Formalizing "unless"

The formalization of "unless" is a problematic case. To see why one might formalize "P unless Q" as "(P∨Q)" one might look at the following example. Suppose you are a student and your teacher says to you:

It is true that you might fail even if you take the exam. That is quite possible. But what I am sure is this. You won’t pass unless you take the exam.

It seems that the teacher is not promising that you will pass if you take the exam. Rather, he is saying that if you do not take the exam, then you will not pass. So this should be formalized as follows :

( You do not take the exam → you will not pass )

In SL, it can be shown easily that (X→Y) is logically equivalent to (~X∨Y). You can draw their truth-tables to check for yourself. So the conditional above is actually equivalent to :

( You take the exam ∨ you will not pass )

which is the same as

( You will not pass ∨ you take the exam )

As we have said, even if you take the exam, you might not pass. This is the case with the above sentence as it is true when both disjuncts are true. It also entails that if you pass, then you have taken the exam, which seems to be the correct implication of the original remarks.

However, formalization of “unless” using disjunction seems less appropriate in other contexts. Suppose Ann and Beth are discussing whether they should go to a party, and Ann says “I won’t go unless you go.” Beth does not want to go but on hearing Ann's remark, decided to go to the party. However, it turns out that Ann decided not to go after all, even though she knows that Beth is going. Now we might think that Beth is justified in feeling that she has been let down, that Ann has failed to deliver her promise. But if Ann's statement were to be formalized as a disjunction, then there is no implication that if Beth were to go, then Ann would too. So in such a case, one might argue (though others might disagree) that Ann's remark should instead be formalized as :

( I will go to the party ↔ you will go to the party )

Such examples show that formalization might not be a straightforward matter, and that it might be argued that the connectives of ordinary language differ in meanings from the connectives of formal logic in rather subtle ways. When it comes to formalization then, one has to look at the context rather carefully to determine the correct reading. However, even if formal logic does not allow for perfect translation of natural language, these examples show that it is an important tool in helping us understand the meanings of natural language through comparison and contrast.


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