Conditional statements (i.e., statements of the form “If p, then q”, where p and q are statements) are staples of logic and are used widely in mathematics, science, philosophy, and everyday life. How do we judge whether such statements are true or not? Normally this is achieved by looking at whether each of the statements p and q are true or not, and whether the consequence follows from the antecedent. But it is not always that simple.
In a post examining this question, Jason Rosenhouse examines a simple statement “If it rains, then I will go to the movies”. In everyday language, the truth of the statement is determined by actual facts. If in fact it rained and I did go the movies, then the statement is true. If it rained and I did not go to the movies, then it is false. But how do we judge if the statement is true or not if it does not rain? In everyday life, we would then say that the truth of the statement cannot be tested.
But according to the rules of classical logic we cannot dodge the question that way but must assign a truth value based on the truth values of the statements p and q alone. The rules for doing so are as follows:
If p and q are true, then the statement is true.
If p is false, then the statement is true whether q is true or false.
If p is true and q is false, then the statement is false.
Rosenhouse gives some amusing examples of seemingly absurd statements that are deemed to be true by following these rules, such as “If I am not in France, then I am not in Spain” (assuming you are not actually in either country), “If Santa Claus exists, then the Moon is made of green cheese”, and “If Neil Armstrong had not walked on the moon, then no one else would have”.
Since all mathematical theorems are at root based on such statements, then according to the rules of mathematics, we need to assign truth values there as well. Rosenhouse says that when it comes to the kinds of statements that mathematicians care about, no problems arise. But these rules do cause problems for formal logicians and philosophers and they have developed a vast literature to address such questions.