Australasian Journal of Philosophy Vol. 87, No. 3, pp. 375–387; September 2009 ARE NECESSARY AND SUFFICIENT CONDITIONS CONVERSE RELATIONS? Gilberto Gomes Claims that necessary and sufficient conditions are not converse relations are discussed, as well as the related claim that If A, then B is not equivalent to A only if B . The analysis of alleged counterexamples has shown, among other things, how necessary and sufficient conditions should be understood, especially in the case of causal conditions, and the importance of distinguish- ing sufficient-cause conditionals from necessary-cause conditionals. It is concluded that necessary and sufficient conditions, adequately interpreted, are converse relations in all cases. I. Introduction The purpose of this paper is to counter claims that necessary and sufficient conditions are not converse relations—claims, that is, that ‘ A is a sufficient condition for B’ is not the converse of ‘ B is a necessary condition for A’ [Wilson 1979; McCawley 1993; Sanford 2003; Brennan 2003]. In the context of this discussion, conditionals of the form If A, then B are taken to express that A is a sufficient condition for B, and those of the form A only if B to express that B is a necessary condition for A, at least within a certain set of circumstances. 1 If the relations are converse, then If A, then B and A only if B should be equivalent. This equivalence has likewise been challenged by the authors just cited and also by Lycan [2001]. Two relations R and R0 are converses of each other if xRy is equivalent to Downloaded By: [Gomes, Gilberto] At: 00:18 27 June 2009 yR 0x for all values of x and y [Russell 1937: 25] . Being a necessary condition is traditionally considered as the converse of being a sufficient condition. If the truth of A is sufficient for the truth of B, then the truth of B is necessary for the truth of A and vice versa. When the relation between A and B is 1One might question the assumption that conditionals involve necessary and sufficient conditions with the argument that most theories of conditionals have the consequence that ‘ A & B ’ entails ‘If A, then B’, while the simple truth of A and B does not entail that A is a sufficient condition for B. For example, it is true that elephants are mammals and that London is the capital of England, but this does not entail that the former is a sufficient condition for the latter. It must be noted, however, that according to such theories, a conditional formed with these sentences must be considered true: ‘ If elephants are mammals, then London is the capital of England ’. My response, then, is that it is as counterintuitive to accept this conditional as true as it would be to accept that its antecedent is a sufficient condition for its consequent. Theories of conditionals are controversial and the consequence mentioned above should not be considered sufficient to invalidate the strong intuition that a conditional of the form ‘If A, then B’ implies that the truth of A is sufficient for the truth of B. The usefulness of analysing conditionals of the forms ‘If A, then B’ and ‘ A only if B’ in terms of necessary and sufficient conditions is exemplified in Varzi [2005] and Gomes [2006]. Australasian Journal of Philosophy ISSN 0004-8402 print/ISSN 1471-6828 online Ó 2009 Australasian Association of Philosophy http://www.tandf.co.uk/journals DOI: 10.1080/00048400802587325 376 Gilberto Gomes logical, mathematical, or purely conceptual, there is no question about necessary and sufficient conditions being converse relations. For example, the truth of ‘Bruno is an Italian pianist’ is a sufficient (but not necessary) condition for the truth of ‘Bruno is a pianist’. Conversely, the truth of ‘Bruno is a pianist’ is a necessary (but not sufficient) condition for the truth of ‘Bruno is an Italian pianist’. In the form of conditional statements, we should have: ‘If Bruno is an Italian pianist, then he is a pianist’, equivalent to ‘Bruno is an Italian pianist only if he is a pianist’. There are cases, however, that have called into question this traditional conception [Wilson 1979; McCawley 1993; Lycan 2001; Sanford 2003; Brennan 2003]. It has been argued that examples like the ones discussed below show the failure of such a view. Contrary to these claims, it is argued here that, rather than demonstrating that necessary and sufficient conditions are not converse relations, these examples help to show how these relations should be understood. It will be seen that most of the anomalous cases involve causal, rather than logical, mathematical or purely conceptual relations. The analysis of these examples has been