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.
I tend to think that the stranger examples in categories like this stem from natural language not being precise in the same ways as logic (and math). Formal conditional statements have established properties regarding formal truth-value, etc.; English(e.g.) statements in the form “if p then q” can refer to all sorts of different concepts. These can be statements of intent or conditional plans (“If it rains, I’ll go to the movies.”), general overlap in categories (“If I’m at home, I’m more relaxed.”), etc…
Not that this stops cases where formal logical properties lead to what look like odd conclusions when translated back into natural language from being fun, of course.
Working mathematicians, not full-time logicians, might say that among other ways to resolve this matter, one could consider the fact that both “it rains” and “I shall [at some future time] go to the movies” are truth-valued functions (with domains the weather and some human’s behavior respectively, and there’s also time-dependence) rather than the kind of propositions about which the propositional calculus itself talks. In this context the p.c. only tells you how to combine truth values, so although your if-then statement looks like an implication, it is really a non-constant truth-valued function with a rather complicated domain. Note that many handy theorems really have a similar form. “If the real square matrix A is symmetric, then A is similar over the reals to a real diagonal matrix.” If it isn’t symmetric, or if you restrict the field of coefficients--well, maybe you can diagonalize it and maybe you can’t; but here the truth value of the theorem is the constant T (A being the only free variable in both the protasis and the apodosis).
This is a similar point I try to get my students to realize when I teach this unit in class.
“If today is Monday, then we don’t have class.”
Is that a general statement that says the class doesn’t meet on Monday? Or is that statement meant to imply for just this week, when Monday is a holiday? As M points out, everyday English can be ambiguous.
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I also like giving them the raven paradox as something to think about at home.
Thanks for the post, Mano. I’m starting to get some ideas how to revise some of my in class group work assignments.
Well, anything that has ravens in it has to be good. 😎 My own feeling about the paradox is that the Bayesian approach is the most convincing, but this may be due to bias. (Back when dinosaurs roamed the earth, Leonard Savage was on my reading committee.)
I tend to take the tack that if you have an apparent paradox, what you’ve actually got is a quirk of language. Just because you can make a grammatically correct sentence doesn’t mean it actually corresponds to anything. For that matter, just because you can arrange a series of propositions according to a formal logic system still doesn’t mean that it corresponds to anything. The solution to the raven paradox is that there isn’t one. If the statement “All ravens are black” is true (it’s not, but that’s beside the point), then the statement “Everything that is not black is not a raven.” is also true. However, if you are trying to determine whether the statement “All ravens are black” is true, non-raven objects cannot be relevant; the rules of finding evidence are not the same as the rules of formal logic, and it is a mistake to treat them as though they are.
When I was learning formal logic, the professor was very emphatic that we refer to the logical construct you’ve described as “arrow”. So (I’m not in France) -> (I’m not in Spain). It makes a lot more sense if you divorce it from “if-then”. Really, what we’re talking is just a mathematical operator, so conflating it with our natural concept of implication is not helpful.
Definitely quirks of language. Of course, in a sense language is made of quirks, especially in terms of unsuitability to exact definition; it’s all vague, and usage will be slightly different from person to person…because language was invented to be close enough rather than exact. And “close enough” is far enough from “exact” that all sorts of paradoxical things slip through the gap.
in natural language semantics if-then statements are usually treated as quantificational statements over possible worlds (or possible situations, events, etc.). it’s easy to understand when there’s an overt quantificational adverb, as in: if it rains, i always/sometimes/never go to the movies.