Language, Proof and
Logic
The Logic of Quantifiers
Chapter 10
10.0
First-order vs. second-order
First-order: x [Loves(max,x)  Loves(x,max)]
Second-order: P [P(max)  P(julia)]
Mixed: P [xP(x)  xP(x)]
10.1.a
Tautologies and quantification
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)]
10.1.b
Tautologies and quantification
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]
The truth-functional form
of the previous sentence
A quantified sentence of FOL is said to be a tautology iff its
truth-functional form is a tautology.
10.1.c
Tautologies and quantification
FO sentence
xCube(x)  xCube(x)
[yTet(y) zSmall(z)]  zSmall(z)
xCube(x)  yTet(y)
xCube(x)  Cube(a)
x(Cube(x)  Cube(x))
x[Cube(x)  Small(x)]  xDodec(x)
Its truth-functional form
10.2.a
First-order validity and consequence
Propositional logic
First-order logic
General notion
Tautology
Tautological consequence
Tautological equivalence
FO validity
FO consequence
FO equivalence
Logical truth
Logical consequence
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:
1.
2.
= is very common and universal
= allows us to express many quantified noun phrases, such as
“exactly one”, “at least 5”, “at most 7”, etc.
10.2.b
First-order validity and consequence
Are the following sentence FO valid?
xSameSize(x,x)
xCube(x)  Cube(b)
[Cube(b)  b=c]  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)
xB(x,x)
xTove(x)  Tove(b)
xT(x)  T(b)
[Tove(b)  b=c]  Tove(c)
[T(b)  b=c]  T(c)
[Cheten(b)  Menshe(b,c)]  Cheten(c)
[C(b)  M(b,c)]  C(c)
10.2.c
First-order validity and consequence
Are the conclusions of the following arguments FO consequences of the
premises?
x[Tet(x)  Large(x)]
Large(b)
Tet(b)
 xLarger(x,a)
xLarger(b,x)
Larger(c,d)
 xLoves(x,moriarty)
xLoves(scrooge,x)
Loves(romeo,juliet)
Larger(a,b)
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
10.3.a
First-order equivalence and DeMorgan’s laws
Are the following pairs FO equivalent? Are they also tautologically so?
[xCube(x) yDodec(y)] vs. xCube(x)  yDodec(y)
x[Cube(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.
Substitution of a (sub)wff by a FO equivalent one preserves FO
equivalence. Hence, e.g., we have:
x[Cube(x)  Small(x)]
 x[Cube(x)  Small(x)]
 x[Cube(x)  Small(x)]
 x [Cube(x)  Small(x)]
10.3.b
First-order equivalence and DeMorgan’s laws
Remember that  is a “big conjunction” and  a “big disjunction”.
Hence no wonder that DeMorgan’s laws extend to quantifiers:
xP(x)  xP(x)
 xP(x)  xP(x)
Example: Bring the following formula down to an equivalent formula
where negation is only applied to atoms:
x[P(x)  Q(x)]
10.4
Other quantifier equivalences
x[P(x)Q(x)]  xP(x)xQ(x)
x[P(x)Q(x)]  xP(x)xQ(x)
x[P(x)Q(x)]  xP(x)xQ(x)
x[P(x)Q(x)]  xP(x)xQ(x)
When P has no free occurrences of x (“null quantification”):
xP  P
xP  P
x[PQ(x)]  PxQ(x)
x[PQ(x)]  PxQ(x)
When y does not occur in P(x) (“replacing bound variables”):
xP(x)  yP(y)
xP(x)  yP(y)