fruitful. It has shown (1) the importance of distinguishing causal from non-causal conditionals and of distinguishing two types of causal conditionals (sufficient-cause conditionals and necessary-cause conditionals); (2) how to spell out the converse relations that hold in the case of causal conditionals; and (3) that necessary and sufficient conditions, adequately interpreted, are converse relations in all cases. II. Sufficient Cause and Necessary Effect Suppose that Jean tells Jack: (1) If you touch me, I’will scream. Downloaded By: [Gomes, Gilberto] At: 00:18 27 June 2009 Although Jack’s touching Jean appears as a sufficient condition for her screaming, Brennan [2003] claims that to take her screaming as a necessary condition for his touching her seems to get the dependencies back to front. This seems reasonable at first sight. Jack’s touching Jean is sufficient to cause her screaming, but her screaming is not a necessary cause of his touching her. However, if we focus on the truth of the propositions, we shall see that the truth of the proposition that Jean will scream is necessary for the truth of the proposition that Jack touched her. This is because, if (1) is true, it cannot be the case that Jack touches her and she does not scream. The event described in B is not causally necessary for the event described in A, but the truth of B is necessary for the truth of A, as far as the event described in A is sufficient for producing the event described in B. In a causal conditional such as (1), A’s being a sufficient condition for B means that A is a sufficient cause of B. We may read ‘sufficient condition’ (Jack’s touching Jean) as ‘sufficient cause’, but we should not read ‘necessary condition’ (Jean’s screaming) as ‘necessary cause’. (This is what Necessary and Sufficient Conditions 377 some commentators do when they say that Jean’s screaming is not a necessary condition for Jack’s touching her.) We should rather read it as ‘necessary effect’. Indeed, if an effect of A is necessary, the truth of the proposition that affirms its occurrence ( B) is a necessary condition for the truth of A. Thus, by understanding sufficient and necessary conditions, in this type of conditional, as sufficient cause and necessary effect , they are clearly recognized as converse relations. If A is a sufficient cause of B, B is a necessary effect of A. Another important point to note is that in most cases, a cause is sufficient to produce an effect only within a certain range of circumstances. Accordingly, a causal conditional such as (1) may apply only in a certain context. We can easily imagine circumstances in which Jill would not scream even if Jack touched her, but the conditional is not meant to apply to such circumstances. Consider the following example by Sanford [2003: 175]: (2) If you learn to play the cello, I’ll buy you a cello. Sanford argues that, in this case, you cannot use ‘ A only if B’ as equivalent to ‘If A, then B’, since you would get: (3) You will learn to play the cello only if I buy you a cello. This would have a rather different meaning from (2), since it implies that your learning depends on my buying you the instrument, while in (2) it is the other way round. In fact, the transformation of the sentence to the equivalent ‘ A only if B’ form is possible. We could say, without any change in meaning: ‘It will be true that you have learned to play the cello only if, at that moment, it is also true that I’ll buy you a cello.’ Some adaptations were made to preserve the temporal and causal sequence, but the content and meaning of the conditional were maintained, even if in a Downloaded By: [Gomes, Gilberto] At: 00:18 27 June 2009 clumsy form. What we find is that in (1) and (2) the tenses of the verbs used (present, future) fix a certain temporal (and causal) sequence between A and B, and the ‘only if’ formulation of the conditional may misleadingly imply an inversion of this sequence. In causal terms, if we understand (2) as meaning ‘Your learning to play will be a sufficient cause of my buying you a cello’, the converse would be: ‘My buying you a cello will be a necessary effect of your learning to play’. A similar example, from McCawley [1993: 82], is: (4) If butter is heated, it melts. McCawley contends that the meaning seems reversed if we change it to ‘Butter is heated only if it melts’. It seems to me, however, that this sentence may also be understood in the right sense. It would become clearer by changing the tense: ‘Butter has been heated only if it melts.’ (The cause precedes the necessary effect.) 378 Gilberto Gomes Another example: (5) If John wins the race, we’ll celebrate.