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

Interchange of Quantifiers & General Inference - Lecture 9 (Logic)

Interchange of Quantifiers & General Inference – Lecture 9

Overview

This lecture completes the rules of derivation for first-order predicate logic by introducing the two rules that govern the existential quantifier, then layers two further topics on top of them. First we add existential specification and existential generalization to round out the four quantifier rules. Next we examine the interchange of quantifiers — the equivalence that lets a universal claim be rewritten as a negated existential claim, and vice versa. Finally, general inference makes explicit the restrictions on the quantifier rules that intuition normally enforces silently. Mastering these gives you a complete proof system: every derivation in first-order logic uses only rule P, rule T, rule CP, reductio ad absurdum, and the four quantifier rules.

Key Concepts

Ambiguous names. An ambiguous name denotes one and only one particular object whose name we happen not to know. It is written with a Greek letter (e.g. $\alpha$, $\beta$). This contrasts sharply with a variable ($x, y, z$), which stands for any object in the domain. Convention used throughout: letters at the start of the alphabet ($a, b, c$) are named constants; letters at the end ($x, y, z$) are variables; Greek letters ($\alpha, \beta$) are ambiguously named constants.

Existential specification (the idea). If something satisfies a condition, then some nameable individual satisfies it. From $\exists x\, H(x)$ ("there is a human") we may introduce a fresh ambiguous name and conclude $H(\beta)$.

• Formal statement: if $S$ results from $R$ by substituting, for every free occurrence of $x$ in $R$, an ambiguous name $\beta$ not previously used in the derivation, then $S$ is derivable from $\exists x\, R$.
• Note that the quantifier is not part of $R$; $x$ is free in $R$ alone.

Existential generalization (the idea). Conversely, if a condition holds of some nameable individual or constant, then something satisfies it. From $E(2) \wedge P(2)$ we may conclude $\exists x\,(E(x) \wedge P(x))$.

• Formal statement: if $S$ results from $R$ by substituting a variable $x$ for every occurrence of a name/constant, then $\exists x\, S$ is derivable from $R$.
• You need not know the individual's name to generalize — only that it is some named thing.

The four quantifier rules in summary.

• Universal specification (US): drop $\forall x$, replacing $x$ with any term (including variables).
• Universal generalization (UG): add $\forall x$ (only over variables that are not flagged/free in a premise).
• Existential specification (ES): drop $\exists x$, replacing $x$ with a fresh ambiguous name.
• Existential generalization (EG): add $\exists x$, replacing a name with a variable.

Why ES must use an ambiguous name, not a variable. If you replaced $\exists x\, H(x)$ with a variable $H(y)$, you could then universally generalize to $\forall y\, H(y)$ — concluding "everything is human" from "something is human." Forcing ES to produce a constant $\beta$ blocks this, because UG applies only to variables.

Interchange of quantifiers. Two logical equivalences (proved as theorems in the next chapter; taken on intuition here):

"Everything has the property" says the same thing as "there is nothing lacking it." Mechanically: swap the quantifier, and negate the matrix.

General inference — the four restrictions. Intuition normally prevents these errors, but the rules must state them:

• R1. You may not existentially generalize onto a variable that is free in the formula. From $\alpha > x$ you cannot conclude $\exists x\,(x > x)$.
• R2. You may not generalize in an order that swaps quantifiers. Think of specifying as opening a bracket and generalizing as closing it: an inner bracket opened last must be closed first. If you US then ES, you must EG before you UG. This stops $\forall x\,\exists y\,(x<y)$ from collapsing into the false $\exists y\,\forall x\,(x<y)$.
• R3. You may not universally specify onto a bound term. In $\forall x\,\exists y\,(x<y)$ you cannot replace $x$ with the already-bound $y$.
• R4. You may not existentially generalize onto a bound term, for the same reason.

A flagged variable is a free variable introduced via a premise (including a conditional premise); UG is forbidden on flagged variables until that premise is discharged.

Worked Example

Claim: All humans are mortal; there are humans; therefore there are mortals — i.e. $\{\forall x\,(H(x)\to M(x)),\; \exists x\, H(x)\} \vDash \exists x\, M(x)$.

The key move is to specify the existential premise first (introducing an ambiguous name), then specify the universal premise onto that same name.

Lines 5–6 depend on premises 1 and 2, completing the proof. Had we done US before ES (specifying line 1 to a variable first), we would have been forced to invent the name in the wrong order and could not have legitimately discharged it — which is exactly why ES on the existential premise comes first.

Summary

• The existential quantifier gets two rules: ES removes $\exists$ by introducing a fresh ambiguous name (a constant whose name is unknown, written $\beta$); EG adds $\exists$ by replacing a name with a variable.
• An ambiguous name denotes exactly one object whose identity is unknown — categorically different from a variable, which ranges over the whole domain.
• In multi-quantifier proofs, specify the existential quantifier first, then specify universals onto the same name, do your propositional work, and generalize in reverse order.
• Interchange of quantifiers: $\forall x\,\varphi \equiv \neg\exists x\,\neg\varphi$ and $\exists x\,\varphi \equiv \neg\forall x\,\neg\varphi$ — swap the quantifier and negate the matrix.
• General inference adds four guard-rails: no generalizing onto a free variable, no order that swaps quantifiers, and no specifying or generalizing onto a bound term.
• The complete proof system needs only seven essential rules — P, T, CP, US, UG, ES, EG — with reductio ad absurdum kept as a convenient (but derivable) eighth.