For 42 – 47: UD = everything;

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basket

43 The only pear in the basket is rotten

44 There are at least two apples in the basket

45 There are two (and only two) apples in the basket

46 There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rotten

44 There are at least two apples in the basket

45 There are two (and only two) apples in the basket

46 There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx

44 There are at least two apples in the basket

45 There are two (and only two) apples in the basket

46 There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basket

45 There are two (and only two) apples in the basket

46 There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basketxy(Ax & Nxb & Ay & Nyb

45 There are two (and only two) apples in the basket

46 There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basketxy(Ax & Nxb & Ay & Nyb & xy )

45 There are two (and only two) apples in the basket

• There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the

basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basketxy(Ax & Nxb & Ay & Nyb & xy )

45 There are two (and only two) apples in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) & xy )

• There are no more than two pears in the basket

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the

basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basketxy(Ax & Nxb & Ay & Nyb & xy )

45 There are two (and only two) apples in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) & xy )

• There are no more than two pears in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) )

47 there are at least three apples in the basket

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basketxy(Ax & Nxb & Ay & Nyb & xy )

45 There are two (and only two) apples in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) )

• There are no more than two pears in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) )

• there are at least three apples in the basket xyz (Ax & Ay & Az & Nxb & Nyb & Nzb & xy & xz & zy )

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten;

b = the basket

42 There are apples and pears in the basketx(Px & Nxb) & x(Ax & Nxb)

43 The only pear in the basket is rottenx(Px & Nxb & Rx & y(Py & Nyb y=x) )

44 There are at least two apples in the basketxy(Ax & Nxb & Ay & Nyb & xy )

45 There are two (and only two) apples in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) )

• There are no more than two pears in the basketxy(Ax & Nxb & Ay & Nyb & z(Az & Nzb (z=y z=x) )

• there are at least three apples in the basketxyz (Ax & Ay & Az & Nxb & Nyb & Nzb & xy & xz & zy )

• there are at most three apples in the basketxyz (Ax & Ay & Az & Nxb & Nyb & Nzb &

& w(Aw & Nwb w=x w=y w=z) )

SL

Truth value assignments

SL

Truth value assignments

PL

Interpretation

SL

Truth value assignments

PL

Interpretation

Giving an interpretation means defining:

UD

SL

Truth value assignments

PL

Interpretation

Giving an interpretation means defining:

UD

Predicates

SL

Truth value assignments

PL

Interpretation

Giving an interpretation means defining:

UD

Predicates

Constants

SL

Truth value assignments

PL

Interpretation

Giving an interpretation means defining:

UD

Predicates

Constants

Of course, we do not define variables

Truth values of PL sentences are

relative to an interpretation

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

Fx = x is human

a = Socrates

• Bab

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

Fx = x is human Fx = x is handsome

a = Socrates a = Socrates

• Bab

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

Fx = x is human Fx = x is handsome

a = Socrates a = Socrates

• Bab

Bxy = x is bigger than y

a = Himalayas

b = Alpes

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

Fx = x is human Fx = x is handsome

a = Socrates a = Socrates

• Bab

Bxy = x is bigger than y

a = Himalayas a = Himalayas

b = Alpes b = the moon

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

Fx = x is human Fx = x is handsome

a = Socrates a = Socrates

• Bab

Bxy = x is bigger than y

a = Himalayas a = Himalayas a = Himalayas

b = Alpes b = the moon b = Himalayas

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

Fx = x is human Fx = x is handsome

a = Socrates a = Socrates

• Bab

Bxy = x is bigger than y

a = Himalayas a = Himalayas a = Himalayas

b = Alpes b = the moon b = Himalayas

No constant can refer to more than one individual!

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

• Bab

• ~xFx

UD = food

Fx = x is in the fridge

Truth values of PL sentences are

relative to an interpretation

Examples:

• Fa

• Bab

• ~xFx

UD = food

Fx = x is in the fridge

UD = everything

Fx = x is in the fridge

Extensional definition of predicates

Predicates are sets

Extensional definition of predicates

Predicates are sets

Their members are everything they are true of

Extensional definition of predicates

Predicates are sets

Their members are everything they are true of

Predicates are defined relative to a UD

Extensional definition of predicates

Predicates are sets

Their members are everything they are true of

Predicates are defined relative to a UD

Example:

UD = natural numbers

Ox = x is odd

O = {1,3,5,7,9, ...}

Extensional definition of predicates

Predicates are sets

Their members are everything they are true of

Predicates are defined relative to a UD

Example:

UD = natural numbers

Ox = x is odd

Ox = {1,3,5,7,9, ...}

Bxy = x>y

Bxy = {(2,1), (3,1), (3,2), ...}

Extensional definition of predicates

Predicates are sets

Their members are everything they are true of

Predicates are defined relative to a UD

Example:

UD = natural numbers

Ox = x is odd Bxyz = x is between y and z

Ox = {1,3,5,7,9, ...} Bxyz = {(2,1,3), (3,2,4), ...}

Bxy = x>y

Bxy = {(2,1), (3,1), (3,2), ...}

Extensional definition of predicates

Predicates are sets

Their members are everything they are true of

Predicates are defined relative to a UD

Example:

