Removing Some of the Mystery of the Material Conditional in Symbolic Logic

By James Johnson

There are certain features of the material conditional that cause problems for beginning logic students, specifically with regard to truth-value assignments.  This entry is designed to help explain the structure of the material conditional, explain why it will receive certain truth-values based on the truth-values of its constituent parts, and why some of these resulting truth-values may seem counterintuitive.

Consider the following material conditional statement:  If I talk to Charles, then I will tell him he’s in charge.  The basic structure of the statement is the antecedent, ‘If I talk to Charles’, followed by the consequent, ‘I will tell him he’s in charge’.  The same holds for all cases of material conditionals, the ‘if’ part of the statement is always the antecedent, and the ‘then’ part is always the consequent.

The following scenario using the above example should explain how truth-values are assigned to material conditionals.  Imagine you are about to leave for work, and on your way out the door you say to me, “Make sure to tell Charles that he is in charge.”  I reply, “If I talk to Charles, then I will tell him he’s in charge.” 

Following this exchange, different possible outcomes could occur in which I will have either spoken truthfully, or falsely. One possibility is that I do in fact talk to Charles, and when I do I immediately tell him that he’s in charge.  In this possibility I have spoken truthfully.  The antecedent and consequent are both true in this case, and so the conditional is true as well. Another possibility is that I talk to Charles and decide that I won’t tell him he’s charge. Instead I tell him that I’m in charge.  In this case I have told a lie, so while the antecedent is true, the consequence is false, and the conditional on the whole is false. There is also the possibility that I don’t get a chance to talk to Charles at all.  In this case there is no way for me to tell him he’s in charge. Hence I have spoken truthfully, since the condition of the antecedent was if I talk to Charles, so I have spoken truthfully no matter what the consequent is, since I didn’t talk to him.  Thus the conditional on the whole is still true.  And the same holds for any case in which the antecedent is false, no matter what the consequent is. 

While in logic conditionals with false antecedents are always true, there are certain expressions in English that make this truth of logic seem counterintuitive. Consider this conditional statement (borrowed from Tom Ryckman): If the ice cube is dropped into boiling water, then it won’t melt.  And let’s say that the ice cube is not dropped into water, but is instead placed back into the freezer.  The outcome makes the antecedent false, so the conditional, logically speaking, is true. Notice, however, that based on the meaning of words in the conditional itself, one would be inclined to say that no matter what happens to the ice cube in actuality, the original conditional statement here looks to be false.  The moral of this story is that while the material conditional serves its purpose for logic, it cannot capture all sentences of English that are in this ‘if-then’ form.

For an in-depth treatment of the material conditional, see sect. 7.1 in LPL.