Translating Disjunctions
Disjunctions are typically indicated by "or." Remember that we're using the inclusive meaning of "or," not the exclusive sense. (See the rules of truth section to review this point.) Here's a straightforward translation:
| Translation: | Commentary: |
| Walter will go for a walk or eat an apple. | English sentence to be translated |
| A: Walter will go for a walk B: Walter will eat an apple. |
Sentence letter abbreviations for simple sentences |
| A or B | Replacement of simple sentences by sentence letters |
| (A v B) | Full translation |
Again, when we have more than one type of operator, we always need to be aware of the
scope of the operators. Here we review the translation from the last section where we have a
disjunction where one of the disjuncts is a conjunction. Note that the word
"either" at the beginning of the sentence indicates that the main operator of
the sentence is the wedge:
| Translation: | Commentary: |
| Either Fred will run five miles, or he will go home and take a nap. | English sentence to be translated |
| A: Fred will run five miles. B: Fred will go home. C: Fred will take a nap. |
Sentence letter abbreviations for simple sentence |
| A or (B and C) | Replacement of simple sentence by sentence letter |
| (A v (B & C)) | Full translation |
Sentences which contain "neither ... nor" can be translated as disjunctions with negation. "Neither" works like a negated "either." So a "neither...nor" sentence is the negation of a disjunction:
| Translation: | Commentary: |
| She had neither friends nor enemies. | English sentence to be translated |
| A: She had friends B: She had enemies. |
Sentence letter abbreviations for simple sentence |
| It's not the case that either A or B | Replacement of simple sentence by sentence letter |
| ~(A v B) | Full translation |
Another word which means "or" is "unless". The simplest way to translate "unless" is simply as "or":
| Translation: | Commentary: |
| I'll go to the store unless there is milk. | English sentence to be translated |
| A: I will go to the store B: There is milk |
Sentence letter abbreviations for simple sentence |
| A unless B | Replacement of simple sentence by sentence letter |
| (A v B) | Full translation |
Sometimes people balk at this translation, and the reason is, that it's natural to want to translate "unless" propositions as conditionals. But what is the appropriate translation as a conditional? "Unless" indicates the antecedent of a conditional, but that antecedent also must be negated. So the correct translation of "I'll go to the store unless there is milk" is "If there is no milk, then I'll go to the store", i.e. (~B ⊃ A).
Now: Verify via truth-tables that (A v B) and (~B ⊃A) are logically equivalent.
