Philosophy Vol. 3 Nos. 2-3 Pp. 153-165 AptiMu|y 1973

RELEVANT DEDUCTION AND MINIMALLY INCONSISTENT SETS*

KEITH LEHRER

Some valid arguments contain premisses that are inessential for establishing the truth of the conclusion by deduction. I shall offer an analysis of relevant deduction in which all of the premisses are essential. Different conceptions of relevance have concerned philosophers and logicians, so it is important to have a clear preanalytic conception in order to test the success of the analysis.~ The concept of relevance I shall attempt to explicate is the following: a relevant deductive argument is a valid argument in which knowledge of the truth of each and every premiss is required to establish the truth of the conclusion by deduction from the premisses.

To clarify the project, I shall consider three kinds of arguments containing premisses whose truth need not be known to establish the truth of the conclusion by deduction from the premisses. All three arguments will he valid in standard logic. It is logically impossible that the premisses should be true and the conclusion false. Equivalently, the set of statements consisting of the premisses and the denial of the conclusion is logically inconsistent. The first is an argument in which the premisses of the argument are contradictory.

An earlier version of this paper was pres0nted at the American Philosophical Association Meetings, Western r)Ivision, in St. Louis, May $, 1970. Work on the paper was partially supported by a National Science Foundation Grant.

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First Argument

John is tall John is not tall Therefore, the moon is older than the earth.

In this argument, knowledge of the truth of the premisses is not needed and is not relevant to establishing the truth of the conclusion. This is due solely to the fact that the premisses are contradictory and thus cannot possibly be known to be true. Knowledge of the truth of such premisses is irrelevant because it is impossible. Having failed to appreciate this consideration, some logicians suppose that the following argument has a premiss that is relevant to the conclusion. 2

John is tall and John is not tall Therefore, John is tall and John is not tall.

But this is a mistake. It is founded on the idea that every statement is relevant to itself, and this idea is erroneous. Know-

ledge of the truth of the premiss of this argument is as impos- sible as knowledge of the truth of the premisses of the preceding argument, and, consequently, knowledge of the truth of the premiss cannot be relevant to establishing the truth of the conclusion.

Second, we have arguments which contain a superfluous premiss, where some subset of the premisses will suffice for the deduction of the conclusion. Consider the following argument.

Second Argument

The moon is older than the earth If John is tall, Mary is short John is tall Therefore, Mary is short.

Not all the premisses are relevant for the deduction of the conclusion because the first is otiose and the second and third will suffice. Thus knowledge of the truth of the first premiss is

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not needed to establish the truth of the conclusion by deduc- tion from the premisses.

Thirdly, arguments with tautological conclusions are not relevant.

Third Argument

The moon is older than the earth Therefore, either John is tall or John is not tall.

In this case, knowledge of the truth of the premiss is irrelevant to establishing the truth of the conclusion by deduction. Again, however, the implications of this have not been appreciated by philosophers and logicians. For this argument will remain one in which knowledge of the truth of the premiss is not needed to establish the truth of the conclusion by deduction no matter what premiss we substitute for the present one. The first premiss of this argument is superfluous for just the same reason that the first premiss was superfluous in the preceding argu- ment, namely, that it is not needed in the argument for deducing the conclusion. When an argument has a tautology as a conclusion, then no premisses are required for the deduction of the conclusion.

This fact is clarified by standard systems of natural deduction, but it follows from the traditional conception of a valid argument as one in which it is impossible for the premisses to be true and the conclusion false and such that the set of premisses together with the denial of the conclusion is logically inconsistent. An argument with a tautological conclusion, no matter what the premisses are, must be valid in terms of these notions because the conclusion cannot possibly be false and the denial of the conclusion is itself logically inconsistent. Thus, arguments with tautological conclusions are such that know- ledge of the truth of any premiss is unneeded to establish the truth of the conclusion by the argument.

