Phil 120 week 1 class 1 July 5th 2004

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SL Truth value assignments
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SL Truth value assignments
PL Interpretation
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SL Truth value assignments
PL Interpretation Giving an interpretation means
defining UD
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SL Truth value assignments
PL Interpretation Giving an interpretation means
defining UD Predicates
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SL Truth value assignments
PL Interpretation Giving an interpretation means
defining UD Predicates Constants
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SL Truth value assignments
PL Interpretation Giving an interpretation means
defining UD Predicates Constants Of course, we
do not define variables
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Truth values of PL sentences are relative to an
interpretation
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• Truth values of PL sentences are
• relative to an interpretation
• Examples
• Fa
• Fx x is human
• a Socrates
• Bab

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• 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

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(No Transcript)
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• 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

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• 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

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• 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

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• 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!

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• Truth values of PL sentences are
• relative to an interpretation
• Examples
• Fa
• Bab

• ?xFx
• UD food
• Fx x is in the fridge

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• 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

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Extensional definition of predicates Predicates
are sets
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Extensional definition of predicates Predicates
are sets Their members are everything they are
true of
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Extensional definition of predicates Predicates
are sets Their members are everything they are
true of Predicates are defined relative to a UD
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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, ...
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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 xgty Bxy (2,1), (3,1),
(3,2), ...
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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 xgty Bxy (2,1), (3,1), (3,2), ...
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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 xgty Bxyz y is between x and z Bxy
(2,1), (3,1), (3,2), ... Bxyz (1,2,3),
(2,3,4), ...
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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
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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
UD All positive integers Ax x is even Bxy
x is bigger than y Cx x is prime m 2 n 1
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Truth-values of quantified sentences
Birds fly UD birds ?xFx
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Truth-values of quantified sentences
Birds fly UD birds ?xFx Fa Fb Fc Ftwooty
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Truth-values of quantified sentences
Birds fly UD birds UD everything ?xFx ?x(Bx
? Fx) Fa Fb Fc Ftwooty
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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
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Truth-values of quantified sentences
Birds fly Some birds dont fly UD1
birds UD2 everything UD1 ?xFx ?x(Bx ?
Fx) ?xFx Fa Ba ? Fa Fb Bb ? Fb Fc Bc ?
Fc Ftwooty Btwootie ? Ftwootie
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Truth-values of quantified sentences
Birds fly Some birds dont fly UD1
birds UD2 everything UD1 ?xFx ?x(Bx ?
Fx) ?xFx Fa Ba ? Fa Ftwootie Fb Bb ?
Fb Fc Bc ? Fc Ftwooty Btwootie ?
Ftwootie
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Truth-values of quantified sentences
Birds fly Some birds dont fly UD1
birds UD2 everything UD1 ?xFx ?x(Bx ?
Fx) ?xFx Fa Ba ? Fa Ftwootie Fb Bb ?
Fb Fc Bc ? Fc UD2 ?x(Bx
Fx) Ftwooty Btwootie ? Ftwootie Bt Ft
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Truth-values of quantified sentences
?xFx Fa Fb Fc ...
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Truth-values of quantified sentences
?xFx Fa Fb Fc ... ?xBx Fa ? Fb ? Fc ? ...
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Truth-values of quantified sentences
(?x)(Ax ? (?y)Lyx)
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Truth-values of quantified sentences
(?x)(Ax ? (?y)Lyx) UD1 positive integers Ax x
is odd Lxy x is less than y
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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
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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)
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Va (?x) (Lxa ? Exa) UD1 positive
integers Vx x is even Lxy x is larger than
y Exy x is equal to y a2 UD2 positive
integers Vx x is odd Lxy x is less than y Exy
x is equal to y a1 UD3 positive integers Vx
x is odd Lxy x is larger than or equal to y Exy
x is equal to y a 1
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Quantificational Truth, Falsehood, and
Indeterminacy
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.
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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)
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Quantificational Truth, Falsehood, and
Indeterminacy
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
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Quantificational Truth, Falsehood, and
Indeterminacy
Show that the following is quantificationally
indeterminate (Ac Ad) (?y) Ay
A sentence P of PL is quantificationally
indeterminate if and only if P is neither
quantificationally true nor quantificationally
false.
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Quantificational Equivalence and Consistency
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.
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Quantificational Entailment and Validity
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.