"Unless"

The English term "unless" can be tricky to translate. For example,

The Reds will win unless their starting pitcher is injured.

If we use the constant "R" to stand for the atomic proposition, "the Reds will win" and "S" to stand for the atomic proposition, "the Reds' starting pitcher is injured," how would we translate this sentence using truth functional connectives? Think about what the sentence is saying (think carefully). Is the sentence asserting that the Reds will win? No; it is only saying that

The Reds will win as long as their starting pitcher isn't injured.

"As long as" denotes a conditional statement. In particular, what follows the "as long as" phrase is a sufficient condition, and as we have seen, a sufficient condition is always the antecedent of a conditional. But notice that the sufficient condition also contains a negation. Thus, the correct translation of this sentence is:

~S ⊃ R

One simple trick you can use to translate sentences which use the term "unless" is just substitute the phrase "if it's not the case that" for the "unless". But another trick is just to substitute an "or" for the "unless". Although it may sound strange in English, a disjunction will always capture the truth functional meaning of "unless". Thus, we could also correctly translate the sentence like this:

S v R

In the next section we will show how we can prove that these two sentences are equivalent using a truth table.