"Only If" Conditionals

During the first week of the course, you learned that the use (and position) of the word "only" can have a significant impact on logical implications of a claim, and that is equally true for a conditional premise.

In the section on Conditionals, we saw that the the claim, "If P, then Q," resulted in two valid conclusions: when P is true, Q is true; and when Q is not true, P is not true. Thus, an argument based on the conditional premise, "The team will win if Yankl scores," can reach the valid conclusions: "when Yankl scores, the team wins"; and "when the team does not win, Yankl does not score."

But the results of a very similar conditional are quite different. Consider, "The team will win only if Yankl scores." Clearly, this suggests that "when the team wins, Yankl scores," and "when Yankl does not score, the team does not win." Notice that these conclusions are the converses of those made when the conditional had no "only" in it. As a result, we can say that "The team will win only if Yankl scores," is logically equivalent to "Yankl scores if the team wins," or "If the team wins, then Yankl scores." Thus, "P only if Q" is equivalent to "If P then Q."

Notice that the "only" need not appear immediately before the "if" in the premise to have this effect, providing that "only" is being used as an adverb and not as an adjective. Thus, "The team will win only if Yankl scores" is logically equivalent to "The team will only win if Yankl scores," but not to "The team will win if only Yankl scores," in the sense that "Yankl and no one else scores." In the previous paragraph, the original conditional was shown to be equivalent to "If the team wins, then Yankl scores"; but this does not indicate that "Yankl and no one else scores," only that "at least Yankl scores." In other words, the "only" in "only Yankl" is an adjective, and therefore does not produce the "only if" effect. "The team will win if only Yankl scores" means that "If Yankl and no one else scores, then the team will win."

Be careful, however, because the usage of "only" can sometimes be ambiguous. While "The team will win if only Yankl scores" seems fairly clearly to mean "Yankl and no one else," what about the conditional, "The team might win if only Yankl would score"? Now the "only" seems to be part of the subjunctive mood expressed by "would score," and suggests neither "only if Yankl scores," nor "if Yankl and no one else scores."

Such problems in meaning aside, remember that P only if Q is equivalent to If P then Q, and so, in a way, "only if" can be replaced by "then."