PREDICATE LOGIC: RULES

Simple non-relational statements

Ms (Socrates is mortal)

Compound non-relational statements (contain &, v, →, ~)

Ac & Jb (Claire is an acrobat and Bob is a juggler)

Simple two-place relational predicates (predicate-subject-object)

Lbc (Bob loves Claire)

Simple three-place relational predicates

Gcfb (Claire gave Fido to Bob)

Compound two-place relational predicates (contain &, v, →, ~)

Rbj → Rjb (if Bob respects Joe then Joe will respect Bob)

Compound three-place relational predicates (contain &, v, →, ~)

Gdbc & Gcfb (Donna gossiped about Bob to Claire and Claire gave Fido to Bob)

Symmetrical relationship

Mbc → Mcb (if Bob is married to Claire, then Claire is married to Bob)

Asymmetrical relationship

Fej → ~Fje (if Ed is the father of Joe, then Joe is not the father of Ed)

Nonsymmetrical relationship

Abc → (Acb v ~Acb) (if Bob admires Claire, then Claire may or may not admire Bob)

Transitive relationship

(Tje & Tea) → Tja (if Jill is taller than Ella, and Ella is tall than Agnus, then Jill is taller than Agnus)

Intransitive relationship

(Fej & Fja)→ ~Fea (if Ed is the father of Joe, and Joe is the father of Al, then Ed is not the father of Al)

Nontransitive relationship

(Abs & Asj)→(Abj v ~Abj) (If Bob admires Sue and Sue admires Joe, then Bob will/won’t admire Joe)

A: all S is P (all students are people)

∀x(Sx → Px)

E: no S is P (no student is a pelican)

∀x(Sx → ~Px)

I: some S is P (some students are pilots)

Ǝx(Sx & Px)

O: some S is not P (some students are not partiers)

Ǝx(Sx & ~Px)

Identity predicate

t=c (Mark Twain is Samuel Clemens)

The only

Gj & ∀x(Gx → x=j) (Joe is the only guitarist in town)

Only

Sj & ∀x(Sx → x=j) (Only Joe survived)

No . . . except

Pj & Sj & ∀x[(Px & Sx) → x=j] (No pilots survived except Joe)

All . . . except

Pj & ~Sj & ∀x[(Px & ~x=j) → Sx] (All pilots survived except Joe)

Superlatives

Tj & ∀x[(Tx & ~x=j) → Gjx] (Joe is the greatest trombonist)

At most

∀x ∀y(x=y) (At most one thing exists)

At least

Ǝx Ǝy(~x=y) (At least two things exist)

Exactly

Ǝx ∀y(x=y) (Exactly one thing exists)

Definite Description

Ǝx[Axi & ∀y(Ayi → y=x) & x=h] (Homer is the author of the Iliad)