Translating Conditionals
Conditionals are a bit more of a challenge to translate than the other sentences we've dealt with so far, because order matters: we have to make sure we identify the antecedent of the conditional and put it before the horseshoe, as well as the consequent of the conditional, which is placed after the horseshoe. While in the most standard form, the antecedent of a conditional in English occurs before the consequent, this need not be the case. The key thing is to locate the word or words which indicate the antecendent, such as "if", and the word or phrase which indicates the consequent, such as "then." In the following example, the consequent occurs before the antecedent:
| Translation: | Commentary: |
| I'll have to pay a fine if I run the red light. | English sentence to be translated |
| A: I run the red light. B: I'll have to pay a fine. |
Sentence letter abbreviations for simple sentences |
| B if A | Replacement of simple sentences by sentence letters |
| (A ⊃ B) | Full translation |
The word "when" sometimes indicates the antecedent of a conditional, and sometimes it means "at the same time as." For example, in the sentence "Ignat gets spasms when he's running.", "when" clearly means "while." In contrast, in the sentence "Ignat will be graduate when he completes 132 units." we can reasonably translate "when" as "if", even though the "when" might also be taken to mean "after." Propositional logic is tenseless. We don't interpret words like "when" and "and" in temporal terms, and so when words are used to express time, we can't use our connectives to express temporal relationships.
Sentences which have complex antecedents or consequents can be trickier to translate. Consider the sentence: "If Bill retires he will write a book or he will go on the lecture circuit." Is this sentence a conditional or a disjunction? The way to answer this question is to ask another: What's the consequent of the conditional in this sentence? Is it just "He will write a book" or is it "He will write a book or go on the lecture circuit." Sometimes punctuation provides a hint. For example, if there were a comma after "book", that would indicate that the disjunction is the consequent. But even without the comma, the natural interpretation is that this whole sentence is a conditional, with the distinjunction as the consequent.
| Translation: | Commentary: |
| If Bill retires he will write a book or he will go on the lecture circuit. | English sentence to be translated |
| A: Bill retires. B: He will write a book. C: He will go on the lecture circuit. |
Sentence letter abbreviations for simple sentence |
| If A then either B or C | Replacement of simple sentence by sentence letter |
| (A ⊃ (B v C)) | Full translation |
An important use of the conditional occurs in sentences which make reference to necessary or sufficient conditions. Suppose someone says: "Oxygen is necessary for fire." How would you translate this as a conditional, if you were told that one simple component is "There is fire." and the other is "There is oxygen."? Here's the correct translation:
| Translation: | Commentary: |
| Oxygen is necessary for fire. | English sentence to be translated |
| A: There is oxygen. B: There is fire. |
Sentence letter abbreviations for simple sentence |
| If B then A. | Replacement of simple sentence by sentence letter |
| (B ⊃ A) | Full translation |
Why did we translate it this way? Consider the other possibility, (A ⊃ B). That says "If there is oxygen, there is fire. But that means that oxygen is sufficient for fire, that with just oxygen, there's fire. But our original sentence was that oxygen is necessary for fire, but it didn't claim that oxygen is sufficient.
In general, when a condition is presented as necessary, make the sentence expressing that condition the consequent of the conditional. When the condition is presented as sufficient, make it the antecedent of the conditional.
A common mistake in dealing with conditionals is the reversal of the antecedent and the consequent. We know from our truth-tables that (A ⊃ B) and (B ⊃ A) do not mean the same thing. They are not true under the same conditions. That is, they have different truth-values. So it's crucial to make sure that you know which part is the antecedent and which is the consequent. Consider the difference between the following conditionals:
| If you'll join me, I'll go to the store. | If I'll go to the store, then you'll join me. |
| A: You'll join me B: I'll go to the store (A ⊃ B) |
A: You'll join me B: I'll go to the store (B ⊃ A) |
The meaning of these two sentences is radically different. The second is coercive, the first is not! Order matters in conditionals!
The exercise, Translating Necessary and Sufficient Conditions, will give you practice with the conditional.
