals.

Rule I (the only new rule). Both principles are bundled into a single inference rule:

• If $x = y$ and $S$ is an open formula containing $x$, you may derive $T$, where $T$ results from $S$ by replacing $x$ with $y$.
• Furthermore, you may always conclude $x = x$.

Open formulas only. Rule I applies only where the variable is open (not bound by a quantifier). Substituting into a bound variable breaks: from $x = y$ and $\exists x\,(x \neq y)$ you would get the absurd $\exists x\,(x \neq x)$.

Equivalence relations. A two-place relation $R$ is an equivalence relation when it is:

• Reflexive — $x\,R\,x$ for all $x$.
• Symmetric — $x\,R\,y \Rightarrow y\,R\,x$.
• Transitive — $(x\,R\,y \wedge y\,R\,z) \Rightarrow x\,R\,z$.

The lecture proves identity has all three, so equality is an equivalence relation. (Contrast $\leq$: reflexive and transitive but not symmetric.) Equivalence relations recur everywhere — congruence in geometry, isomorphism of sets, etc.

Quantifier theorems (now derived, not assumed). The earlier definition of $\exists$ in terms of $\forall$ is actually provable from the rules:

• $\exists x\,P(x) \;\equiv\; \neg\,\forall x\,\neg P(x)$
• $\neg\,\forall x\,P(x) \;\equiv\; \exists x\,\neg P(x)$
• $\neg\,\exists x\,P(x) \;\equiv\; \forall x\,\neg P(x)$

A negation pushed across a quantifier flips $\forall \leftrightarrow \exists$ and negates the inner predicate.

Operating on both sides. Identity justifies the school-algebra move of applying the same operation to both sides of an equation, for any operator $\circ$ for which the result is defined (a term):

• $\forall x\,\forall y\,\forall z\;\big(x = y \Rightarrow x \circ z = y \circ z\big)$
• $\forall x\,\forall y\,\forall z\;\big(x = y \Rightarrow z \circ x = z \circ y\big)$

The interpretation is valid only when $x \circ z$ is genuinely a term, which is why dividing both sides by $z$ requires $z \neq 0$ ($x \div 0$ is undefined, hence not a term).

Groups (the next axiomatic system). A group is a set $G$ with an operation $\circ$ satisfying:

• Associativity — $x \circ (y \circ z) = (x \circ y) \circ z$.
• Identity element — there exists $e \in G$ with $x \circ e = x$ for all $x$.
• Inverses — for every $x$ there exists $x^{-1}$ with $x \circ x^{-1} = e$.

Examples: $(\mathbb{R}, +)$ with identity $0$ and inverse $-x$; $(\mathbb{R}\setminus\{0\}, \times)$ with identity $1$ and inverse $1/x$. Subtraction on $\mathbb{R}$ fails (not associative); division fails the same way.

Worked Example

Theorem (symmetry of identity): prove $\forall x\,\forall y\,(x = y \Rightarrow y = x)$.

Assume the antecedent, introduce the law of identity, then substitute using Rule I:

The trick is line 3: in $x = x$, replacing the first $x$ with $y$ (licensed by $x = y$) yields $y = x$. Because the final line rests on no premises, this is a theorem of logic — true in any axiomatic system built on top of logic, never something algebra has to assume.

A second application: the apartment proof. Symbolizing "the criminal was in the apartment" as the constant $c$ ($A c$), with premises that anyone in the apartment was in town ($\forall x\,(Ax \Rightarrow Tx)$), anyone in Mexico was not ($\forall x\,(Mx \Rightarrow \neg Tx)$), and Barnes was in Mexico ($Mb$), one derives $\neg Ab$. Assuming $b = c$ for a reductio, Rule I rewrites $Ac$ into $Ab$, contradicting $\neg Ab$ — forcing the conclusion $b \neq c$ (Barnes is not the criminal). This proof cannot be done without identity.

Summary

• Identity adds the equality symbol and exactly one rule (Rule I), completing first-order logic; equality is now a logical symbol and may not be reinterpreted as another predicate.
• Rule I bundles the Law of Identity ($x = x$) and the Principle of Extensionality (equals substitute for equals), but applies only to open (unbound) formulas.
• Identity is an equivalence relation: reflexive, symmetric, and transitive — a pattern shared by congruence, set isomorphism, and other "sameness" relations.
• A statement provable with no remaining premises is a theorem of logic, valid in every system built on logic — including symmetry and transitivity of $=$.
• The quantifier-negation equivalences ($\exists x\,P(x) \equiv \neg\forall x\,\neg P(x)$, etc.) are derivable from the rules, not extra assumptions.
• Identity justifies applying the same operation to both sides of an equation, provided the result stays a term (hence no division by zero); this leads into group theory, the next axiomatic system.