Truth Tables
A truth table allows us to know with certainty if a deductive argument is valid or not.
Remember, an argument is valid if and only if it is
impossible
for the premises to be true
and the conclusion false.
A truth table gives us every possible combination of truth
values for the variables and as a result we can determine if it is ever possible for the
premises to be true and the conclusion false.
Another way of saying it is, if the premises
are true does the conclusion necessarily have to be true?
This is, for the purposes of this
class, what a truth table allows us to discover.
To understand truth tables you must first understand the truthfunction of some
statements.
Implication:
An implication is a statement which says “p implies q” or “If p, then q” (p
is called the antecedent and q is called the consequent).
So we need to ask ourselves,
when is a statement of this form true and when is it false?
If I say “If I have a cat, then I
have a mammal”, when is the statement false?
It is false when in fact I do have a cat, but
do not have a mammal.
To put it another way, when my having a cat is true and when
my having a mammal is false.
We can say then, an implication is false when the
antecedent “p” is true and the consequent “q” is false.
Anytime an implication is of the
following truth variable structure: True implies false, then we can say the implication is
itself false.
We can say it is false that the antecedent
implies
the consequent.
Any other time we say the implication is still true.
If I have a cat and I have a mammal,
then it is true “If I have a cat, then I have a mammal” (true implies true).
If I do not have
a cat and I do not have a mammal, then the implication is still true that If I did have a cat,
then I would have a mammal (false implies false).
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 Fall '08
 Hicks
 Logic, Logical connective

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