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II

We have three kinds of valid deductive arguments in which knowledge of some of the premisses is not needed to establish the truth of the conclusion by the argument. Such arguments are not relevant deductive arguments. A relevant deductive argument is one in which knowledge of the truth of each and every premiss is needed to establish the truth of the conclusion by deduction from the premisses. I shall now give a precise analysis of the concept of a relevant deductive argument as thus characterized. Let us first say that a set of statements is inconsistent if and only if a contradiction may be deduced from the set in standard logic. A valid deductive argument is one in which the set of statements consisting of the premisses and the denial of the conclusion is inconsistent. Some inconsistent sets are such that when some statement is deleted from the set the remainder is still inconsistent. To put it another way, some inconsistent sets have proper subsets which are inconsistent. On the other hand, there are inconsistent sets which are such that once any statement is deleted from the set the remainder is consistent. Such sets are inconsistent, but every proper subset of the set is consistent. These sets I call minimally inconsistent sets.

In terms of the notion of a minimally inconsistent set we may define a relevant deductive argument. An argument is a relevant deductive argument if and only if the set of statements consisting of the premisses and the denial of the conclusion (or any truth functional equivalent) is a minimally inconsistent set. Relevant deductive arguments as so defined exclude the three kinds of arguments mentioned above.

First, arguments with contradictory premisses are not relevant no matter what the conclusion. In such arguments the set consisting of the premisses and the denial of the conclusion (or any truth functional equivalent) will not be minimally inconsistent. The reason is that a proper subset of the set consisting of the premisses alone is inconsistent. It is a corollary of this result that any argument with a contradictory conclusion

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will also fail to be a relevant argument, because any inconsistent set containing a tautology is such that the proper subset obtained by deleting the tautology from the set will also be inconsistent.

Secondly, in valid arguments with superfluous premisses the set consisting of the premisses and the denial of the conclu- sion will not be minimally inconsistent because the proper subset from which the superfluous premisses are deleted will also be inconsistent. Thus such arguments are not relevant deductive arguments. For example, in the second argument given above, when the second and third premisses are taken together with the denial of the conclusion, the resultant set is inconsistent even though it does not contain the first premiss. The argument consisting of the second and third premisses together with the conclusion is relevant.

Thirdly, no argument with one or more premisses and a tautological conclusion will be a relevant deductive argument. This is a consequence of the fact that a set consisting of the denial of the conclusion (or any truth functional equivalent) will not be minimally inconsistent if it contains any other state- ment. The set consisting of merely the denial of the conclusion (or any truth functional equivalent) will be inconsistent, because that statement is a contradiction.

However, a tautological statement by itself would be a relevant deductive argument on our definitiou if we allow something as an argument which lacks premisses. Traditionally, the notion of an argument without premisses would have seemed quite nonsensical, but the development of the technique of natural deduction admits such arguments. It matters little whether we impose the ad hoc requirement that an argument must contain one or more premisses or whether we drop this requirement. However, to say that every tautology is such that knowledge of each and every one of the null set of premisses of the argument is needed to deductively establish the truth of the tautological conclusion is sufficiently bizarre to exclude this alternative. Thus by an argument we shall mean a set of two or more statements, one of which is designated as the conclusion.

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As so specified, no argument with a tautological conclusion will be a relevant deductive argument.

To sum up. By a relevant deductive argument I mean one in which knowledge of the truth of each and every premiss is needed to establish the truth of the conclusion by the argument in question. I then defined a relevant deductive argument as a set of two or more statements one of which is designated as a conclusion such that a set of statements consisting of the premisses of the argument and the denial of the conclusion of the argument (or any truth functional equivalent) is a minimally inconsistent set of statements.A set of statements is minimally inconsistent if and only if it is inconsistent but such that no proper subset of it is inconsistent.

