SL Truth value assignments
description
Transcript of SL Truth value assignments
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