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Notre Dame Journal of Formal Logic Volume 22, Number 2, April 1981
Relevance: A Fallacy?
JOHN P. BURGESS
Introduction Responding to Harvey's theories about the circulation of the blood, Dr. Diafoirus argues (a) that no such theory was taught by Galen, and (b) that Harvey is not licensed to practice medicine in Paris. Plainly there is something wrong with a response of this sort, however effective it may prove to be in swaying an audience. For either or both of the allegations (a) and (b) might well be true without Harvey's theory being false. So Diafoirus' argument can serve only to divert discussion from the real question to irrelevant side- issues. The traditional term for such diversionary debating tactics is "fallacy of relevance". In recent years this traditional term has come to be used in a quite untraditional sense among the followers of N. D. Belnap, Jr., and the late A. R. Anderson. (All citations of these authors are from their masterwork [ 1 ], and are identified by page number.) According to these self-styled "relevant logicians", it is items (IA) and (IIA) in the accompanying table that constitute the archetypal "fallacies of relevance".
Table
(I) p or q (II) not both p and q notp p q n o t q
(IA) pvq (IIA) ~(p&q) ~P P q ~q (IB) p + q (IIB) ~(poq) ~P P q ~q
Received May 28, 1980; revised July 23, 1980
98 JOHN P. BURGESS
(In the table ~, &, and v stand for truth-functional negation, conjunction, and disjunction, respectively.) These forms of argument, say Anderson and Belnap, are "simple inferential mistake[s], such as only a dog would make" (p. 165). The authors can hardly find terms harsh enough for those who accept these schemata; they are called "perverse" (p. 5) and "psychotic" (p. 417). Needless to say, (IA) and (IIA), which can be traced back at least to Chrysippus, were not traditionally regarded as fallacious. The Anderson-Belnap notion of "relevance", whatever it may amount to, must be something quite different from the traditional notion, which "was central to logic from the time of Aristotle" (p. xxi). And yet the authors declare their so-called "relevant logic" to be a commonsense philosophy, in accord with the intuitions of "naive freshmen" (p. 13) and others who have not been "numbed" (p. 166) by a course in classical logic. Moreover, whereas other dissident logicians (e.g., intuitionists) hold that some forms of argument always accepted and used without question by mathematicians in their proofs are in fact untrustworthy, Anderson and Belnap are at some pains to explain (pp. 17-18 and 261-262) that their brand of nonclassical logic does not conflict with the practice of mathe- maticians, but only with the classical logician's account o/that practice. In view of the fact that everyday arguments and mathematical proofs abound in instances of (I) and (II), one may wonder how Anderson and Belnap could hope to reconcile their rejection of (IA) and (IIA) with the claim that their "relevant logic" is compatible with commonsense and accepted mathe- matical practice. The answer is that the authors believe that the ordinary- language argument patterns (I) and (II) should be represented as expressions of the "intensional" schemata (IB) and (IIB), which are relevantistically accept- able, and not of the "extensional" schemata (IA) and (IIA), which the rele- vantists reject. The compound p + q appearing in (IB) is supposed to be an "intensional disjunction" stronger than the truth-functional p v q in that mutual relevance of p and q is required for its truth. This p + q is not entailed by /?, nor even by p & q, since it might be false even though p and q were both true (or even necessary). This happens in the case of irrelevant pairs such asp = "Bach wrote the Coffee Cantata" and q = "The Van Allen belt is doughnut-shaped" (p. 30). Dually, the p o q of (IIB) is an "intensional conjunction", better called "co- tenability" or "nonpreclusion", a compound weaker than the truth-functional p & q in that mutual irrelevance of p and q is sufficient for its truth. This p o q does not entail p, nor even p v q, since it might be true even though p and q were both false (or even impossible). The relevantists' claim that (IB) and (IIB) best represent (I) and (II) admits of two formulations: a stronger and a weaker. The stronger claim would be that the ordinary-language "or" and "and" literally mean + and o rather than v and &. The weaker claim would be that anyone basing an argument on the premise that p or q, or that not both p and q, will at least be in a position to assert that p + q, or ~(p o q) as the case may be. (The latter claim is weaker than the former because even if "or" and "and" meant v and &, it might still be that arguments of form (IA) and (IIA) could always be avoided in practice because in any instance where one might wish to argue from p v q or from ~(p & q) the stronger premises p + q and ~(p o q) would be available.)
100 JOHN P. BURGESS
Example 1, Argument: During the course of a game of Mystery Cards, Wyberg
hears von Eckes ask Zeemann, "Is it the deuce of hearts and the queen of
clubs?" He hears Zeemann reply "No". Later in the game he manages to figure
out that it is the deuce of hearts. He argues: It isn't both the deuce of hearts
and the queen of clubs; but it is the deuce of hearts; so it isn't the queen of
clubs. He goes on to use this information to win the game.
