Philosophy 120
Introduction to Logic

Translating "All" and "Only"

It's easy to make mistakes when translating "all" and "only" into FOL. Even if you know that a sentence like Only freshmen are eligible will be translated by an FOL sentence of the form ∀x (F(x) → G(x)), you may still be confused about whether "freshmen" goes into the antecedent (replacing F) or the consequent (replacing G).

Do you remember the simple rule? The noun phrase governed by "only" goes into the consequent.

Probably for the same reason, it's also easy to make mistakes in reasoning with sentences containing "only." People sometimes treat the "only" as if it means "all," and hence make erroneous inferences. Here's a good example.

Catbert is clearly offering the following argument:

Only the top performers leave for higher pay.
You are still here (i.e., you did not leave for higher pay).
Therefore, your performance is average at best (i.e., you are not a top performer).

Of course, this argument is invalid. Just because all of those who leave for higher pay are top performers (which is what the first premise means) does not entail that all of those who are top performers leave for higher pay. So the fact that Dilbert did not leave does not mean that he is not a top performer.

If Catbert had said "All the top performers leave for higher pay" his conclusion about Dilbert would have followed. But he didn't say that. He confused "all" and "only," and hence (from an FOL perspective) committed the fallacy of denying the antecedent. In saying to Dilbert "You're still here" he was denying the antecedent, rather than the consequent, of the conditional

∀x (LeaveForHigherPay(x) → TopPerformer(x)).

So what Dilbert should have said to Catbert is not "That's not fair!" but "That's not valid!"

Copyright © 2014, S. Marc Cohen