Section 3: Other symbolization

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We have now introduced all of the connectives of SL. We can use them together to translate many kinds of sentences. Consider these examples of sentences that use the English-language connective ‘unless’:

27. Unless you wear a jacket, you will catch cold.
28. You will catch cold unless you wear a jacket.

Let \(J\) mean ‘You will wear a jacket’ and let \(D\) mean ‘You will catch a cold.’

We can paraphrase sentence 27 as ‘Unless \(J\), \(D\).’ This means that if you do not wear a jacket, then you will catch cold; with this in mind, we might translate it as ¬\(J\) → \(D\).

It also means that if you do not catch a cold, then you must have worn a jacket; with this in mind, we might translate it as ¬\(D\) → \(J\). Which of these is the correct translation of sentence 27? Both translations are correct, because the two translations are logically equivalent in SL.

Sentence 28, in English, is logically equivalent to sentence 27. It can be translated as either ¬\(J\) → \(D\) or ¬\(D\) → \(J\).

When symbolizing sentences like sentence 27 and sentence 28, it is easy to get turned around. Since the conditional is not symmetric, it would be wrong to translate either sentence as \(J\) → ¬\(D\). Fortunately, there are other logically equivalent expressions. Both sentences mean that you will wear a jacket or— if you do not wear a jacket— then you will catch a cold. So we can translate them as \(J\)∨\(D\). (You might worry that the ‘or’ here should be an exclusive or. However, the sentences do not exclude the possibility that you might both wear a jacket and catch a cold; jackets do not protect you from all the possible ways that you might catch a cold.)

If a sentence can be paraphrased as ‘Unless \(\mathcal{A}\), \(\mathcal{B}\),’ then it can be symbolized as \(\mathcal{A}\)∨\(\mathcal{B}\).