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q, the flags or subscripts assigned to the premises must be passed over to the
conclusion, thus keeping track of the reasoning thread. We could then expect that within
Rthat natural deduction framework
pUq would be deduced from {p,q} provided the
Rsubscripts of both premises are inherited by the conclusion, that is to say provided
pUq
Ris assigned all the subscripts of
p and all the subscripts of
q. But no, things cannot
Rbe like that. For then, within system R we could prove
p.q.pUq, and therefore
R
p.qp " which, in the colourful expression of Dunn, one of the champions of the
school, would be equivalent to washing dirty money through the third world. Within
Rsystem E that dismal outcome is averted, even with Adjunction formulated as we have
Rsjust done, but a different if less damaging result ensues: we could then prove
pq.R\ p.pUq, which is the principle of reduced factor, RF for short. From which the whole
RG!Factor would be easily deduced:
pq.pUr.qUr.
Factor is a principle which has been studied by the Australian relevantists,
specially by Sylvan and by Sylvan & Urbas in a joint publication. Those authors have
Rx$shown that Factor leads to irrelevance when joined to the axioms of system E but not
necessarily so when several of those axioms are sufficiently watered down.
R&Within the framework of system E of entailment, Factor (or equivalently RF) immediately lead to validating as a theorem the principle of implied selfimplications, PII,
R(
pq.rr: any selfimplication is implied by any implication. True, that result is very
very far from the banned VEQ. A system with PII may nonetheless comply with some
of the relevantist strictures. For instance it may fail to have a strongest formula, one,
RQ+
p, that is, such that for all formulae
q:
pq. It may have the Ackermann property,Q+ o.,,55Rwhich means that for no implicative
q is
pq a theorem, when
p is a sentential
Rvariable. In fact a system with PII (or equivalently " within the framework of E " with
Factor) may remain miles away from classical logic. However, such a system is no
longer a relevant system. For a minimal, necessary (although not sufficient) condition
Rof relevance is that no formula
pq is a theorem if
p and
q share no common
variable.
^
3." Three Ways Out " or the Reasons for a Gradualistic
[Appropriation#
It is a widespread tendency in human behaviour to be content with nothing less
than grand thoroughgoing principles, adherence to which can afford stable situations.
Most of the time, things are less straightforward and more complicated than we had
fancied. The all or nothing criterion is likely to lead us astray.
That seems to me the case as regards the idea of relevance. It was a nice idea.
It appealed to some qualms a number of students have felt over the years when first
becoming acquainted with classical logic and the apparently whimsical turns of classical
inference. Yet implementing the idea up to the hilt leads to such amazing results that
you ought to wonder if the price is right.
There are several ways out. One is offered by what I have called radical relevantism, that of M)ndez and Avron. The word may not be apposite, since in some important (and relevant) respects it amounts to jettisoning a nuclear part of the original
relevantist plan, namely having systems with the Ackermann property, and avoiding that
a logical axiom follows from a contingent sentence; in other words, claiming that from
RG
p alone nothing follows except
p (and
NNp and
pUp " and also
pVq and
qVp
etc?); in particular nothing follows which we can also know from other sources " as logical theorems can be, through the study of logic presumably. (Radical relevantists fall
Rback on the converse Ackermann property: no theorem of the form
pqr, where
r
is a sentential variable. In some important sense, as will emerge below, that kind of
approach is the dual of the one I shall be suggesting towards the end of the paper.)