Example 1, Analysis: Let p = "The mystery red card is the deuce of hearts",
q = "The mystery black card is the queen of clubs". Zeemann's hint is no more
and no less than that ~(p & q). Her statement is made on purely truth-
functional grounds: She sees the queen in her hand. Her statement is not made
on the basis of any "relevance" between p and q: The two mystery cards were
chosen entirely independently of each other. Zeemann is justified in denying a
truth-functional conjunction, but would not be justified in denying coten-
ability. Since the premise ~(p o q) is not available to Wyberg, his argument is
an instance of (II) that can be neither read as nor replaced by an instance
of(IIB).
Had Wyberg been a relevantist, unwilling to make a deductive step not
licensed by the Anderson-Belnap systems E and R, he would have been unable
to eliminate the queen of clubs from his calculations, and would have lost the
game. A relevantist would fare badly in this game and others, and in game-like
situations in social life, diplomacy, and other areas—unless, of course, he
betrayed in practice the relevantistic principles he espouses in theory.
Background to Example 2: Dr. Zeemann has just been awarded her degree for
a dissertation in number theory. Her main result is a proof that every natural
number n either has a certain property A{n) or else has a certain other property
B(n). As written up in her thesis, the proof is by induction on n, as follows:
Case n - 0: We show that ^4(0). [Here follows a proof.]
Case n = 1: We show that JS(1). [Here follows a proof.]
Case n > 2: We assume as induction hypothesis that either A(n - 1) and
A(n - 2), or A{n ~ 1) and Bin - 2), or B(n - 1) and A(n - 2), or else
B(n - 1) and Bin - 2). [Here follows a proof treating each of the four
cases separately.]
She remarks that the famous d'Aubel-Hughes Conjecture would imply that
B(0), whereas the equally famous conjecture of MacVee would imply that ^4(1),
but reports that she has no light to shed on these old conjectures.
Commentary: Before proceeding, let us note that, following the universal
practice of mathematicians, Zeemann has taken her proof that ^4(0) to dispose
of the case n = 0 of the general theorem that for all n, either A in) or Bin). In
other words, she argues from the premise ^4(0) to the conclusion that^l(O) or
B(Q). This is worth mentioning because relevantistically inclined writers have
been known to claim that no one ever seriously argues from p to p or q. Indeed,
in everyday conversation we are, in R. C. Jeffrey's words, "at a loss to know
what the motive could be" for someone to pass from p to the longer and less
informative statement that p or q. "Knowing the premise, why not assert it,
rather than the conclusion?" However, in mathematics we often have good
RELEVANCE: A FALLACY? 101
reason to say less than we know: We will assert less than we could about the
cases n = 0 and n = 1 in order to incorporate these cases in a generalization
holding for all values of n. Now the inference from ^4(0) to ^4(0) or 5(0) is only
valid if "or" is taken as v rather than +. Hence Zeemann's theorem must be
formalized as (n)(A(n) v B(n)), not (n)(A(n) + B(n)). This means that any argu-
ment of form (I) in which the major premise is supplied by Zeemann's Theorem
will be an instance of (I) that can be neither read as nor replaced by an instance
of (IB). Let us proceed to examples.
Example 2a, Argument: Zeemann applies her work to give bounds to the
number of solutions to Tiegh's Equation, thus:
Tiegh himself has shown that the number t of solutions to his equation is
< 13. Now a little elementary algebra shows that we cannot have ^4(0-
Hence by our main result, we must have B(t). But no n with 5 < « < 16
can satisfy 5(rc), as is clear from some more elementary algebra. Hence
r < 4.
Example 2a, Analysis: This is a typical mathematical argument of form (I). The
premise A(t) or B{t) must be represented as a truth-functional, not an "inten-
sional", disjunction. The "unknown" t might for all we know be equal to 1,
and no "relevant" connection has been established between yl(l) and 5(1)—
indeed, as Zeemann herself reports, she has been unable to establish anything
about 5(1).
Example 2b, Argument: Professor Wyberg has been working for years on the
celebrated conjecture of von Eckes, but has got no further than showing that
the conjecture follows from the assumption that 5(1), a result he considers not
worth publishing. Just recently he has given up work on von Eckes' Conjecture
in disgust, and has turned to other matters. In particular, he has just refuted
an old conjecture of MacVee by proving that ~^4(1). Now he reads an an-
nouncement of Zeemann's result. The details of her proof are not available—it
takes years for theses to come out in print—but he recognizes the significance
of her results. In particular, they enable him to prove von Eckes' Conjecture
at last. He writes a set of notes, "A Proof of von Eckes' Conjecture", with the
following structure: First comes his proof that ~^4(1). Second comes a linking
passage:
And so we see that the MacVee Conjecture fails. Now Zeemann has
recently announced the result that for all n, either A(n) or B(n). Hence we
must have 5(1). We now proceed to put this fact to good use.
