Translating Biconditionals
Biconditionals are easier to translate than conditionals, because order doesn't matter, and so all you have to do is recognize the English equivalent of the triple-bar operator. The most common form in which biconditionals appear is with the phrase "if and only if." For example:
| Translation: | Commentary: |
| Bertha will fly to Brazil if and only if airfares remain reasonably priced. | English sentence to be translated |
| A: Bertha will fly to Brazil. B: Airfares remain reasonably priced. |
Sentence letter abbreviations for simple sentences |
| A if and only if B | Replacement of simple sentences by sentence letters |
| (A ≡ B) | Full translation |
Another phrase which is translated by the triple-bar is "just in case":
| Translation: | Commentary: |
| Matt will mulch just in case Wynn weeds. | English sentence to be translated |
| A: Matt mulches B: Wynn weeds. |
Sentence letter abbreviations for simple sentence |
| A if and only if B | Replacement of simple sentence by sentence letter |
| (A ≡ B) | Full translation |
The biconditional is also used when a condition is said to be both necessary and sufficient for something. Remember that we said that necessary conditions are rewritten as the consequent of a conditional, and sufficient conditions as the antecedent. If something is necessary and sufficient, then it is expressed as both the antecedent and the consequent of a conditional. But how can something be both the antecdent and consequent of a conditional? Easy: A biconditional is equivalent to a conjunction of two conditionals, one of which is the "reverse" of the other. Here's an example, though sadly, the sentence in question is probably true:
| Translation: | Commentary: |
| Making a good impression is necessary and sufficient for getting into the movies. | English sentence to be translated |
| A: One makes a good impression. B: One gets into the movies. |
Sentence letter abbreviations for simple sentence |
| A if and only if B | Replacement of simple sentence by sentence letter |
| (A ≡ B) | Full translation |
