The Language The Language PP

8 9 (8x) (9y) (8z) 9 8

Quantificational LogicQuantificational Logic

Quantificational Logic:Quantificational Logic is the logic of sentences involving quantifiers, predicates, and names. We will study the properties which arguments and sentences have in virtue of their quantificational structure.

Predicates and Singular TermsPredicates and Singular Terms

• Socrates is human• Plato is human• Socrates is bald• The inventor of Velcro is bald• The inventor of Velcro kills Plato• Socrates kills Socrates• Seven is less than twenty• Socrates introduces Plato to the inventor of Velcro• The least prime number is between seven and twenty

Singular TermsSingular TermsSingular Term:

A singular term is a word or phrase which designates or is supposed to designate some individual object. Natural language singular terms are either proper nouns or definite descriptions (a phrase which is supposed to designate an object via a unique description of it).

From the previous slide:• Socrates• Plato• the inventor of Velcro• seven• twenty• the least prime number

PredicatesPredicatesPredicate:

A predicate is a series of words with one or more blanks which yields a sentence when all its blanks are filled with singular terms. Conversely, we could think of a predicate as what remains after removing one or more singular terms from a sentence.

From the earlier slide:• ___ is human• ___ is bald• ___ kills …• ___ is less than …• ___ introduces … to - - -• ___ is between … and - - -

Predicate Letters and ConstantsPredicate Letters and Constants

• We will use uppercase letters A-Z (with or without super- and subscripts) as predicates of P

• We will use lowercase letters a-v (with or without subscripts) as constants of P to represent proper nouns (definite descriptions will receive a special analysis)

• We will use lowercase letters w-z (with or without subscripts) as variables (to be discussed later)

Interpretation in Interpretation in PP

Interpretation in P:An interpretation in P consists of 3 components:

1) a non-empty universe of discourse (UD) specifying the range of the variables (i.e., the things about which we’ll be talking)

2) an interpretation of the predicate letters, either via a translation key, or a specification of extensions

3) an interpretation of the constants via an assignment of objects from the UD to constants such thata) every individual constant is interpreted, and

b) no individual constant is allowed more than one interpretation

An InterpretationAn Interpretation

B: is bald

H: is human

K: kills L: is less than

B: is between and

e: eleven

l: Plato

p: Plato

s: Socrates

t: twenty

v: seven

UD: All people and all positive integers

Pronouns and QuantifiersPronouns and Quantifiers

You see, a pronoun was made to take the place of a noun

’Cause saying all those nouns over and over can really wear you down!

—Albert Andreas Armadillo in “Rufus Xavier Sarsaparilla” Schoolhouse Rock

Eliminable PronounsEliminable Pronouns(Pronouns of Laziness)(Pronouns of Laziness)

• Socrates was human, and he killed himself• Socrates was human, and Socrates killed Socrates

• Da Vinci painted the Mona Lisa and now he is considered a great artist and it is considered a masterpiece

• Da Vinci painted the Mona Lisa and now Da Vinci is considered a great artist and the Mona Lisa is considered a masterpiece

• Rufus Xavier Sarsaparilla found a kangaroo that followed him home and now it belongs to him

• Rufus Xavier Sarsaparilla found a kangaroo that followed Rufus Xavier Sarsaparilla home, and now the kangaroo that Rufus Xavier Sarsaparilla found belongs to Rufus Xavier Sarsaparilla

Ineliminable PronounsIneliminable Pronouns

• Any student caught cheating will have her name published and she will be expelled? Any student caught cheating will have Susie’s

name published and Susie will be expelled? Any student caught cheating will have any

student’s name published and any student will be expelled

• Every thing is such that, if it is a student and it is caught cheating, then it will have its name published and it will be expelled

Ineliminable PronounsIneliminable Pronouns

• Some student was caught cheating and she had her name published and she was expelled– Some student was caught cheating and Susie’s had

Susie’s name published and Susie was expelled– Some student was caught cheating and some

student had some student’s name published and some student was expelled

• Some thing is such that, it is a student and it was caught cheating and it had its name published and it was expelled

