Name: Answer Key
Exam II
Part 1 (20 pts)
(a) Explain what it
means to say that a sentence in the predicate calculus is semantically true.
Also state what it means for a sequent to be semantically valid in predicate
logic.
A sentence
in the predicate calculus is semantically true iff
the sentence is true in very interpretation of the predicates and very domain.
A sequent G ├
j in the predicate calculus is semantically true iff whenever a model thinks that every sentence in G is
true, then it thinks that j is true.
(b) Why can’t we use
interpretations to show that a sentence is semantically true?
One cannot
use this method to show that a sentence is semantically true, since there are infinitely many
interpretations one can assign to the predicates in the sentence and also,
there are infinitely many ways to define the domain.
(c) Recall the
conceptual connections between quantifiers and sentential connectives. Given
the following interpretation, eliminate the quantifier in the sentence, $x(Fx & Gx), by
rewriting it using the appropriate sentential
connectives:
UD : { Julie, Tom,
Mike }
j : Julie
t : Tom
m : Mike
(Fj & Gj) Ú (Ft & Gt)
Ú (Fm & Gm)
(d) For formula (i) and (ii) below, state the scope of each quantifier.
(i) $xFx & "xGx
The scope
of $ is Fx
and the scope of " is Gx
(ii) "x(Px→$yRxy)
The scope
of " is Px→$yRxy and the
scope of $ is Rxy
Part II ( 10 pts ): Given the description of the predicates and
relations translate the two sentences below into the appropriate
symbolism.
(a) Bxy: x bores y
Sxy: x is
a student of y
Px: x is a professor
Lxy: x
listens to y
(i) Some professors do not listen to any of their
students.
$x(Px & "y(Syx → ~Lxy)
(ii)
Every professor bores some of his or her students.
"x(Px → $y(Syx & Bxy))
Part III ( 20 pts): For the sentences below give interpretations that show that the sequents are not semantically valid.
(a) "xFx → "xGx ├ "x(Fx→Gx)
Take the following:
UD = The
natural numbers
Fx = x is prime
Gx = x < 0
Explanation: Clearly "xFx
is false in this case, since there are numbers that are not prime,
for example 6. So, then "xFx→"xGx
is true. Now, note that "x(Fx→Gx) is false
since there is a number n such that ~(Fn→Gn),
for example 5. So, the above
interpretation shows that the sequent is invalid.
(b) ~"xFx ├ "x~Fx
Take the following:
UD = The
natural numbers
Fx = x is prime
Explanation: Clearly ~"xFx
is true in this case, since for example ~F8 holds. But, "x~Fx is not true since there are natural numbers such that Fn,
for example F5. So, then the sequent is
shown to be invalid.
Part
IV (30 pts): Prove the following sequents
using the four quantifier rules and primitive or derived rules of the
propositional calculus.
Commonly used propositional
rules:
DeMorgan’s Law: ~(P v Q)
┤├ ~P & ~Q
Commutativity: P
v Q ┤├ Q v P
Distribution: P
& (Q v R) ┤├
(P & Q) v (P & R)
Transposition: P
→ Q ┤├ ~Q → ~P
Law of excluded middle ├
P v ~P
Negated antecedent ~P ├ P→R
Negated arrow ~(P→Q) ┤├ P & ~Q
(a) $x(Fx v Gx) ├ $xFx v $xGx
1 (1) $x(Fx Ú Gx) A
2 (2)
3 (3) Fa A
3 (4) $xFx 3, $-intro
3 (5) $xFx
Ú $xGx 4, Ú-intro
6 (6) Ga A
6 (7) $xGx 6, $-intro
6 (8) $xFx
Ú $xGx 7, Ú-intro
2 (9) $xFx
Ú $xGx 2, 3, 5, 6, 8 Ú-elim
1 (10) $xFx
Ú $xGx 1, 2, 9 $-elim
(b) "x($yFyx→"zFxz) ├ "y"x(Fyx→Fxy)
1 (1) "x($yFyx → "zFxz) A
1 (2) $yFyb → "zFbz 1,
UE
3 (3) Fab A
3 (4) $yFyb 3,
$-intro
1, 3 (5) "zFbz 2,
4 MPP
1, 3 (6) Fba 5,
UE
1 (7) Fab→Fba 3,
6 CP
1 (8) "x(Fax→Fxa) 7,
UE
1 (9) "y"x(Fyx→Fxy) 8,
UE
(c) "xFx ├ ~$xGx ↔ ~($x(Fx & Gx) & "y(Gy→Fy))
Hint:
In one of the directions use transposition.
See homework 11
problem #3
Part
V (20 Pts): Prove the following sequents using the
any primitive or derived rules for propositional logic and any primitive rule
or application of QE (Quantifier Exchange) of predicte
logic.
QE rules: ~"xFx ┤├ $x~Fx
"xFx┤├ ~$x~Fx
"x~Fx┤├ ~$xFx
~"x~Fx┤├ $xFx
(a) "xFx → "xGx ├ $x(Fx → Gx)
1 (1) "xFx → "xGx A
2 (2) ~$x(Fx→Gx) A
2 (3) "x~(Fx→Gx) 2, QE
2 (4) ~(Fa→Ga) 3,
UE
2 (5) Fa & ~Ga 4,
negated arrow (above)
2 (6) Fa 5,
&-elim
2 (7) "xFx 6,
UI
1,2 (8) "xGx 1,7
MPP
2 (9) ~Ga 5,
&-elim
2 (10) $x~Gx 2,
$-intro
2 (11) ~"xGx 10,
QE
1,2 (12) "xGx & ~"xGx 8,
11 &-intro
1 (13) $x(Fx→Gx) 2, 12
RAA
(b) ~"x(P & Fx)├ (P→
~"xFx)
1 (1) ~"x(P
& Fx) A
2 (2) ~(P→ ~"xFx) A
2 (3) P & "xFx 2,
Negated arrow
2 (4) P 3, &-elim
2 (5) "xFx 3,
&-elim
2 (6) Fa 5,
UE
2 (7) P & Fa 4,
6 &-intro
2 (8) "x(P
& Fx) 7,
UI
1, 2 (9) ~"x(P
& Fx) & "x(P & Fx) 1, 8 &-intro
1 (10) P→ ~"xFx 2, 9 RAA