Third follows the derivation of von Eckes' Conjecture from 5(1).
Example 2b, Analysis: Since what is established by Zeemann is just ^4(1) v
5(1), not ^4(1) +5(1), we have here another mathematical instance of (I) that
can be neither read as nor replaced by an instance of (IB). It is a slightly
atypical instance. Had he known the details of Zeemann's work, had he known
that she actually proves 5(1) outright, Wyberg would surely have just cited the
fact that 5(1) from her thesis, rather than give the roundabout argument that
he did. But this is not to say that the proof of von Eckes' Conjecture that
RELEVANCE: A FALLACY? 103
p or knew q though one has now forgotten which. (And paradoxically, the acquisition of more information could threaten one's right to assert p + q: if one's informant decides to provide more specific information, if the value of m is settled, if one's memory improves, one may suddenly lose the right to assert p + q.) Relevantism would reduce to the position that (IA) is valid when and only when one's grounds for asserting p v q are something other than the simple knowledge that q. Such a position, however, looks suspiciously like a confusion of the criteria for the validity of a form of argument with the criteria for its utility, a confusion of logic with epistemology. Indeed, some writers have been willing to dismiss the whole relevantistic movement as a simple case of confusion between the logical notion of implica- tion and the methodological notion of inference. The following (unpublished) remarks of G. Harman on this point will bear quoting:
By reasoning or inference I mean a process by which one changes one's views, adding some things and subtracting others. There is another use of the term 'inference' to refer to what I will call 'argument', consisting in premises, interme- diate steps, and a conclusion. It is sometimes said that each step of an argument should follow from the premises or prior steps in accordance with a 'rule of inference'. I prefer to say 'rule of implication', since the relevant rules do not say how one may modify one's views in various contexts. Nor is there a very direct connection between rules of logical implication and principles of infer- ence. We cannot say, for example, that one may infer anything one sees to be logically implied by one's prior beliefs. Clearly one should not clutter up one's mind with many of the obvious consequences of things one believes. Furthermore, it may happen that one discovers that one's beliefs are logically inconsistent and therefore logically imply everything. Obviously, one ought not to respond to such a discovery by believing as much as one can. Some philosophers and logicians [the reference is to Anderson and Belnap] have imagined that the remedy here is a new logic in which logical contradictions do not logically imply everything. But this is to miss the point that logic is not directly a theory of reasoning at all.
And indeed if "relevance" is taken to be something subjective and relative (according to the proposal discussed above), I do not see how the relevantists could escape Harman's charge that they confuse implication and (useful) inference. I do not, however, believe that the authors of [1] understand by "rele- vance" something subjective. What little they tell us about the nature of "relevance" (e.g., pp. 32-33, where they quote with approval from several sources) strongly suggests that it is a matter of meaning. Certainly their com- monest charge against classical logic (first raised on p. xxii and repeated ad nauseum) is that it ignores "intension" and meaning. Meaning, however, is something that, generally speaking, will be the same for Wyberg as it is for Zeemann. That relevance is meant to be a semantical, and hence impersonal, notion, and not a matter of individual psychology, is further suggested by the relevantists' criticisms of T. J. Smiley (p. 217), who is faulted for "episte- mologizing" and "psychologizing" the logical notion of entailment. Thus if the authors of [ 1 ] intend by "relevance" something less than objective, they are highly remiss in failing to alert their readers to the fact; while if "relevance" is
104 JOHN P. BURGESS
supposed to be impersonal, then the claim that the relevantistic position is
(even in a weak sense) compatible with commonsense and accepted mathe-
matical practice succumbs to the counterexamples presented above.
In closing, let me reiterate that I have been concerned here solely with the
original Anderson-Belnap account of "relevant" logic, and with their claim
that their systems E, R, etc., are in better agreement with commonsense than is
classical logic. I have not been concerned with other rationales for developing
these systems, nor with the possibility of imposing interpretations on them that
were not originally intended by their authors. (It has been suggested, for
instance, that some of the formalisms created by the relevantists might be
useful in developing a logic of ambiguity, or of truth-in-fiction.) Workers in
category theory, one of the least constructive branches of modern mathematics,
have found certain technical uses for intuitionistic logic; but no one imagines
that this vindicates Brouwer's philosophy of mathematics. Similarly, the
discovery of serendipitous applications of some of the formalisms created by
Anderson and Belnap would not justify the claim that their logical systems
are accurate formalizations of current mathematical practice. Still less could it
justify the abusive tone of their remarks about classical logicians.
REFERENCE
[1] Anderson, A. R. and N. D. Belnap, Jr., Entailment: The Logic of Relevance and Necessity, Princeton University Press, Princeton, New Jersey, 1975.
Department of Philosophy Princeton University Princeton, New Jersey 08544