Example QuantifiersExample Quantifiers

(8x) Universal x-quantifier:all (objects, things) x are such that…for all x…every (thing) x is such that…

(9x) Existential x-quantifier:there exists at least one (thing) x such that…(for) some x…at least one x is such that…

The Symbols of The Symbols of PP

Predicate Letters:

Individual Terms:Individual Constants:

a, b,…, v, a1,…, v1, a2,…

Individual Variables:w, x, y, z, w1, x1, y1, z1, w2,…

The Symbols of The Symbols of PP

Truth-Functional Connectives:: Æ Ç ! $

Quantifier Symbols:8 9

Punctuation Marks:( )

MetavariablesMetavariables

Metavariables:

Usually ‘t’ will range over individual terms, ‘a’ over individual constants, and ‘x’ over individual variables.

Expressions and Quantifiers of Expressions and Quantifiers of PP

Definition 5.2.1 (Expression of P). An expression of P is any finite sequence of the symbols of P.

Definition 5.2.2 (Quantifier of P). Where x ranges over individual variables, expressions of the form (8x) are called universal quantifiers of P, while expressions of the form (9x) are called existential quantifiers of P.

Well-Formed Formulas of Well-Formed Formulas of PPDefinition 5.2.3 (Well-Formed Formula of P). Where P and Q range over

expressions of P, ranges over predicate letters of P, t ranges over individual terms of P, and x ranges over individual variables of P,

1) Any expression of form t1,…,tn is a wff of P2) If P and Q are wffs of P, then

a) :P is a wff of Pb) (P Æ Q) is a wff of Pc) (P Ç Q) is a wff of Pd) (P ! Q) is a wff of Pe) (P $ Q) is a wff of P

3) If x is an individual variable of P and P is a wff of P which i) contains at least one occurrence of x and ii) contains no x-quantifiers, then

a) (8x)P is a wff of Pb) (9x)P is a wff of P

4) Nothing is a wff of P unless it can be shown so by a finite number of applications of clauses 1) through 3)

Syntactic ConceptsSyntactic Concepts

Bound Variable, Free Variable:An occurrence of a variable x in a wff P is bound iff it is within the scope of an x-quantifier. An occurrence of a variable is free iff it is not bound.

Open Wff:A wff of P is open iff it contains at least one free occurrence of a variable.

Closed Wff, Sentence of P:A wff of P is closed iff it contains no free occurrences of variables. We also call such wffs sentences of P.

Interpreting QuantifiersInterpreting Quantifiers

• Where Fx is a wff with only instances of x free:

– A universally quantified wff (8x)Fx is true iff the condition expressed by the immediate subcomponent Fx is true of (satisfied by) every object in the UD.

– An existentially quantified wff (9x)Fx is true iff the condition expressed by the immediate subcomponent Fx is true of (satisfied by) at least one object in the UD.

UD: The marbles in Fred’s Collection

B: is blue

G: is green

R: is red

Y: is yellow

C: is cracked

S: is scratched

B: is bigger than S: is smaller than

g: The Green Giant

o: Old Yeller

r: Big Red

s: Sky

UD: Fred, Susie, Fred’s marbles and Superballs, Susie’s marbles and Superballs

M: is a marble

S: is a Superball

B: is blueG: is greenR: is redY: is yellowC: is cracked

S1: is scratchedB: belongs to

H: bounces higher than L: is larger than S: is smaller than

G: gives to

f: Fredg: The Green Gianto: Old Yellerr: Big Reds: Susie

UD: The people in Kate’s Office

A: is an accountant

C: is a cashier

D: is diligent

L: is lazy

U: is upbeat

L: likes W: works for

f: Fred

k: Kate

UD: Positive Integers

E: is even

O: is odd

P: is prime

G: is greater than L: is less than E: times is even

O: times is odd

P: times is prime

T: times equals

a1: one

a2: two

a3: three

a4: four

an: n

Simple Square of OppositionSimple Square of Opposition

Everything is F(8x)Fx

:(9x):Fx

Contraries(cannot both be true)

