1.3 Translation of English to Sentential Wffs

translation scheme	Definition. A TRANSLATION SCHEME for the language of sentential logic is a pairing of sentence letters with sentences of a natural language. The sentences in a translation scheme should be logically simple. That is, they should not contain any of the words corresponding to the sentential connectives.

logical form	Definition. The LOGICAL FORM of a sentence of a natural language relative to a translation scheme is given by its translation into a wff of sentential logic according to that translation scheme.
	Example. Under the translation scheme P: John does well at logic Q: Bill is happy The sentence If John does well at logic, then Bill is happy has the logical form (P->Q).
	Comment. English provides many different ways of stating negations, conditionals, conjunctions, disjunctions, and biconditionals. Thus, many different sentences of English may have the same logical form.

stylistic variants	Definition. If two sentences of a natural language have the same logical form relative to a single translation scheme, they are said to be STYLISTIC VARIANTS of each other.
	Comment. There are far too many stylistic variants of negations, conjunctions, disjunctions, conditionals, and biconditionals to list here. The following is a partial list of stylistic variants in each category.

negations	Let P translate the sentence `John is conscious.' Here are a few of the ways of expressing ~P: John is not conscious. John is unconscious. It is not the case that John is conscious. It is false that John is conscious.

conditionals	Stylistic variants whose logical form is (PHI -> PSI), where PHI is the antecedent and PSI is the consequent include the following: If PHI, PSI. PHI only if PSI. PHI is a sufficient condition for PSI. PHI is sufficient for PSI. PSI provided that PHI. Provided that PHI, PSI. PSI on the condition that PHI. PSI is a necessary condition for PHI. PSI is necessary for PHI. Whenever PHI, PSI. PSI if PHI. Given that PHI, PSI. In case PHI, PSI. PHI only on the condition that PSI.

conjunctions	Variants with logical form (PHI & PSI) include the following: PHI and PSI. Both PHI and PSI. PHI, but PSI. PHI, although PSI. PHI as well as PSI. Though PHI, PSI. PHI, also PSI.

disjunctions	Variants with logical form (PHI v PSI) include these: PHI or PSI. Either PHI or PSI. PHI unless PSI.
	Comment. `PHI unless PSI' is also commonly translated as (~PSI -> PHI). The proof techniques introduced in section 1.4 can be used to show that this is equivalent to (PHI v PSI).

biconditionals	Variants having the logical form (PHI <-> PSI) include the following: PHI if and only if PSI. PHI is equivalent to PSI. PHI is necessary and sufficient for PSI. PHI just in case PSI.

neither...nor...	English sentences of the form `Neither PHI nor PSI' have the logical form ~(PHI v PSI), or, equivalently, (~PHI & ~PSI).

tenses	Comment. In English, the sentences `Mary is dancing' and `Mary will dance' have different meanings because of the tenses of their respective verbs. In some cases, when one is analyzing arguments it is important to preserve the distinction between tenses. In other cases, the distinction can be ignored. In general, a judgment call is required to decide whether or not tense can be safely ignored.
	Example. Consider the following two arguments: A. If Mary is dancing, John will dance. Mary is dancing. Therefore, John is dancing. B. If Mary dances, John will dance. If John dances, Bill will dance. Therefore, if Mary dances, Bill will dance. In A, if the difference between `John will dance' and `John is dancing' is ignored, then the argument will look valid in translation. But this seems unreasonable on inspection of the English. In B, ignoring the difference between `John will dance' and `John dances' also makes the argument valid in translation. In this case, however, this seems reasonable. In the translation exercises that follow, assume that tense distinctions may be ignored.

Exercise 1.3	Translate the following sentences into the language of sentential logic. Translation scheme for 1-20 P: John dances. Q: Mary dances. R: Bill dances. S: John is happy. T: Mary is happy. U: Bill is happy.
1*	John is dancing but Mary is not dancing.
2*	If John does not dance, then Mary will not be happy.
3*	John's dancing is sufficient to make Mary happy.
4*	John's dancing is necessary to make Mary happy.
5*	John will not dance unless Mary is happy.
6*	If John's dancing is necessary for Mary to be happy, Bill will be unhappy.
7*	If Mary dances although John is not happy, Bill will dance.
8*	If neither John nor Bill is dancing, Mary is not happy.
9*	Mary is not happy unless either John or Bill is dancing.
10*	Mary will be happy if both John and Bill dance.
11*	Although neither John nor Bill is dancing, Mary is happy.
12*	If Bill dances, then if Mary dances John will too.
13*	Mary will be happy only if Bill is happy.
14*	Neither John nor Bill will dance if Mary is not happy.
15*	If Mary dances only if Bill dances and John dances only if Mary dances, then John dances only if Bill dances.
16*	Mary will dance if John or Bill but not both dance.
17*	If John dances and so does Mary, but Bill does not, then Mary will not be happy but John and Bill will.
18*	Mary will be happy if and only if John is happy.
19*	Provided that Bill is unhappy, John will not dance unless Mary is dancing.
20*	If John dances on the condition that if he dances Mary dances, then he dances.
	Translation scheme for 21-25 P: A purpose of punishment is deterrence. Q: Capital punishment is an effective deterrent. R: Capital punishment should be continued. S: Capital punishment is used in the United States. T: A purpose of punishment is retribution.
21*	If a purpose of punishment is deterrence and capital punishment is an effective deterrent, then capital punishment should be continued.
22*	Capital punishment is not an effective deterrent although it is used in the United States.
23*	Capital punishment should not be continued if it is not an effective deterrent, unless deterrence is not a purpose of punishment.
24*	If retribution is a purpose of punishment but deterrence is not, then capital punishment should not be continued.
25*	Capital punishment should be continued even though capital punishment is not an effective deterrent, provided that a purpose of punishment is retribution in addition to deterrence.

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