Modern Logic

PHIL109 Modern Logic

Spring 2017
Take
Home Quiz #2
Due:
3/2
3
/2017
Name:
1.
What
information must
each line of a proof
provide
in order
to demonstrate unequivocally
that the conclusion is
logically entailed by the premises?
(mark
all that apply below)
[5
pts]
A statement, which is either an assumption made for the sake of proof, or a validly derived inference.
A
line number, which
is
provides
a
way to
identify each line in the proof.
An
assumption set, which list
s
all the assumptions o
n which the given statement
depends.
An
annotation, which
justifies the statement as either
an assumption or
a validly derived
inference
.
An
XKCD
comic strip
satirizing
the principle of explosion
.
2.
A
statement that is tautologous is logically true, whereas a statement that is contingent is…
[
2
pts]
i)
logically false
.
ii)
sometimes true and sometimes false
.
3.
Two
statements
that are
logically equivalent
share
the same truth value on each row
of a truth table
, where
as two
statements that are contradictory
[
2
pts]
iii)
have opposite truth values
on each row
.
iv)
have
at least one row on which
the truth values are all true
.
4.
Identify the conclusion of the
following argument
:
[2
pts]
Either the butler
did it
or the cook did it, unless it was the doctor. If it was done with a knife
,
the cook did
it.
But it was
no
t done with
a knife.
So
,
if the butler didn’t do it, the doctor did.
i)
Either the butler
did it
or the cook did it, unless it was the doctor.
ii)
If it was done with a knife
,
the cook did it.
iii)
But it was
no
t done with a knife.
iv)
So
,
if the butler didn’t do it, the doctor did.
5.
Is the wff an atomic proposition,
conjunction, disjunction, conditional, biconditional,
or
negat
ion?
[2
0 pts]
i)
Q
atomic
ii)
P
&~S
conjunction
iii)
~R
T
conditional
iv)
(
P
&
Q
)
Ú
R
disjunction
v)
~P
~Q
biconditional
vi)
(
P
Ú
Q
)
& S
conjunction
vii)
~(R
(T
Ú
~S
)
)
negation
viii)
S
(R
Q
)
conditional
ix)
~(S
Q)
Ú
(~S
Ú
Q)
disjunction
x)
~(~(T
Ú
P
)
Ú
~(P & Q
))
negation
6.
Complete the truth table below
,
then
st
ate whether the sequent
valid
by marking the correct box
.
[8
pts]
P
Q
R
P
Q
Q
(R
& P
)
P
R
i)
The sequent is valid.
ii)
The sequent is invalid.
7.
Match the primitive rule of inf
erence to its abbreviation
.
[18
pts]
i)
Given a statement that is a disjunction,
A
Ú
B
(at line
m
), and another statement (at line
n
) that is a denial of
one of its disjuncts, you may conclude the other disjunct.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
ii)
Given both a statement and its denial (at lines
m
and
n
), you may conclude the denial of the assumption from
which the contr
adiction is derived (at line
k
)
.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
iii)
A
Given a statement that is a biconditional,
A
B
(at line
m
), you may conclude either
A
B or B
A
.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
iv)
Given a statement that is a conditional,
A
B
(at line
m
), and another statement that is its antecedent,
A (at line
n
), you may conclude the consequent of the conditional, B.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
v)
Given two statements A and B (at lines
m
and
n
), you may conclude their conjunction.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
vi)
Given a statement, A (at line
m
), you may conclude any disjunction having
A as a disjunct.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
vii)
Given two conditional statements having the forms A
B
and B
A
(at lines
m
and
n
), you may conclude a
biconditional statement with A and B as
its constituents.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
viii)
Given a statement, B
(at line n), you may conclude a conditional having B as the consequent if the antecedent
appears in the proof as an assumption, A (at line
m
), and B is derivable from A.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA
ix)
Given a statement that is a conjunction, A & B (at line
m
), you may conclude either conjunct.
&I
&E
Ú
I
Ú
E
E
I
I
E
RAA