The results of this inquiry are that no relevant deductive argument has a contradictory or tautological conclusion, none has contradictory or tautological premisses, and none has any premiss that is unneeded for the deduction of the conclusion in the argument. In effect, we have excluded all deductive argu- ments from the circle of relevance except those all o f whose

premisses could be jointly true or false and whose conclusion could also be true or false. Thus, the set of premisses and the conc lus ion must be contingent, and, therefore, reievant arguments contain exclusively contingent statements and sets of premisses. 3 This result is not surprising, for it is only with respect to arguments with contingent conclusions that know- ledge of the truth of premisses is needed to establish the truth of the conclusion, and it is only the knowledge of the truth of other contingent statements that is needed to establish the truth of the conclusion by deduction. The foregoing results are more in line with standard logic that the results of most other investigators interested in relevant deduction. Indeed, my results do not represent any departure from standard logic or standard notions of entailment or validity.

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III

Having analy.zed the concept of relevant deduction infor- mally characterized at the outset, I now wish to consider some objections to what I have said. First, it might seem that the following argument should be accepted as a relevant deductive argum en t:

Fourth Argument

Mary is short If John is tall, then Mary is short John is tall Therefore, Mary is short.

The reason for regarding this as a relevant deductive argument is that all the premisses are such as may be employed in the deduction of the conclusion in such a manner that they are essential premisses of the deduction. Thus, if the conclusion is deduced from the first premiss, then that premiss is essential in the deduction. On the other hand, if the conclusion is deduced from the second and third premisses, then both of those premisses are essential in the deduction.

My reply to this argument is twofold. First, the concept of relevant deduction alluded to in the objection is different from the one I have characterized in which knowledge of the truth of all the premisses is required to establish the truth of the conclu- sion by deduction from the premisses. In the foregoing argu- ment, the first premiss is not required to establish the truth of the conclusion by deduction from the premisses, because that conclusion may be deduced from the other two premisses, and those two premisses are unrequited because the conclusion may be deduced from the first premiss. Secondly, the sense in which the fourth argument is relevant can be analyzed in terms of the concept of relevance I have already explicated. The kind of relevance possessed by the fourm argument I shall call weak relevance. In a weakly relevant argument all the premisses of the argument may be employed in some deduction of the conclu- sion in such a way that knowledge of the truth of those pre-

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misses is required to establish the truth of the conclusion by that deduction. An argument is a weakly relevant deductive argument if and only if each premiss of the argument belongs to some proper subset of premisses such that an argument consis- ting of that subset of premisses and the conclusion of the argument is a relevant deductive argument in the sense defined earlier. This is equivalent to saying that an argument is a weakly relevant deductive argument if and only if each premiss of the argument belongs to some proper subset of premisses such that a set consisting of that subset of premisses and the denial of the conclusion (or some truth functional equivalent) is a minimally inconsistent set. Of course, the argument consisting of the first premiss and the conclusion of the fourth argument and the argument consisting of the second and third premisses and the conclusion of the fourth argument are relevant deductive argu- ments. Hence, the fourth argument is a weakly relevant

deductive argument. A second objection is based on the fact that alternative

analyses of relevant deduction and pure entailment offered by logicians do not require that the premisses and conclusions of relevant deductive arguments be contingent statements. 4 Indeed, the notions of relevant deduction developed by such logicians are intended to apply to reductio ad absurdum arguments and other forms of deductive arguments demonstra- ting that some statement is a theorem. The concept these logicians are attempting to characterize is somewhat different from the one I characterized at the outset of this paper, but it might nevertheless be contended that the concepts developed by these logicians may be substituted for the concept I have analyzed thereby rendering my efforts superfluous. It is this objection I must meet.

The concept developed by the logicians in question cannot replace the concept analyzed above. On the accounts they offer, it is a feature of relevant deduction that every statement may be relevantly deduced from itself. 5 This common feature of their analyses renders them inadequate for the purpose at

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hand, because it has the consequence that every contradiction may be relevantly deduced from itself. It is obvious that no contradiction is essential to establislting its own truth for the simple reason that it cannot possibly be true. Therefore, no contradictory conclusion can be relevantly deduced from a contradictory premiss in a sense of relevance requiring that knowledge of the truth of each of the premisses of the argument be essential for establishing the truth of the conclusion by deduction from the premisses. Thus, the concepts of relevant deduction and pure entailment developed by the logicians cited cannot be substituted for the concept of relevant deduction I have analyzed.

