The Meaning of
the Material Conditional
Philosophy 202 Fall 2011 |
The
material conditional as we have defined it is the most counterintuitive of
connectives. This is due in part to
the fact that we are not clearly aware of the logic of conditional claims
even though we frequently make them.
This connective is counterintuitive because a conditional sentence is
true when the antecedent of that sentence is false. What gives? A. This is the truth condition of the
material conditional: P Q P
→ Q
-- -- -------- T T T T F F F T T F F T Rows 3 and 4 are the source of the
problem; rows 1 and 2 are right on the mark—they express what we clearly
understand about conditional claims.
What should we do with rows 3 and 4?
Keep in mind that we must be able to give our connective a truth-table
definition, and the binary character of this connective
forces us to use a four row truth-table; furthermore, we must put
either "T" or "F" in rows 3 and 4; therefore, we have to
determine what truth values to put in those rows. There are three other alternative
definitions, and to those we now turn. B. P Q P
→ Q
-- -- -------- T T T T F F F T F F F F The obvious alternative would be to put
"F" in rows 3 and 4.
However, if this is the definition we give, then (P → Q)
⇔ (P
⋀ Q).
But think about the reasoning this endorses: if you know (P → Q), then you know both P and you know Q;
however, this conflicts with the hypothetical character of (P → Q). Consider the following example: "If I flipped the switch, the lights
are on." We would like to say
that this sentence can be true even when I have not flipped the switch. If so, then the truth of this conditional
claim does not imply the truth of the antecedent and the consequent; in fact,
it doesn't imply the truth of either.
This is due to the logical character of conditional claims: they assert a relationship between the
truth of one sentence (viz., the antecedent) and the truth of another (viz.,
the consequent) that can obtain even when one or both of the claims made by
the sentences are in fact false.
C. P Q P
→ Q
-- -- -------- T T T T F F F T T F F F Another definition is found at the left. This is equivalent to Q. Thus, if you knew (P → Q), you would know Q. But, it seems intuitively clear that we can know (P → Q) without therefore knowing Q. When we know that a conditional claim is true, we know something that seems to depend on P; however, on this definition there is no dependence on P. P is not involved in the reasoning this definition sanctions.
Consider the light switch example once
again. If you know that claim is true,
then do you know that the lights are on?
Or consider another example:
"If three whales are swimming under the Golden Gate Bridge, then
at least one whale is swimming under the Golden Gate Bridge." If I know this to be true—which it
certainly is—then do I know that there is at least one whale swimming under
the Golden Gate Bridge? It seems that
we would want to answer in the negative to both of these questions, which is
a sign that this definition is inadequate. D. P Q P
→ Q
-- -- -------- T T T T F F F T F F F T This is the last alternative. Unfortunately, this also sanctions
reasoning that doesn't seem in line with the few intuitions we have about (P
→ Q). Here, knowing (P
→ Q) and anything at all about P tells us something
about Q. But (P → Q) only explicitly tells us about Q when we have P;
when we have ¬P, why couldn't we also have Q? Consider first the light example. If I know that the conditional claim is
true and that I have not flipped the switch, then do I know that the lights
are not on? The answer has to be
"no", since someone else could have flipped the switch and turned
them on. Consider second the whale example.
If I know that conditional claim and I also know that there are not
three whales swimming under the Golden Gate Bridge, do I also know that there
is not at least one whale swimming under the bridge? No, since the antecedent could be false and
the consequent true if two whales are swimming there. Again, this definition of the connective
sanctions inferences that we do not want to make—inferences that conflict with
intuitions we have about the logic of conditional claims. Thus,
the only remaining definition is the one we have adopted. It is counterintuitive, but that is the only drawback. Note that there are no sanctioned patterns
of reasoning when Q is true or when P is false. |