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


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






b = the basket


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






b = the basket


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






b = the basket



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





b = the basket



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


• There are no more than two pears 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




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


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




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) )



b = the basket




45 There are two (and only two) apples 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 )


b = the basket




45 There are two (and only two) apples 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


PL

Interpretation

SL


PL

Interpretation

Giving an interpretation means defining:

UD

SL


PL

Interpretation


UD

Predicates

SL


PL

Interpretation


UD

Predicates

Constants

SL


PL

Interpretation


UD

Predicates

Constants

Of course, we do not define variables

Truth values of PL sentences are

relative to an interpretation



Examples:

• Fa

Fx = x is human

a = Socrates

• Bab



Examples:

• Fa

Fx = x is human Fx = x is handsome

a = Socrates a = Socrates

• Bab



Examples:

• Fa



• Bab

Bxy = x is bigger than y

a = Himalayas

b = Alpes



Examples:

• Fa



• Bab


a = Himalayas a = Himalayas

b = Alpes b = the moon



Examples:

• Fa



• Bab


a = Himalayas a = Himalayas a = Himalayas

b = Alpes b = the moon b = Himalayas



Examples:

• Fa



• Bab


a = Himalayas a = Himalayas a = Himalayas

b = Alpes b = the moon b = Himalayas

No constant can refer to more than one individual!



Examples:

• Fa

• Bab

• ~xFx

UD = food

Fx = x is in the fridge



Examples:

• Fa

• Bab

• ~xFx

UD = food


UD = everything


Extensional definition of predicates

Predicates are sets


Predicates are sets

Their members are everything they are true of


Predicates are sets


Predicates are defined relative to a UD


Predicates are sets



Example:

UD = natural numbers

Ox = x is odd

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


Predicates are sets



Example:


Ox = x is odd

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

Bxy = x>y

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


Predicates are sets



Example:


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), ...}


Predicates are sets



Example:


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


Birds fly

UD = birds

xFx

Fa

Fb

Fc

:

Ftwooty

:


Birds fly

UD = birds UD = everything

xFx x(Bx Fx)

Fa

Fb

Fc

:

Ftwooty

:


Birds fly

UD = birds UD = everything

xFx x(Bx Fx)

Fa Ba Fa

Fb Bb Fb

Fc Bc Fc

: :

Ftwooty Btwootie Ftwootie

: :


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

: :


: :




xFx x(Bx Fx) x~Fx

Fa Ba Fa ~Ftwootie

Fb Bb Fb

Fc Bc Fc

: :


: :




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

: :


xFx

Fa & Fb & Fc & ...


xFx

Fa & Fb & Fc & ...

xBx

Fa Fb Fc ...


(x)(Ax (y)Lyx)


(x)(Ax (y)Lyx)

UD1: positive integers

Ax: x is odd

Lxy: x is less than y


(x)(Ax (y)Lyx)


Ax: x is odd



Ax: x is even



(x)(Ax (y)Lyx)


Ax: x is odd



Ax: x is even


(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.


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.


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.


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

Documents

For 42 – 47: UD = everything;