A third objection is based on the consideration that my analysis of relevant deduction has the consequence that two arguments with logically equivalent sets of premisses and the same conclusion may be such that one of the arguments is a relevant deductive argument and the other is not. For example, if P and Q are logically contingent and logically independent statements, then an argument having P as one premiss, and Q as a second premiss, with Q as the 'conclusion, is not a relevant deductive argument. But an argument having as a single premiss the conjunction P and Q and Q as a conclusion, will be a rele- vant deductive argument. And, surely, it might be objected, since the content of the premisses is the same in both cases, the set of premisses consisting of the statement P and the statement Q having the same logical consequences as the set of premisses consisting of the statement P and Q, one argument is relevant if and only if the other is.

The initial reply to this objection is that in the argument containing the single premiss, P and Q, knowledge of the truth of that premiss is essential to establishing the truth of that conclusion by deduction from the premiss. On the other hand, the argument containing two premisses, P, and Q is such that knowledge of the truth of one of the premisses, P, is not essential to establishing the truth of the conclusion by deduc- tion from the premisses. Thus, it is a feature of the concept characterized at the outset that arguments with the same conclu-

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sion and logically equivalent sets of premisses may be such that one is relevant and the other is not. Hence it is no objection to my analysis of that concept that it has such consequences.

A determined objector might find the preceding reply less than satisfactory. He might argue 'as follows: surely a more useful concept of relevant deduction to concern ourselves with is one in which knowledge of all the logical content of all the premisses is essential to establishing the truth of the conclusion by deduction. For, in the argument containing the single premiss, P and Q, the content of P is as irrelevant to establishing the truth of Q by deduction as is the content of P in the argument where P is a premiss. Thus it is the logical content of the premisses, where this is understood as the set of logical consequences of the premisses, and not the mere formulation of the premisses, that should determine whether the argument is relevant.

The preceding objection embodies an intuitively plausible idea, but like other such plausible ideas it has a completely absurd and unacceptable consequence. If we require that a relevant deductive argument be one in which all of the content of each and every premiss be essential to establishing the truth of the conclusion by deduction, we will arrive at the absuid result that the only relevant deductive arguments are those containing a single premiss, namely, the statement that is the conclusion of the argument. Only arguments of the form P, therefore P, would be relevant. The reason is that every set of premisses from which P may be deduced has the same content as the conjunction of those premisses, and the conjunction, since it has P as a logical consequence, has the same content as the conjunction of itself with P. (I am here assuming that a set of statements or a single statement has the same content as another set of statement or statement if and only if they are logically equivalent in the sense that they have the same deductive consequences.) 6 Thus any set of premisses that has P as a logical consequence has the same content as a conjunction C of the premisses which in turn has the same content as the conjunction of C and P. But the only content of the conjunc-

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tion of C and P that is essential for establishing the truth of P by deduction from that conjunction is the content of P itself. Any additional content would be superfluous. Therefore, if we require that a relevant argument be one in which knowledge of all the content of all the premisses is essential for establishing the truth of the conclusion by deduction from the premisses, then only arguments of the form, P, therefore, P, will turn out to be relevant. Such a notion of relevant deduction would be as absurd as it is useless.

Finally, someone might object that even in those arguments which are relevant deductive arguments in the sense analyzed, knowledge of the truth of the premisses may not be essential for establishing the truth of the conclusion. The reason is that one may know something else, not expressed in the premisses at all, by which one can establish the truth of the conclusion. Or one may know that conclusion is true without employing any premisses whatever.

The reply to this objection is that the sense of relevant deduction I have analyzed is one in which knowledge of the truth of each and every premiss is essential for establishing the truth of the conclusion by deduction from the premisses. Though there may be other ways of establishing the truth of a conclusion than by means of the relevant deductive argument in question, it remains the case that knowledge of the truth of each and every premiss is essential for establishing truth in one special way, namely, by deducing the conclusion from the premisses.

