### Propositional Logic Part 2

```Use a truth table to determine the validity or invalidity of this argument.
“Martin is not buying a new car, since he said he
would buy a new car or take a Hawaiian vacation
and I just heard him talking about his trip to
Maui.”
C or H
H
_
First, translate into standard form
Not C
Now , translate into symbols
CvH
H
_
~C
2
CvH
H
_
~C
Now, build a truth table.
“C” and “H” each need a column and will
serve as references.
Next, we need a column for
each premise and the
conclusion.
C
T
T
F
F
H
T
F
T
F
C v H ~C
3
CvH
H
_
~C
C
T
T
F
F
H
T
F
T
F
Fill in the truth values for the first premise based
on the rule of disjunction:
A disjunction is false if and only if both
disjuncts are false.
CvH
T
T
T
F
~C
F
F
T
T
Our truth table now tells
us whether or not the
argument is valid.
What do you think?
4
CvH
H
_
~C
Is it possible for the premises to be true and
the conclusion false?
C
T
T
F
F
H
T
F
T
F
CvH
T
T
T
F
H
T
F
T
F
~C
F
F
T
T
“Martin is not buying a new car, since he said he
would buy a new car or take a Hawaiian vacation and I
just heard him talking about his trip to Maui.”
5
O → (H & S)
~H v ~S
_
~O
O
T
T
T
T
F
F
F
F
H
T
T
F
F
T
T
F
F
S
T
F
T
F
T
F
T
F
~H
F
F
T
T
F
F
T
T
There are no cases where the premises
are true and the conclusion false; this is a
valid argument. Can you provide an
interpretation?
~S
F
T
F
T
F
T
F
T
H & S O → (H&S) ~H v ~S
T
T
F
F
F
T
F
F
T
F
F
T
T
T
F
F
T
T
F
T
T
F
T
T
~O
F
F
F
F
T
T
T
T
6
If side A has an even number, then
side B has an odd number, but side A does not have
an even number. Therefore, side B does not have
an odd number.
If side A has an even number, then
side B has an odd number, but side A does not have
an even number. Therefore, side A has an even
number.
If the theory is correct, then we will have
observed squigglyitis in the speciment. However, we
know the theory is incorrect. Therefore, we could not
have observed squigglyitis in the specimen.
If the theory is correct, then we will have
observed dilation in the specimen. Therefore, since we
did not observe dilation in the specimen, we know the
theory is not correct.
If we observe dilation in the specimen,
then we know the theory is correct. We observed
dilation, so the theory is correct.
What conclusion does the
following demonstrate to
be true and how do you
know it is true?
P  R.
R  S.
P.
/
If
the antecedent of a
conditional appears on
another line of the
argument, the consequent
of the conditional may be
How do you know the
following are true:
P&Q
P
P&Q
Q
A
conjunction is true if and
only if both conjuncts are
true; therefore, if a
conjunction is true, either
and both conjuncts are
also true.
3.
R & S.
S  P.
If the above is true, what
else is true?
S.
P.
What should the next line of
the argument line be and
how do you know?
MP.
(P & Q)  R.
S.
S  ~R.
~R.
~(P & Q)
Now what should the next
line be?
Modus Tollens (MT): If a
consequent of a conditional
that appears as one
premise appears appears as
the negation of that
consequent in another
premise, then the negation
of the antecedent should
appear as the next line in
the argument.
4.
P→Q
~P → S
~Q
/∴ S
How do you know S is true
if the premises are true?
~P 1, 3,
S 2, 4,
7.
~S
(P & Q) → R
R→S
/∴ ~(P & Q)
How do you know ~(P & Q)
is true if the premises are
true?
~R 1, 3,
~(P & Q) 4, 2,
8.
P → ~(Q & T)
S → (Q & T)
P
/∴ ~S
~(Q & T) 1, 3,
~S 2, 4,
P→Q
R→S
PvR
/∴ Q v S
How do you know Q v S is
true?
The disjunction of
the antecedents of any
two conditionals allows
the derivation of the
disjunction of their
consequents.
2.
P→S
PvQ
Q→R
/∴ S v R
How do you know S v R is
true?
1, 2, 3,
5.
(P v Q) → R
Q
/∴ R
How do you get to R?
): The truth
of one disjuct makes the
entire disjunction true.
1.
R→P
Q→R
/∴ Q → P
How we get to Q → P?
You
can create a new
conditional from two
other conditionals,
provided that the
antecedent of one existing
conditional appears as the
consequent of the other
existing conditional.
9.
(P v T) → S
R→P
RvQ
Q→T
/∴ S
Talk us to S.
2, 3, 4,
1, 5,
6.
1, 3, MP
~P
~(R & S) v Q
~P → ~Q
/∴ ~(R & S)
~(R & S) 2, 4, Disjunctive
Argument (DA): The
negative of one disjunct in
a disjunction implies the
truth of the other
disjunct.
```