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Predicate Logic

Predicates, names, variables, and the quantifiers ∀ and ∃ — capturing "all" and "some".

Sentence logic can’t see inside a sentence. “All circles are blue” and “Some circle is blue” both become single atoms — yet one entails the other, which sentence logic can’t explain. Predicate logic adds the missing structure.

Names ( $a, b, c$ ) denote particular objects; variables ( $x, y, z$ ) are placeholders.
Predicates ( $Px$ , $Bx$ , $Rxy$ ) express properties and relations: $Bx$ = “x is blue”, $Rxy$ = “x relates to y”.
Quantifiers bind variables: $\forall x$ (“for all x”) and $\exists x$ (“there exists an x”). Teller writes these $(x)$ and $(\exists x)$ .

A statement is evaluated in a model: a domain of objects plus an interpretation of each predicate. Pick predicates below and watch the universal and existential claims come out true or false over a fixed domain.

LetK(x) = "x is a",C(x) = "x is".

a K that is C (witness) a K that is not C (counterexample)

∀x (K(x) → C(x))every circle is blue: FALSE

∃x (K(x) ∧ C(x))some circle is blue: TRUE

At least one circle is not blue (red ring) — that single counterexample makes the universal FALSE.

The two translation patterns

The single most important habit in predicate logic — and the easiest thing to get wrong:

“Every K is C” is $\forall x\,(Kx \rightarrow Cx)$ — universal with a conditional. (Writing $\forall x\,(Kx \land Cx)$ would say “everything is a K and is C”, which is far stronger.)
“Some K is C” is $\exists x\,(Kx \land Cx)$ — existential with a conjunction. (Writing $\exists x\,(Kx \rightarrow Cx)$ is almost always wrong: it is satisfied by any non-K.)

A universal is falsified by a single counterexample (a K that is not C); an existential is verified by a single witness (a K that is C) — exactly the rings the visualizer draws.

Multiple quantifiers and order

Relations need several quantifiers, and order matters:

$\forall x\,\exists y\,Lxy$ — “everyone loves someone (possibly different people)”.
$\exists y\,\forall x\,Lxy$ — “there is someone whom everyone loves (one person)”.

The second implies the first, but not the reverse. Swapping two quantifiers of the same type ( $\forall x\forall y$ vs $\forall y\forall x$ ) is harmless; swapping $\forall$ with $\exists$ changes the meaning.

Takeaways

Predicate logic adds names, variables, predicates/relations, and the quantifiers $\forall$ , $\exists$ .
Truth is relative to a model (domain + interpretation).
“Every K is C” = $\forall x(Kx \rightarrow Cx)$ ; “Some K is C” = $\exists x(Kx \land Cx)$ .
A universal falls to one counterexample; an existential needs one witness; quantifier order matters.

References

Paul Teller, A Modern Formal Logic Primer, Prentice Hall (1989) — free at tellerprimer.ucdavis.edu. Curriculum follows the primer; explanations and examples here are original.