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 42There are apples and pears in the basket 43The only pear in the basket is rotten 44There are at least two apples in the basket 45There are two (and only two) apples in the basket 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten 44There are at least two apples in the basket 45There are two (and only two) apples in the basket 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx 44There are at least two apples in the basket 45There are two (and only two) apples in the basket 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket 45There are two (and only two) apples in the basket 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket  x  y(Ax & Nxb & Ay & Nyb 45There are two (and only two) apples in the basket 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket  x  y(Ax & Nxb & Ay & Nyb & x  y ) 45There are two (and only two) apples in the basket 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket  x  y(Ax & Nxb & Ay & Nyb & x  y ) 45There are two (and only two) apples in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) & x  y ) 46There are no more than two pears in the basket 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket  x  y(Ax & Nxb & Ay & Nyb & x  y ) 45There are two (and only two) apples in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) & x  y ) 46There are no more than two pears in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) ) 47there 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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket  x  y(Ax & Nxb & Ay & Nyb & x  y ) 45There are two (and only two) apples in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) ) 46There are no more than two pears in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) ) 47there are at least three apples in the basket  x  y  z (Ax & Ay & Az & Nxb & Nyb & Nzb & x  y & x  z & z  y )

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 42There are apples and pears in the basket  x(Px & Nxb) &  x(Ax & Nxb) 43The only pear in the basket is rotten  x(Px & Nxb & Rx &  y(Py & Nyb  y=x) ) 44There are at least two apples in the basket  x  y(Ax & Nxb & Ay & Nyb & x  y ) 45There are two (and only two) apples in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) ) 46There are no more than two pears in the basket  x  y(Ax & Nxb & Ay & Nyb &  z(Az & Nzb  (z=y  z=x) ) 47there are at least three apples in the basket  x  y  z (Ax & Ay & Az & Nxb & Nyb & Nzb & x  y & x  z & z  y ) 48there are at most three apples in the basket  x  y  z (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 humanFx = x is handsomea = Socrates Bab

Truth values of PL sentences are relative to an interpretation Examples: Fa Fx = x is humanFx = x is handsomea = 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 humanFx = x is handsomea = Socrates Bab Bxy = x is bigger than ya = Himalayas b = Alpesb = the moon

Truth values of PL sentences are relative to an interpretation Examples: Fa Fx = x is humanFx = x is handsomea = Socrates Bab Bxy = x is bigger than y a = Himalayasa = Himalayas a = Himalayas b = Alpesb = the moon b = Himalayas

Truth values of PL sentences are relative to an interpretation Examples: Fa Fx = x is humanFx = x is handsomea = Socrates Bab Bxy = x is bigger than y a = Himalayasa = Himalayas a = Himalayas b = Alpesb = 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 oddBxyz = 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 oddBxyz = x is between y and z Ox = {1,3,5,7,9,...} Bxyz = {(2,1,3), (3,2,4),...} Bxy = x>yBxyz = 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 integers Ax: x is odd Bxy: x is bigger than y Cx: x is prime m: 2 n: 1 Truth-values of compound sentences

(An & Bmn)  ~ Cn UD: All positive integers Ax: x is odd Bxy: x is bigger than y Cx: x is prime m: 2 n: 1 Truth-values of compound sentences UD: All positive integers Ax: x is even Bxy: x is bigger than y Cx: x is prime m: 2 n: 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 = birdsUD = everything  xFx  x(Bx  Fx) Fa Fb Fc : Ftwooty :

Truth-values of quantified sentences Birds fly UD = birdsUD = everything  xFx  x(Bx  Fx) FaBa  Fa FbBb  Fb FcBc  Fc: FtwootyBtwootie  Ftwootie:

Truth-values of quantified sentences Birds flySome birds don’t fly UD 1 = birdsUD 2 = everythingUD 1  xFx  x(Bx  Fx)  x~Fx FaBa  Fa FbBb  Fb FcBc  Fc: FtwootyBtwootie  Ftwootie:

Truth-values of quantified sentences Birds flySome birds don’t fly UD 1 = birdsUD 2 = everythingUD 1  xFx  x(Bx  Fx)  x~Fx FaBa  Fa~Ftwootie FbBb  Fb FcBc  Fc: FtwootyBtwootie  Ftwootie:

Truth-values of quantified sentences Birds flySome birds don’t fly UD 1 = birdsUD 2 = everythingUD 1  xFx  x(Bx  Fx)  x~Fx FaBa  Fa~Ftwootie FbBb  Fb FcBc  FcUD 2 ::  x(Bx & ~Fx) FtwootyBtwootie  FtwootieBt & ~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) UD 1 : positive integers Ax: x is odd Lxy: x is less than y

Truth-values of quantified sentences (  x)(Ax  (  y)Lyx) UD 1 : positive integers Ax: x is odd Lxy: x is less than y UD 2 : positive integers Ax: x is even Lxy: x is less than y

Truth-values of quantified sentences (  x)(Ax  (  y)Lyx) UD 1 : positive integers Ax: x is odd Lxy: x is less than y UD 2 : positive integers Ax: x is even Lxy: x is less than y (  x)(  y)(Lxy & ~Ax)

Va & (  x) (Lxa  ~ Exa) UD 1 : positive integers Vx: x is even Lxy: x is larger than y Exy: x is equal to y a:2 UD 2 : positive integers Vx: x is odd Lxy: x is less than y Exy: x is equal to y a:1 UD 3 : positive integers Vx: x is odd Lxy: x is larger than or equal to y Exy: x is equal to y a: 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