The Logic of Quantifiers

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The Logic of Quantifiers
Language, Proof and Logic
Chapter 10
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First-order vs. second-order
10.0
First-order ?x Loves(max,x) ?
Loves(x,max) Second-order ?P P(max) ?
P(julia) Mixed ?P ?xP(x) ? ?xP(x)
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Tautologies and quantification
10.1.a
Are the following arguments valid? Are they
tautologically so? ?x Cube(x) ? Small(x)
?x Cube(x) ?x Cube(x)
?x Small(x) ?x
Small(x)
?x Cube(x) ? Small(x) ?x Cube(x) ?
Small(x) ?x Cube(x) ?x
Cube(x)
?x Small(x) ?x Small(x)
?x Cube(x) ? Small(x)
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Tautologies and quantification
10.1.b
Are the following sentences logically true? Are
they also tautologies? ?x Cube(x) ? ?Cube(x)
?x Cube(x) ? ?x?Cube(x)
?xCube(x) ? ?x?Cube(x)
?x Cube(x) ? ??xCube(x)
?y(P(y)?R(y)) ? ?x(P(x)?Q(x)) ?
??x(P(x)?Q(x)) ? ??y(P(y)?R(y))
?y(P(y)?R(y)) ? ?x(P(x)?Q(x)) ?
??x(P(x)?Q(x)) ? ??y(P(y)?R(y))
A?B? ?B??A
A quantified sentence of FOL is said to be a
tautology iff its truth-functional form is a
tautology.
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Tautologies and quantification
10.1.c
FO sentence
Its truth-functional form
?xCube(x) ? ??xCube(x)
?yTet(y) ??zSmall(z) ? ?zSmall(z)
?xCube(x) ? ?yTet(y)
?xCube(x) ? Cube(a)
?x(Cube(x) ? ?Cube(x))
?xCube(x) ? Small(x) ? ?xDodec(x)
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First-order validity and consequence
10.2.a
Propositional logic First-order
logic General notion Tautology
FO validity
Logical truth Tautological
consequence FO consequence
Logical consequence Tautological
equivalence FO equivalence
Logical equivalence
FO validity, consequence and equivalence mean
those that hold solely in virtue of the
truth-functional connectives, quantifiers and
identity.

• Two reasons for treating identity as a logical
  symbol
• is very common and universal
• allows us to express many quantified noun
  phrases, such as
• exactly one, at least 5, at most
  7, etc.

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First-order validity and consequence
10.2.b
Are the following sentence FO valid? ?xSameSize(x
,x) ?xCube(x) ? Cube(b) Cube(b) ? bc ?
Cube(c) Small(b) ? SameSize(b,c) ? Small(c)
To test FO validity, consequence
or equivalence Use nonsense
Replace by meaningless symbols
?xBolshe(x,x) ?xTove(x) ? Tove(b) Tove(b) ?
bc ? Tove(c) Cheten(b) ? Menshe(b,c) ?
Cheten(c)
?xB(x,x) ?xT(x) ? T(b) T(b) ? bc ?
T(c) C(b) ? M(b,c) ? C(c)
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First-order validity and consequence
10.2.c
Are the conclusions of the following arguments FO
consequences of the premises?
?xTet(x) ? Large(x) ?Large(b) ?Tet(b)

• ? ?xLarger(x,a)
• ?xLarger(b,x)
• Larger(c,d)
• Larger(a,b)

• ? ?xLoves(x,moriarty)
• ?xLoves(scrooge,x)
• Loves(romeo,juliet)
• Loves(moriarty,scrooge)

Generally, recognizing FO (non)validity is not
easy. No computer can ever be able to
successfully handle every instance of this
problem!
You try it, p. 273
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First-order equivalence and DeMorgans laws
10.3.a
Are the following pairs FO equivalent? Are they
also tautologically so?
??xCube(x) ??yDodec(y) vs. ??xCube(x) ?
??yDodec(y)
?xCube(x) ? Small(x) vs. ?x?Small(x) ?
?Cube(x)
We say that two wffs are FO equivalent iff, when
you replace their free variables by new names,
the resulting sentences are FO equivalent. Substi
tution of a (sub)wff by a FO equivalent one
preserves FO equivalence. Hence, e.g., we have
?xCube(x) ? Small(x) ? ?x?Cube(x) ?
Small(x)
? ?x?Cube(x) ? ??Small(x)
? ?x ?Cube(x) ?
?Small(x)
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First-order equivalence and DeMorgans laws
10.3.b
Remember that ? is a big conjunction and ? a
big disjunction. Hence no wonder that
DeMorgans laws extend to quantifiers Example
Bring the following formula down to an
equivalent formula where negation is only
applied to atoms
??xP(x) ? ?x?P(x) ? ?xP(x) ? ?x?P(x)
??xP(x) ? Q(x)
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Other quantifier equivalences
10.4
?xP(x)?Q(x) ? ?xP(x)??xQ(x) ?xP(x)?Q(x)
? ?xP(x)??xQ(x)
?xP(x)?Q(x) ? ?xP(x)??xQ(x) ?xP(x)?Q(x) ?
?xP(x)??xQ(x)
When P has no free occurrences of x (null
quantification)
?xP ? P ?xP ? P
?xP?Q(x) ? P??xQ(x) ?xP?Q(x) ?
P??xQ(x)
When y does not occur in P(x) (replacing bound
variables)
?xP(x) ? ?yP(y) ?xP(x) ? ?yP(y)