R3The second way out is offered by deep relevantism, socalled, whose main
champion is Richard Sylvan. Its general plan " except perhaps as concerns Factor and
Rsome other isolated principles " is to further weaken system E. There are a number
of features separating Sylvan's philosophical enterprise from the original relevantist idea,
not all of which are directly related to his weakening proposal. Such a proposal can be
independently defended on the ground that it is more cautious to assert less: if we can
implement logical systems sufficiently useful for reasoning keeping clear of the more
controversial principles, it is reasonable to refrain from asserting those principles: since
R#logic is a priori and acquired through (considered or reflective) intuition, the more controRh$vertible a principle is, the less likely it turns out to count as a genuine a priori, analytic,
nonfactual truth. The other main idea in Sylvan's own enterprise seems to me a quite
different and even, to some extent, opposite idea. The founding fathers loathed
contradictions as much as the classicist, and never for a minute thought that a contradiction could be true at all; their objection to classical logic was not that, in virtue of
the Cornubia rule, it could lead from a true contradiction to an utterly false conclusion,
but that it leads from statements taken as premises to a statement taken as
[pseudo]conclusion which in fact has nothing to do with the premises. As against that
point of view, Sylvan has been led little by little to the idea that there can be true contra+
o.,,55Ԯdictions, and in fact that there are. Now, that may be the case, and I am in fact sure it
is the case. Yet, in an important sense, this runs against relevantism as initially conceived. For if the Cornubia rule is to be rejected on the ground that a contradiction can
be true after all, the classical view that what is [utterly] impossible implies everything is
not challenged: you only displace the bounds of the impossible. Of course, you can be
both a relevantist and a believer in true contradictions. Still, you are then bound carefully
to sort out your grounds for each of your departures from classical logic, or from any
logic you happen to take as your starting point. Finally, and most significantly for our
present concerns, Sylvan has developed a very different approach to a logic of
reasoning, which is at variance with the classical outlook in a much more radical way
than even mainstream relevantism does. In so doing, he renounces the claim that RL
is in general a logic of reasoning. Yet, canvassing the pros and cons of Sylvan's plan
for a logic of reasoning goes beyond the scope of the present paper.
A third way out is provided by a gradualistic appropriation of the relevantist plan.
`Reappropriation' seems to me the right word, since gradualism and relevantism have
not been bed fellows, their leanings taking them apart from one another. The main idea
of gradualism has little to do with relevantist concerns. In fact it consists of recognizing
that there are degrees of truth and, accordingly, since what is to some extent true is
true, there are true contradictions, but that in so much as the general classical view of
logic can be adapted to such an acceptance of degrees of truth, it can and must remain
unchanged in all other respects. In particular, gradualism has been keen on keeping,
alongside a weaker or natural negation, a strong, classical negation, endowed with the
reading `not8at all', through which in fact systems of gradualistic logic are conservative
extensions of CL, which maintain not only all classical theorems, but also all classical
inference rules (provided the translation of classical negation is strong negation, of
course). Thus, the gradualistic approach relinquishes the Cornubia rule for natural or
weak negation but keeps it for strong or classical negation. However, if you have a
strong negation, you are bound to countenance inferences which fall afoul of the
relevantist constraints. You can no longer pride yourself on being relevant in that sense.
RNot because you avoid
pUNpq if you accept
pUpq, `' being strong negation.
Your logical choice may have a number of reasons to recommend it, but not the general
unqualified principle of relevance.
This state of affairs probably explains why thus far no bridges have been built
between the two schools. Their original motivations kept them quite apart. Gradualism
has remained adamant in its closeness to the classicist's main ideas and in fact it has
been developed with forceful allegiance to an extensionalistic, Quinean approach to
many philosophical subjects. The idea of degrees of truth is compatible with
extensionalism, and in fact is the only ground on which Quine himself has contemplated
R#abandoning CL (in his What Price Bivalence?
, JP, 78/2 [febr 1981], pp.90ff).
However, logic has a number of surprises to offer. One of them is that gradualism is a not so distant relative of relevantism, which is going to become clear
R&through some mending (in fact a powerful strengthening) of system E. Needless to say,
the kind of moderate, middle of the road approach I am going to sketch as the final part
of this paper entails renouncing the unqualified main tenets of the founding fathers and
probably of most relevantist thinkers. A number of inferences which do not conform to
the relevantist constraints have to be accepted. To such extent, the main motivation of
the relevantist movement " to capture a logic of reasoning, in a somehow puritanicalD+o.,,55sense of the word " seems to me hard to retrieve. Yet, something in the neighbourhood emerges, something through which the relevantist enterprise is vindicated all the
same.
^1
4." A Gradualistic Construal of SubscriptAssignment, and how
[Oto Strengthen System E#
The relevantist implementation of natural deduction techniques consists in assigning subscripts to the premises and thus keeping track of the thread of the argument.
