Truth Table, and Logic Arguments Review
Truth tables assign a good/bad/neutral rating as to
how a conditional promise is kept or broken depending upon whether or not the
specified condition was satisfied. If the condition of a promise did not
materialize, then a good (true) rating is given by default because the promise
was not broken.
THE Promise:
Paul promised that if he got is $30K bonus, then he
would buy Quincy a new car.
The tables show the validity of different
statements dependent upon the truth value of the given parts.
Note: There are four possible T/F combinations for any two statements. In terms of the analogy each says the following:
Paul Quincy
p q
T T Paul gets the $ and buys Quincy the car. Ipso facto, Paul has fulfilled his promise and is a good guy.
T F Paul gets the $ but does not buy the car for Quincy as he promised. Paul has broken his promise and is not considered to be a good guy.
F T Paul does not get the $ but buys Quincy the
car anyway. Obviously Paul is either rich or perhaps stupid and in debt, but
he did not break his promise. Paul is not a bad guy so he is considered to
be a good guy by default (default truth values are shown below in lower case).
F F Paul does not get the $ and cannot buy the car for Quincy. Paul did not break his promise, so is not a bad guy and is considered to be a good guy by default
| P | Q |
Implication p
|
Contrapositive ~q
|
Inverse
(q
|
Law of Excluded Middle ~( p^ ~p) |
Converse
~p
|
Indirect Proof ((~p
|
Indirect Proof (((p
|
||||||||||||||||||||||||
|
T |
T |
t |
T |
t |
f |
T |
f |
t |
T |
t |
T |
T | F | F |
f |
t |
f |
f |
t |
t |
F |
f |
t |
t |
t |
t |
t |
F |
f |
t |
t |
t |
|
T |
F |
t |
F |
f |
t |
F |
f |
f |
t |
t |
T | T | F | F |
f |
t |
t |
f |
t |
f |
T |
t |
T |
t |
t |
f |
f |
F |
t |
t |
t |
t |
|
F |
T |
f |
t |
t |
f |
T |
t |
t |
F |
f |
T | F | F | T |
t |
F |
f |
t |
T |
t |
F |
f |
t |
f |
f |
t |
t |
F |
f |
t |
t |
t |
|
F |
F |
f |
t |
f |
t |
T |
t |
f |
t |
f |
T | F | F | T |
t |
T |
t |
t |
F |
f |
F |
t |
t |
f |
f |
t |
f |
T |
t |
T |
T |
t |
Two Comments:
|
That the Implication and Contrapositive are
tautologies. Their truthvalues are identical. ((p -> q )<-> (~q->~p)) t t t T f t f t f f T t f f f t t T f t t f t f T t t t |
A direct proof such as one that follows directly
from a definition or other previously proved statements trivially satisfies
the following tautology. ((p -> q ) <-> (p->q)) t t t T t t t t f f T t f f f t t T f t t f t f T f t f |