Page 305 - Encyclopedia of Philosophy of Language

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A standard form of explicit definition consists in speci- fying necessary and sufficient conditions for the truth of a sentence. The notions of necessity and sufficiency are sometimes defined simply by reference to the 'truth-functional' account of the conditional. A con- ditional sentence of the form 'if p, then q' is false, according to classical logic, only when the antecedent (the ^/-clause) is true and the consequent (the then- clause) is false. In all other cases, the conditional is true. It appears to follow that if a conditional is true and has a true antecedent, then its consequent will be true. The truth of the antecedent, then, might be said to be sufficient for the truth of the consequent, and the truth of the consequent said to be, in its turn, necessary for the truth of the antecedent. The latter usage reflects the fact that equivalent to 'if p, then q' is the form 'if not q, then not p.' A necessary condition, in other words, is one whose truth is a sine qua non of the truth of the other condition (for a discussion, and criticism, of definitions in terms of necessary and sufficient conditions seeFamily Resemblance).
1. Problems with a Truth-functional Account Standardly, the vocabulary of necessary and sufficient conditions is made to apply to sentences which are related either logically or nonlogically. Thus, to take a case of physically necessary and sufficient conditions, consider the conditional:
This way of explaining the distinction runs up against counterintuitive results if the conditional in the above example is taken to be truth-functional. A sufficient condition is meant to guarantee the truth of a certain further sentence. Now, provided it is true that the sun is shining and true that Jane is eating a cheese cracker, the truth-functional conditional 'If Jane is eating a cheese cracker then the sun is shining' is also true. Yet it would be implausible to claim that the truth of the sentence 'Jane is eating a cheese cracker' guarantees the truth of the sentence 'the sun is shining.'
A similar problem arises if the notions of necessary and sufficient conditions are defined by reference to inferential relations understood in the fashion of classical logic. Although in a deductive argument the truth of the premises is meant to guarantee the truth of the conclusion, there is no requirement that the premises be relevant to the conclusion. Likewise, in the same sort of argument, if the conclusion is false, then at least one of the premises is false: but there may well be no connection of the sort that makes it seem reasonable to describe the truth of the conclusion as necessary for the truth of the premises.
2. AnAlternativetotheTruth-functionalAccount
Rather than defining necessary and sufficient con- ditions by reference to the truth-functional conditional, then, it may be more straightforward to do this in terms of the natural-language use of 'if where the truth of the antecedent is held to provide at least some relevant reason for thinking that the consequent is true, if not an explanation of why the consequent is true. For such conditionals, the truth of the antecedent is sufficient for the truth of the consequent, while the truth of the consequent, in turn, is necessary for the truth of the antecedent. That the car starts gives reason for thinking that there is charge in the battery, and so constitutes a plausible sufficient condition of there being charge in the battery. More- over, the battery's having charge is a necessary condition for starting the car, in that its being dis-
If the car starts there is charge in the battery.
(1)
Here the truth of the condition 'the car starts' is usu- ally said to be sufficient for the truth of the claim 'there is charge in the battery.' Likewise, the truth of 'there is charge in the battery' is a (physically) necess- ary condition for the truth of 'the car starts.' The vocabulary is often used of the physical situations themselves. Thus one could say that the car's starting is a sufficient condition of there being charge in the battery. Likewise, there being charge in the battery is a necessary condition of the car's starting (for if there were not charge in the battery, the car would not start).
Necessary and Sufficient Conditions A. A. Brennan
Necessary and Sufficient Conditions
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