Section 13:
Simple Truth Tables,
Barker, pp. 7783.
***Read the section, then the following additional comments.
In this section we explore what happens to our basic relationships when the simple sentences fall
into various combinations of truth or falsehood.
We will express these possibilities in “truth
tables” which will capture all possible combinations.
To keep things compact, we will express
truth with a capital “T,” and falsehood with a capital “F.”
1.
Negation
What happens when you negate a false sentence?
It becomes true.
What if you negate a true
sentence?
It becomes false. Thus here is a very simple truth table.
P
P
T
F
F
T
The first column exhausts the possibilities for truth and falsehood of P, the second column
indicates what happens when they are negated.
As we can see, their truth value is reversed.
2.
Conjunction
Under what conditions can a conjunction be true?
Remember, a conjunction tells us that both
statements are true.
Thus, a conjunction as a whole can only be true if, in fact, both
statements are true.
So we get this truth table:
P
Q
P & Q
T
T
T
F
T
F
T
F
F
F
F
F
There is only one line on this truth table where the conjunction is true, namely where both
propositions are true.
In all other combinations, the conjunction is false.
3.
Disjunction
When is a disjunction true?
Whenever at least one of the propositions is true.
This fact can
be summarized with the following truth table:
1
LESSON 7
Truthfunctional Logic, Part 2
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P
Q
P v Q
T
T
T
F
T
T
T
F
T
F
F
F
Thus we see that a disjunction is true unless it turns out that both propositions are false.
The last line may cause you to be puzzled a bit.
If a statement says “or,” doesn’t that mean
that one must be true and the other one false?
Sometimes, but not always.
If I say,
The Braves or the Reds will win tonight.
B v R
We need to understand from the context how we are to interpret this statement.
If the Braves
play against the Reds, then one will be true, the other false.
But if they are playing against
other teams and not each other both statements may in fact be true.
We call this distinction
the difference between an exclusive disjunction where both cannot be true and an inclusive
disjunction where both may be true.
Logic is cautious by nature—it never wants to rule out
something unless there is good reason to.
That’s why the normal assumption for a
disjunction is inclusive.
A disjunction is false only if both propositions are false.
4.
Conditional
In what combination of truth and falsehood is a conditional as a whole true?
The results turn out
to be a little surprising, but make sense when you think about the meaning of a conditional. This
time, let us follow the lines of the basic truth table with an example.
If it’s a rose then it has thorns.
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 Spring '11
 BrentKelly
 Logic, Logical connective, Barker

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