IV

In conclusion, I should like to mention briefly my primary philosophical motivation for analyzing the concept of relevant deduction. The concept of relevant deduction was developed for application to certain epistemological questions concerning justification and justificatory argument. A number of philoso- phers have argued that certain beliefs should be regarded as

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i n n o c e n t un t i l p r o v e n gui l ty , or, m o r e l i t e ra l ly , t h a t t h e y are

r e a sonab l e o r j u s t i f i ed un t i l p r o v e n u n j u s t i f i e d o r u n r e a s o n a b l e .

Th i s n a t u r a l l y raises t he q u e s t i o n o f w h a t sor ts o f a r g u m e n t s

w o u l d p r o v e t h a t a b e l i e f is u n j u s t i f i e d o r un rea sonab l e . I t is m y

c o n t e n t i o n t h a t a r e l evan t d e d u c t i v e a r g u m e n t , o r even a w e a k

r e l evan t d e d u c t i v e a r g u m e n t , each o f w h o s e p remisses is m o r e

p r o b a b l e t h a n the b e l i e f in q u e s t i o n , and w h o s e c o n c l u s i o n

c o n t r a d i c t s o u r be l ie f , is a p r o o f t h a t o u r b e l i e f is u n r e a s o n a b l e

and un jus t i f i ed . On t h e o t h e r hand , i f a b e l i e f is m o r e p r o b a b l e

t h a n at least one p remiss in all r e l evan t d e d u c t i v e a r g u m e n t s

w h o s e c o n c l u s i o n s c o n t r a d i c t t h a t be l ie f , t h e n I c o n t e n d the

b e l i e f is r ea sonab le and jus t i f i ed . H o w e v e r , t he f u r t h e r e labo-

r a t i on and de fense o f these e p i s t e m o l o g i c a l theses is t he sub jec t

fo r a n o t h e r day.

THE UNIVERSITY OF ROCHESTER ROCHESTER, NEW YORK 14627 USA

NOTES

1 A bibliography of articles on this topic is to be found in A.R. Anderson and N.D. Belnap, "Tautological EntaiLments," Philosophical Studies (1962), Vol. XIII, No. 1-2, pp. 9-24, and "The Pure Calculus of Entailment," Journal of Symbolic Logic (1962), Vol. 27, pp. 19-52.

2 Anderson and Belnap, "Pure Calculus," pp. 20, 36, 39,42. 3 Cf. P.T. Geach, "Entailment," Arist. Soc. Sup. 1Iol.32 (195S) pp. 157-72; T.J.

Smiley, "Entailment and Deducibility," Proc. of Arist. Soc., Vol. 59 (1959), pp. 233-254; and G.H. von Wright, Logical Studies (London: Routledge and Kegan Paul, 1957); also "A Note on Entailment," Philosophical Quarterly (1959) Vol. 9, pp. 363-65; J.A. Barker, A Formal Analysis of Conditionals (Carbondale: Southern lUinois University, 1969), pp. 67-70. These authors discuss various concepts of relevant implication. One of the authors, yon Wright, proposes that "A entails B only if there is an a priori way of getting to know A ~B which is not

way of getting to know whether A or whether B." This would suggest that A and B should be contingent. All such proposals differ from the one made here simply because an argument containing a conjunction of those premisses as a single premiss and the same conclusion may be relevant. Thus, the question of whether the argument A, B, therefore, C, is relevant is not equivalent to the question of whether CA & B), therefore, C, is relevant. For a defense of this position see the reply to the third objection in section llI.

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4 Cf. Anderson and Belnap, "Pure Calculus." 5 Ibid., pp. 20, 26, 39, and 42. Systems of other authors which have this conse-

quence are discussed. 6 Cf. P, udolf Carnap, The Logical Syntax of Language, (London: goutledge and

Kegan Paul Ltd., 19371, p. 42.

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