Survey of Math - MAT 140
Page: 1
Arguments and Truth Tables
The text goes a different path for this topic than the path I chose. I don't care which one you use, as long as you can
work through the logic of it all.
In a logical argument, it is assumed that we are given True statements, and we have to determine if these statements
conclusions form an Implication (Valid) or not (Invalid).
Example 1
If you pass this test then you will pass this class
You passed the test.
You will pass the test
symbolically
p!q
p
q
The top compound statement and the middle simple statement are know as True statements. What we need to do is
combine them with an AND to see if the conclusion that they are supposed to have is True also. The argument becomes
a Conditional set up this way:
.T op/ ^ .Middle/ ! Bottom
In this case:
. p ! q/ ^ p ! q
To solve, all we need to do is set up the truth table, the matching colored columns were put together to "get" the answer
between them...:
Where the Blue Columns gave us the #1 answer, then the Red Columns gave us the #2 answer, and nally the Green
Columns gave us the nal #3 Answer. Notice that #3 is ALL Trues, thus this is an implication and the Argument is
VALID
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 2
Example 2
If a rabbit lives in the Arctic then it is white.
My rabbit is white.
)
My rabbit lives in the Arctic.
p!q
q
p
Becomes: . p ! q/ ^ q ! p for the truth table:
Since there is an F in the last Column (green) the argument is INVALID. (not always true)
Example 3
The bear will run away or we will be hurt.
We are hurt.
The bear ran away.
)
p_q
q
p
The argument in symbolic form would be:
. p _ q/ ^ q ! p
And completing a Truth Table we would have:
And since the nal solution has at least one False, we conclude that the Argument is Invalid.
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 3
Example 4
If it snows in Maine then it will rain in Florida
If it rains in Florida then there is a hurricane.
If it snows in Maine then there is a hurricane.
)
You could do a truth table for this (very long) or we could just notice a pattern
First a diagonal must exist, where the simple statements are the same and the argument is in the same order as if the
remaining statements fell from above, then the argument is Valid!
**Note- this "trick" only works for Three or More Conditional Statements. It does not work on OR, or AND
statements!!!!!
Example 5
If George is smart then he studied for his test
If George passes his test then he studied.
If George is smart then he passed his test.
)
p!q
r !q
p!r
There is no diagonal that has the same simple statement, thus the argument is INVALID
Copyright 2007 by Tom Killoran