UD = natural numbers

Ox = x is odd Bxyz = x is between y and z

Ox = {1,3,5,7,9, ...} Bxyz = {(2,1,3), (3,2,4), ...}

Bxy = x>y Bxyz = y is between x and z

Bxy = {(2,1), (3,1), (3,2), ...} Bxyz = {(1,2,3), (2,3,4), ...}

(An & Bmn) ~ Cn UD: All positive integersAx: x is oddBxy: x is bigger than yCx: x is prime

m: 2n: 1

Truth-values of compound sentences

(An & Bmn) ~ Cn UD: All positive integersAx: x is oddBxy: x is bigger than yCx: x is prime

m: 2n: 1

Truth-values of compound sentences

UD: All positive integersAx: x is evenBxy: x is bigger than yCx: x is prime

m: 2n: 1

Truth-values of quantified sentences

Birds fly

UD = birds

xFx

Truth-values of quantified sentences

Birds fly

UD = birds

xFx

Fa

Fb

Fc

:

Ftwooty

:

Truth-values of quantified sentences

Birds fly

UD = birds UD = everything

xFx x(Bx Fx)

Fa

Fb

Fc

:

Ftwooty

:

Truth-values of quantified sentences

Birds fly

UD = birds UD = everything

xFx x(Bx Fx)

Fa Ba Fa

Fb Bb Fb

Fc Bc Fc

: :

Ftwooty Btwootie Ftwootie

: :

Truth-values of quantified sentences

Birds fly Some birds don’t fly

UD1 = birds UD2 = everything UD1

xFx x(Bx Fx) x~Fx

Fa Ba Fa

Fb Bb Fb

Fc Bc Fc

: :

Ftwooty Btwootie Ftwootie

: :

Truth-values of quantified sentences

Birds fly Some birds don’t fly

UD1 = birds UD2 = everything UD1

xFx x(Bx Fx) x~Fx

Fa Ba Fa ~Ftwootie

Fb Bb Fb

Fc Bc Fc

: :

Ftwooty Btwootie Ftwootie

: :

Truth-values of quantified sentences

Birds fly Some birds don’t fly

UD1 = birds UD2 = everything UD1

xFx x(Bx Fx) x~Fx

Fa Ba Fa ~Ftwootie

Fb Bb Fb

Fc Bc Fc UD2

: : x(Bx & ~Fx)

Ftwooty Btwootie Ftwootie Bt & ~Ft

: :

Truth-values of quantified sentences

xFx

Fa & Fb & Fc & ...

Truth-values of quantified sentences

xFx

Fa & Fb & Fc & ...

xBx

Fa Fb Fc ...

Truth-values of quantified sentences

(x)(Ax (y)Lyx)

Truth-values of quantified sentences

(x)(Ax (y)Lyx)

UD1: positive integers

Ax: x is odd

Lxy: x is less than y

Truth-values of quantified sentences

(x)(Ax (y)Lyx)

UD1: positive integers

Ax: x is odd

Lxy: x is less than y

UD2: positive integers

Ax: x is even

Lxy: x is less than y

Truth-values of quantified sentences

(x)(Ax (y)Lyx)

UD1: positive integers

Ax: x is odd

Lxy: x is less than y

UD2: positive integers

Ax: x is even

Lxy: x is less than y

(x)(y)(Lxy & ~Ax)

Va & (x) (Lxa ~ Exa)

UD1: positive integersVx: x is evenLxy: x is larger than yExy: x is equal to y

a:2

UD2: positive integersVx: x is oddLxy: x is less than y

Exy: x is equal to ya:1

UD3: positive integersVx: x is oddLxy: x is larger than or equal to yExy: x is equal to ya: 1

A sentence P of PL is quantificationally true if and only if P is true on every possible interpretation.

A sentence P of PL is quantificationally false if and only if P is false on every possible interpretation.

A sentence P of PL is quantificationally indeterminate if and only if P is neither quantificationally true nor quantificationally false.

Quantificational Truth, Falsehood, and Indeterminacy

A sentence P of PL is quantificationally true if and only if P is true on every possible interpretation.

Quantificational Truth, Falsehood, and Indeterminacy

Explain why the following is quantificationally true.~ (x) (Ax ≡ ~Ax)

A sentence P of PL is quantificationally false if and only if P is false on every possible interpretation.

Quantificational Truth, Falsehood, and Indeterminacy

Explain why the following is quantificationally false:(x)Ax & (y) ~Ay

A sentence P of PL is quantificationally indeterminate if and only if P is neither quantificationally true nor quantificationally false.

Quantificational Truth, Falsehood, and Indeterminacy

Show that the following is quantificationally indeterminate:

(Ac & Ad) & (y) ~Ay

Sentences P and Q of PL are quantificationally equivalent if and only if there is no interpretation on which P and Q have different truth values.

A set of sentences of PL is quantificationally consistent if and only if there is at least one interpretation on which all members are true. A set of sentences of PL is quantificationally inconsistent if and only if it is not quantificationally consistent, i.e. if and only if there is no interpretation on which all members have the same truth value.

Quantificational Equivalence and Consistency

A set of sentences of PL quantificationally entails a sentence P of PL if and only if there is no interpretation on which all the members of are true and P is false.

An argument is quantificationally valid if and only if there is no interpretation on which every premise is true yet the conclusion false.

Quantificational Entailment and Validity