PREDICATE LOGIC

NON-RELATIONAL SUBJECT-PREDICATE STATEMENTS (a.k.a., monadic or one-place predicates)

Simple non-relational statements

Examples

Ms (Socrates is mortal)

Ac (Claire is an acrobat): Ac

Ln (New York City is large)

Subject: for subjects that are individual constants (specific people, places, things), use letters a, b, c . . . r, s, t

Predicate: for the predicate qualities, use capital letters A-Z

The predicate always appears before subject

Compound non-relational statements

Statements that use logical connectives: &, v, →, ~

Examples

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

Gb v Jb (Bob is either a geek or a jock)

Sj → Tj (if John is a student then John pays tuition)

Ab ↔ Sp (Beth will sing alto if and only if Pam will sing soprano)

~Vh (Hitler was not virtuous)

RELATIONAL SUBJECT-PREDICATE STATEMENTS

Two-place and three-place relational predicates

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

Lbc (Bob loves Claire)

Abb (Bob admires himself)

Simple three-place relational predicates

Gdbc (Donna gossiped about Bob to Claire)

Gcfb (Claire gave Fido to Bob)

Compound two-place relational predicates

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

Compound three-place relational predicates

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

Symmetry

Symmetrical relationship (relationship works in both directions)

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

Asymmetrical relationship (relationship does not work in both directions)

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

Nonsymmetrical relationship (relationship may or may not work in both directions)

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

Transitivity

Transitive relationship (relationship is like hypothetical syllogism)

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

Intransitive relationship (relationship is not like hypothetical syllogism)

(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 (relationship may or may not be like hypothetical syllogism)

(Abc & Acj) → (Abj v ~Abj) (If Bob admires Claire, and Claire admires Joe, then Bob may or may not admire Joe)

QUANTIFIERS AND VARIABLES

Variables: use letters u-z

Different variables do not necessarily designate different objects

Choice of variables makes no difference to meaning

Two quantifiers

∀: universal quantifier (“∀x” means “for all x”)

Ǝ: existential quantifier (“Ǝx” means “for some x”)

Four quantification statement forms:

A: all S is P (e.g., all students are people)

∀x(Sx → Px)

For all x, if x is S then x is P

Warning: do not use a conjunction with A statements, since it will not mean the same thing. E.g. ∀x(Sx & Px) means “everything is a student and a person”

E: no S is P (e.g, no student is a pelican)

∀x(Sx → ~Px)

For all x, if x is S then it is not the case that x is P

I: some S is P (e.g., some students are pilots)

Ǝx(Sx & Px)

For some x, x is S and x is P

O: some S is not P (e.g., some students are not partiers)

Ǝx(Sx & ~Px)

For some x, x is S and it is not the case that x is P

Examples of quantified non-relational statements

Frogs are green: ∀x(Fx → Gx)

There is at least one green frog: Ǝx(Fx & Gx)

Green frogs exist: Ǝx(Fx & Gx)

Some frogs are not green: Ǝx(Fx & ~Gx)

Everything is a frog: ∀x(Fx)

Something is a frog: Ǝx(Fx)

Not everything is a frog: ~∀x(Fx)

Nothing is a frog: ∀x~(Fx) -- or, as per replacement rule below, ~Ǝx(Fx)

Everything is a green frog: ∀x(Fx & Gx)

Examples of quantified relational statements

Bob admire nothing: ∀x~(Abx)

Nothing admires Bob: ∀x~(Axb)

There is something which both Bob and Claire admire: Ǝx(Abx & Acx)

If Bob admires himself, then he admires something: Abb → Ǝx(Abx)

A musician admires Bob: Ǝx(Mx & Axb)

Every musician admires Bob ∀x(Mx → Axb)

Everything is a musician that admires Bob: ∀x(Mx & Axb)

IDENTITY PREDICATES

Identity predicate: the symbol =, which means “identical to”

Mark Twain is Samuel Clemens: t=c

George Eliot is not Samuel Clemens: ~e=c (alternatively e≠c)

The only

Joe is the only guitarist in town

Gj & ∀x(Gx → x=j)

Alternative: Gj & ∀x(~x=j → ~Gx)

Joe is the only one who finished

Fj & ∀x(Fx → x=j)

Only

Only Joe survived

Sj & ∀x(Sx → x=j)

Only Mark Twain wrote *Huckleberry
Finn*

Wth & ∀x(Wxh → x=t)

No . . . except

No pilots survived except Joe

Pj & Sj & ∀x[(Px & Sx) → x=j]

No people except Joe love Michelle

Pj & Ljm & ∀x[(Px & Lxm → x=j]

All . . . except

All pilots survived except Joe

Pj & ~Sj & ∀x[(Px & ~x=j) → Sx]

All people except Joe love Michelle:

Pj & ~Ljm & ∀x[(Px & ~x=j) → Lxm]

Superlatives

Joe is the greatest trombonist

Tj & ∀x[(Tx & ~x=j) → Gjx]

At most

At most one thing exists

∀x ∀y(x=y)

At most one student failed

(Sx & Sy) & ∀x ∀y[(Fx & Fy) → x=y]

At least

At least two things exist

Ǝx Ǝy(~x=y)

There are at least two giraffes in the zoo

Ǝx Ǝy[(Gx & Gy) & ~x=y]

Exactly

Exactly one thing exists

Ǝx ∀y(x=y)

There is exactly one giraffe in the zoo

Ǝx Ǝy[(Gx & Gy) & x=y]

Relational identity predicates

If
Mark Twain is Samuel Clemens, then Samuel Clemens wrote *Huckleberry Finn*:
t=c → Wch

No American author is better than Mark Twain: ∀x(Ax → ~Bxt)

Definite Description

Homer is the author of the Iliad

Ǝx[Axi & ∀y(Ayi → y=x) & x=h]

The inventor of the toothbrush was British

Ǝx[Ixt & ∀y(Iyt → y=x) & Bx]

RULES

Summary of Logical symbols

Logical operators: ~ & v → ↔

Predicate symbols: upper case letters (“A” for “is an acrobat”)

Constants: lower case a-t (“b” for “Bob”)

Variables: lower case u-z (use in quantifiers “for all x” “there exists some x”)

Existential Quantifiers: Ǝ (Ǝx “there exists some x”)

Universal Quantifier: ∀ (∀x “for all x”)

Identity: = (is identical to)

Wffs

1. Any atomic formula is a wff

2. If P is a wff, so is ~P

3 if P and Q are wffs, so are (P & Q), (P v Q), (P & Q), (P → Q), (P ↔ Q)

4. If P is a wff containing the name letter a, then any formula of the form ∀bPb/a or ƎbPb/a is a wff, where Pb/a is the result of replacing one or more of the occurrences of a in P by some variable b not already in P

5. The result of writing ‘=’ between any pair of name letters is an atomic wff