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.