DISCUSIONES

*DEDUCIBILITY *IMPLIES *RELEVANCE?*

A CAUTIOUS ANSWER

(ON PROFESSOR ORAYEN’S CRITICISM OF RELEVANT LOGIC)*

RAYMUNDO MORADO

Instituto
de Investigaciones

Filosóficas, UNAM

I believe it to
be an underlying principle of logical theory that when a correct reasoning is
involved the conclusion cannot be just anything; what can be inferred depends
upon what premises we have. We are faced, in a rather imprecise manner, with
the requirement that the premises be relevant to the conclusion if we are to
have a good inference. This is not, of course, a proof that deducibility implies
relevance but an attempt to make explicit what I think is usually considered a
necessary condition for a correct deduction.

But it is not altogether clear what
kind of “relevance” should exist between premises and their conclusion. For
instance, in propositional logic the kind of relevance Anderson and Belnap (A&B)
are interested in is closely related to the notions of “using in a proof” and “variable-sharing”.
*A* is relevant to *B *iff *B* can be inferred
from (not just under) *A*; that is, iff
*A* *could
be used* in a proof of *B * from *A*.
For instance, A&B reject *A **®** *(*B **®** B*)
because they believe that *A* may be
irrelevant to (*B **®** B*) in the sense
that *A* is not used in arriving to (*B **®** B*). With respect
to the requirement of variable-sharing, A&B believe that in order to infer *B* from *A* it is necessary that *A *and
*B *have some common meaning content and
since they also think that in propositional logic commonality of meaning is
carried by commonality of propositional variables, they conclude that *A* and *B* should share at least one propositional variable. Due to this,
formulas of the form (*A & A*) ® *B* are rejected.

Orayen[1]
implicitly accepts this notion of relevance. Perhaps he does so for the sake of
argument but I think he has taken too much for granted. His analysis can be
misleading since it does not consider other theoretical possibilities. What is
discussed in Orayen’s paper is:

1) Does deducibility *in the sense of classical logic* (CL)
imply relevance *in the sense of A&B*?
or rather,

2) Should the intuitive notion of
deducibility (underlying our logical researches) which CL tried to capture,
imply relevance *in the sense of A&B?*

To the first question both A&B and
Orayen answer, “No”, for as we know, variable-sharing is not a CL metatheorem. In
fact, in standard propositional logic *A *can
imply *B *even if *A *and *B* have no common
variable; this may happen whenever *A *is
a contradiction or *B* a tautology.

To the second question Orayen’s answer
is negative while that of A&B is affirmative. I Think Orayen is right: A&B
claim that normal intuitions support their requirement of “A&B-relevance”
for “intuitive deducibility”, but Orayen shows the existence of even more
commonly felt intuitions (those supporting the rules used in Lewis’ argument:
Simplification, Addition and Disjunctive Syllogism) that lead us to accept some
cases of deducibility without A&B-relevance. Against this strategy A&B
had raised objections (centered on a criticism of the Disjunctive Syllogism)
that Orayen seems able to cope with. I believe Orayen makes his point: there
can be acceptable deducibility without implying A&B-relevance.

But in all this discussion we are no
longer trying to find out whether CL is relevant, but whether it needs to be “A&B-relevant”.
A&B cannot be said to be wrong because of demanding that the premises be
relevant to the conclusion, but rather because of believing that “relevance”
has to mean “A&B-relevance”. Orayen tries to prove the existence of some cases of deducibility without any
relevance, but only succeeds in proving the existence of some cases of
deducibility without *some *sort of
relevance, namely A&B-relevance. He disregards that for every true
deducibility relation, even that of CL, a relation closely connected with our
intuitions about relevance must be involved. So it must be possible to trace
some reasonable kind of relevance in CL although not necessarily the one A&B
describe. I do not wish to diminish the importance of the notion of relevance
depicted by A&B; I am just trying to argue that it is not the only possible
one.

But, what kind of relevance can be
found in CL?

To answer this I would like to state
what I call “Ackermann’s dictum”: To say that from *A *we can deduce (in a strong sense) *B* is equal to saying that the content of *B* is a part of the content of *A.**[2]* and
this implies that *A* must be relevant
to *B.*

* *There
is at least one notion of propositional content that satisfies Ackermann’s
dictum with respect to classical deducibility. If we take the content of a
proposition to be the set of state descriptions which falsify that proposition
(following certain ideas of Popper and Wittgenstein) it is easy to show how the
content of a tautology is part of the content of any proposition, which in turn
is part of the content of any contradiction. The set of state descriptions
which falsify a tautology is empty and therefore contained in the set of state descriptions which falsify any proposition
whatsoever; and the set of state descriptions which falsify any proposition
whatsoever is a part of the set of state descriptions which falsify a
contradiction, since every state description does so. Therefore, e.g., *A* is relevant to *B *É* B* and *A *& *-A* is relevant to *B*, if
relevance is understood as the content relation described above. It is easy to
see how the notion of relevance just sketched above holds for these well known
cases of A&B-irrelevance.[3]

Do not be fooled by the oddity of
saying that the content of *B* É *B* is part of the content of any *A* whatever. Notice that the content of a complex proposition is not
only determined by the propositions it contains but also by its syntactic
structure. A&B believe that two propositional formulas share intensional
content iff they share a variable and they do not realize that tautologies and
contradictions are the extreme cases in which the content of a compound formula
is not only affected but determined by the syntactic structure.

To summarize: it is misleading to say
as Orayen does, that Lewis’ argument is an excellent argument in favor of the
existence of deduction without relevance (section 11.3). Lewis’ argument shows
that there is deducibility without A&B-relevance. It does not show that
there is deducibility without any kind of relevance. This would shock our
intuitions and I have hinted at least one sense in which CL satisfies these
intuitions. Since I agree with Orayen that CL is correct in its inferences I
take it to be unfair to say, without some nuances, that it validates some
irrelevant arguments.

* I am indebted to Professor Orayen for his
encouragement to write down these remarks. Without his attitude my own research
would had not developed to its present state. His fair-mindedness makes him
worry more about progress than about defending himself from objections.

[1] “*Deducibility*
Implies *Relevance*? A Negative Answer”,
*Crítica*, vol. XV, No. 43, 44,
April-August, 1983.

[2] Cf. W. Ackermann, “Begründung Einer
Strengen Implikation”, *The Journal of
Symbolic Logic, *vol. 21, number 2, June 1956, p. 113.

[3] After writing this comment I
formally developed the notions of *propositional content* and *relevance* and proved a
metatheorem showing that
CL can satisfy Ackermann’s dictum with respect to those notions. This is the
content of a paper of mine presented at the IV Simposio International de
Filosofía (IIF-México), to be published in the proceedings of the forementioned
Symposium.