Nothing is F(8x):Fx

:(9x)Fx

:(8x):Fx(9x)Fx

Something is F(cannot both be false)

Subcontraries

:(8x)Fx(9x):Fx

Something is not F

Contradictories ImpliesImplies

Quantifiers, Conjunction, Quantifiers, Conjunction, DisjunctionDisjunction

(8x)Fx Æ (8x)Gx , (8x)(Fx Æ Gx)

(8x)Fx Ç (8x)Gx ) (8x)(Fx Ç Gx)

(9x)Fx Æ (9x)Gx ( (9x)(Fx Æ Gx)

(9x)Fx Ç (9x)Gx , (9x)(Fx Ç Gx)

Modern Categorical Square of OppositionModern Categorical Square of Opposition

A: All F are G(8x)(Fx ! Gx)

:(9x)(Fx Æ :Gx)

E: No F are G(8x)(Fx ! :Gx)

:(9x)(Fx Æ Gx)

:(8x)(Fx ! :Gx)

(9x)(Fx Æ Gx)

I: Some F are G

:(8x)(Fx ! Gx)

(9x)(Fx Æ :Gx)

O: Some F is not G

Contradictories

Variations on A and O FormsVariations on A and O Forms

Vagaries of ‘Any’Vagaries of ‘Any’

Quantifiers with Overlapping ScopeQuantifiers with Overlapping Scope

(9x)(9y)…There is an x and there is a y such that…There is a pair x, y such that…

(8x)(8y)…For every x and for every y…For every pair x, y…

(8x)(9y)…For every x there is a y such that…

(9x)(8y)…There is an x such that for every y…

Note: the object denoted by x is not necessarily different from y

Examples of Overlapping ScopesExamples of Overlapping Scopes

1. Everyone loves someone(8x)(9y)Lxy

2. Everyone is loved by someone(8x)(9y)Lyx

3. Someone loves everyone (9y)(8x)Lyx

4. Someone is loved by everyone(9y)(8x)Lxy

a b

c d

a b

c d

a b

c d

a b

c d

Note:4 ‘ 1, 1 — 43 ‘ 2, 2 — 3

(These are not the only models which satisfy their respective sentences. They were chosen to illustrate the above entailments/failures of entailment.)

IdentityIdentity

Identity:By identity we mean numerical identity—that what may appear to be distinct objects are actually one and the same thing. Object x is identical to object y iff x is y.

• We do not mean exactly similar, as identical twins or triplets are exactly similar—since there are two or three of them, they are not numerically identical.

Identity SignIdentity Sign

• To express numerical identity, we introduce a new predicate letter into the symbols of P, the identity sign:

= : is (numerically) identical to

• This predicate letter is receives a fixed interpretation for all interpretations, and so does not need to be explicitly included in any interpretation

• Unlike other two-place predicates the identity sign is written with infix notation, i.e., between the individual terms:a = b a = y x = y

• Rather than using a hook to negate the identity sign, we slash it:a ≠ b a ≠ y x ≠ y

Identity, Definite Descriptions, and Identity, Definite Descriptions, and Numerical QuantificationNumerical Quantification

There are at least two Fs(9x)(9y)(x y Æ (Fx Æ Fy))

There is at least one x and at least one y s.t., x and y are distinct and both are F

There is exactly one F(9x)(Fx Æ (8y)(Fy ! x = y))

There is an x s.t. it is F and for any y, if y is F, then y is the same thing as x

There are exactly two Fs(9x)(9y)((x y Æ (Fx Æ Fy)) Æ (8z)(Fz ! (x = z Ç y = z)))

There is at least one x and at least one y s.t., x and y are distinct and both are F, and for any z, if z is F, then z is either the same thing as x or the same thing as y

The F is G(9x)((Fx Æ (8y)(Fy ! x = y)) Æ Gx)

There is an x s.t. it is F and for any y, if y is F, then y is the same thing as x, and x is G