CHAPTER 6: PREDICATE LOGIC

6.1 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)

6.2 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 to Claire about Bob)

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

Lbc & Abb (Bob loves Claire and Bob admires himself)

Symmetry

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)

Transitivity

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

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

6.3 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)

6.4 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]

6.5 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