Part I: Lesson

To conclude our study of compound propositions, we’ll be looking at the hypothetical proposition in this lesson. In addition to this, we’ll also take a look at what happens when we combine some of the compound propositions we’ve been discussing. 

• Hypothetical Proposition: a proposition consisting of two other propositions, linked together in an if-then structure. The form of this proposition is “if p, then q.”

In the variable notation above, we call p the antecedent and q the consequent. The last of the compound propositions, the hypothetical, links two propositions by asserting that the truth of the antecedent implies the truth of the consequent. It is hypothetical because it states that if, hypothetically, the antecedent is true, then it is the case that the consequent is true. Only the antecedent is hypothetical. These propositions will take the form “If p, then q” in variable notation. For a simple example, consider: “If frogs are amphibians, then they are animals”. If it is true that frogs are amphibians (antecedent), then it is true that they are animals (consequent). 

Truth Conditions for Hypothetical Propositions: The truth of these propositions is a bit more complicated than the prior two compound forms, as they will only be false in situations where the antecedent is true and the consequent turns out false. For example, the hypothetical proposition “If I am a human, then I am an animal” is false only when “I am a human” is true but “I am an animal” is false. This is because the hypothetical seeks to assert that the antecedent cannot be true without the consequent being true. Because of this, the only situation in which the hypothetical is false will be exactly when the antecedent is true, but the consequent still turns out to be false. 

As the table shows, the only situation when the hypothetical proposition, “if p then q,” is false is when p is true and q is false. Strangely enough, this means that whenever p is false, “if p then q” is true no matter whether q is true or false! For example, take p to be “Obama is president” and q to be “unicorns are real,” then the statement “if p then q” is “if Obama is president, then unicorns are real.” Since p is false—Obama is no longer president—then the statement as a whole is true under these logical conditions, even though, as far as we know, unicorns aren’t real. Likewise, “if p then q” is always true when q is true.