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

Terms, Predicates, and Quantifiers - Lecture 5 (Logic)

Overview

This lecture begins the move from propositional logic into first-order predicate logic. Propositional logic could only treat whole sentences as atomic letters joined by connectives, so it could not capture the internal structure of statements like "everything is either red or not red." To expose that structure we introduce three new kinds of symbol — terms, predicates, and quantifiers — and add them to the parentheses and sentential connectives we already had. Along the way we build up the formal vocabulary of the grammar: atomic formulas, formulas, the scope of a quantifier, free vs. bound variables, and finally the precise definition of a sentence. The payoff is the ability to symbolize essentially any English statement and to expose ambiguities that ordinary language hides.

Key Concepts

From propositional to predicate logic. A proposition is anything that is either true or false (not both); a statement is an English sentence that encodes a proposition. Propositional logic symbolized whole statements with capital letters and combined them with the five connectives — not $\neg$, and $\wedge$, or $\vee$, implies $\rightarrow$, equivalent $\leftrightarrow$. Predicate logic keeps the parentheses and connectives but breaks each statement open into terms, predicates, and quantifiers.

Terms. A term names or describes an object — it is the "subject" of a sentence.

• A term is an expression that names or describes exactly one object, either directly or after its variables are replaced by names/descriptions.
• Terms split into constants (particular things, e.g. Isaac Newton, 2, Jack) and variables (universals / classes, e.g. men, humans).
• Notation: lowercase letters early in the alphabet ($a, b, c$) for constants; lowercase letters late in the alphabet ($x, y, z$) for variables.
• Test: "$x + 7$" is a term (replace $x$ with a number and it names one number); "$2x > n$" is not a term (it yields true/false, not an object); "$x^2 + y^2 - 17$" is a term; "the fattest man in state $x$" is a term.

Predicates. A predicate is a proposition-valued function: give it the right number of terms and it returns something true or false.

• A predicate alone is neither true nor false; only after it receives its terms does it become a proposition.
• Notation: capital letters for predicates, followed by the terms. $Bx$ means "$x$ is blue." The variable written inside is just a placeholder ("syntactic sugar") marking where a term goes — the predicate itself is "is blue," written $B$.
• A one-term predicate is a property (e.g. is blue, is greater than three); a two-term predicate is a relation (e.g. loves, taller than, greater than).
• Infix vs. prefix notation: for a relation you may write the predicate between its terms. "Frank is taller than Jack" is $Tfj$ (prefix) or $fTj$ (infix); order matters — $Tfj \neq Tjf$.
• Predicates can be combined with connectives because they ultimately return propositions, letting you define new predicates from old ones (e.g. $W \equiv R \wedge B$, "red and blue").

Quantifiers. Two only:

• Universal $\forall x$ — "for all / for every / for each $x$."
• Existential $\exists x$ — "there exists / there is / some $x$" (read it followed by such that). In logic some means at least one (possibly more).
• A quantifier is a second way to turn a predicate into a proposition (the first being to supply a constant term). $\forall x\, Lxy$ is not a proposition; $\forall x\, \exists y\, Lxy$ ("everybody loves somebody") is.
• Order of quantifiers matters. $\forall x\, \exists y\,(x+y=0)$ is true (each $x$ has its own additive partner), but $\exists x\, \forall y\,(x+y=0)$ is false (no single $x$ works for every $y$).

Defining the existential quantifier. $\forall x\, P(x)$ is a proposition that is true iff $P$ holds independent of the choice of $x$. The existential is then defined in terms of the universal: $\exists x\, P(x) \;:=\; \neg\,\forall x\,\neg P(x)$ "There is a white swan" means "it is not the case that all swans are not white."

Negating quantifiers (the key theorem). To negate a quantified statement, push the negation inward and swap the quantifier: $\neg\,\forall x\, P(x) \equiv \exists x\, \neg P(x) \qquad\text{and}\qquad \neg\,\exists x\, P(x) \equiv \forall x\, \neg P(x)$

Unique existence. $\exists! x\, P(x)$ means "there is exactly one $x$ with $P$": $\exists x\,\big(P(x) \wedge \forall y\,(P(y) \rightarrow y = x)\big)$.

Building the grammar.

• Atomic formula — a single predicate together with the appropriate number of terms (constants and/or variables). No connectives, no quantifiers inside. May or may not be a proposition ($Bs$ is; $Bx$ is not).
• Formula — defined recursively: every atomic formula is a formula; if $R, S$ are formulas so are $\neg R,\ R\wedge S,\ R\vee S,\ R\rightarrow S,\ R\leftrightarrow S$; and if $R$ is a formula and $x$ a variable, then $\forall x\, R$ and $\exists x\, R$ are formulas. Nothing else is a formula. This rules out nonsense strings.
• Scope of a quantifier — the quantifier together with the smallest formula immediately following it. Quantifiers may nest; work from the inside out.
• Free vs. bound — an occurrence of a variable is bound if it lies in the scope of a quantifier over that same variable; otherwise it is free. A variable is a free (bound) variable of a formula if it has at least one free (bound) occurrence — so a variable can be both.
• Sentence — a formula with no free variables. Sentences are exactly the formulas you can call true or false (propositions).

Why this matters: ambiguity. "Men play golf" is, strictly, only the predicate $Mx \rightarrow Gx$ — neither all men nor some men. English silently supplies a quantifier, which is the source of much disagreement (e.g. "do the ends justify the means?" has four distinct quantifier readings). Writing the logical structure makes the intended claim unequivocal.

Worked Example

Symbolize "Some freshmen are not intelligent" (i.e. "not all freshmen are intelligent").

Let $Fx$ = "$x$ is a freshman" and $Ix$ = "$x$ is intelligent." The natural reading is negating a universal; using the negation rule and the equivalence $\neg(A\rightarrow B)\equiv A\wedge\neg B$:

The final form reads "there exists something that is a freshman and is not intelligent." Note the connective must be $\wedge$, not $\rightarrow$: writing $\exists x\,(Fx \rightarrow \neg Ix)$ would be satisfied vacuously by any non-freshman (e.g. a piece of chalk), which is not what the sentence claims.

Summary

• Predicate logic adds terms (constants/variables), predicates (proposition-valued functions: one-term = property, two-term = relation), and quantifiers ($\forall$, $\exists$) to the connectives and parentheses inherited from propositional logic.
• A predicate is true/false only after it receives its terms; variables written inside a predicate are placeholders, and for relations the order of terms matters.
• $\exists x\, P(x)$ is defined as $\neg\,\forall x\,\neg P(x)$; negation flips a quantifier and moves inside: $\neg\forall x\,P \equiv \exists x\,\neg P$ and $\neg\exists x\,P \equiv \forall x\,\neg P$. Quantifier order changes meaning.
• An atomic formula is one predicate plus its terms; formulas are built recursively from atomic formulas via connectives and quantifiers; the scope of a quantifier is the smallest formula following it.
• A variable occurrence is bound if inside a matching quantifier's scope, else free; a sentence is a formula with no free variables — exactly the formulas that are propositions.
• Translating English into this grammar exposes hidden quantifier ambiguities (e.g. "men play golf," "the ends justify the means") that cause much real-world disagreement.