Since reasoning is putting forward grounds for getting some conclusion, the procedure
seems quite reasonable. In fact the relevantist logicians didn't need to invent it, since
it had already been designed even within the framework of CL as a didactic tool. It had
only to be put to a more substantial use.
Does the idea work? Well, within the relevantist program only after a fashion.
The trouble comes with Adjunction, as we have seen. The natural way of countenancing
R
Adjunction fails. It could be enacted as a strengthening of system E, but, as we have
seen, that would entail acceptance of Factor and PII. The relevantist logicians offer us
Rrsome makeshifts; for instance
pUqi can be inferred from the couple of premises
pi
R[and
qi, that is to say both premises have to possess the same subscript. Which in
RDpractical terms means that outside logic nothing can be inferred from premises
p and
R-
q, given independently from one another. Any nonlogical theory has to be provided
with only one axiom, which can be a conjunction of formulae, or else nothing can be
inferred from the separate axioms, unless they are cast in terms allowing use of
implicative MP.
What in effect the relevantists are doing is to reduce Adjunction to a systemic
rule, which is to be applied only to such premises as are logical theorems. For a logic
of general reasoning such a step is a policy of despair. With Adjunction so hamstrung
no bright prospect is opened for reasoning.
RaNow, what if we strengthen E through Factor, thus unshackling Adjunction at the
same time? We have seen that the general principle of relevance will no longer be in
R5operation, since we'll have
pq.rr, the PII. But can something of the initial implementation of natural deduction techniques be rescued all the same? Yes, a lot of it can
be saved. But an overhaul is necessary, and a different interpretation is to be put on the
whole assigning of subscripts.
The interpretation to be now considered is that keeping track of the use of the
premises is a guarantee to the effect that the conclusion is not less true than the
premises. That idea is closely connected with a program put forward by Guccione &
Tortora, two Italian logicians working in the field of manyvalued and fuzzy logic. And
R#surprisingly once at least within the relevantist movement " apropos system RM, which,
granted, is no longer a relevant system of logic " Robert K. Meyer developed similar
ideas. With such an overhaul, the target is no longer that of keeping clear of irrelevancies, but that of avoiding an increase in the degree of falseness of your assertions.
So, let us think of subscripts assigned to the premises as variables ranging over
R(degrees of truth or falsehood. The main idea in now that from
pq and
pi you can
R)conclude
qj provided you jot down that ji (the degree of falseness of the conclusion
R*does not exceed that of the premise). What about the very same implication,
pq?
Doesn't it receive a subscript? All asserted implications receive the same subscript.+o.,,55Their degree of truth is immaterial. In fact there are grounds for regarding implication
as twovalued " which does not mean that the two values must necessarily be the two
classical extremes of complete truth and complete falsity.
Thus implications are special. It seems to me this is as it should, even from
the very same relevantist motivations. After all the founding fathers made much of the
cleavage between facts and entailments. Not that I think they were quite right on that
score either, since entailments are entailmental facts; their looking at entailments as
Rnonfacts is perhaps connected with their acceptance of system R as a relevant logic:
if an implication is, if true, a fact, then the permutation principle is hard to believe: even
if p is relevant for the fact that q is relevant for r, it does not follow that q is relevant for
the fact " if it is one " that p is relevant for r. Likewise, even if the authorities'
carelessness causes that the earthquake causes many damages, it does not follow that
the earthquake causes that the carelessness of the authorities causes many damages.
ROnce we accept Factor " without relinquishing any other E principle or rule ",
R
things begin to be straightened out, and a number of oddities in system E vanish. For
Rinstance, with system E you cannot infer
p.pUq from
pq. Yet, in any theory
Rwherein you have, for some formulae
p and
q, both
pp and
pq you'll also
Rmhave
p.pUq " if Adjunction can be applied to those theorems. How is that possible?
The answer is that inference, in the canonic relevant sense, is not the usual consequence operation. A theory's being closed for some operation - [on sets of of formulae]
is neither a necessary nor a sufficient condition for it to be the case that within such a
theory formulae in -S can be inferred from the set of formulae S. Yet, isn't it really odd
that within a theory in which two implications are theorems which share all their atomic
Rformulae and are in fact very similar,
pq and
p.pUq, the latter cannot be inferred
from the former? What else do we need in order to be able to draw the conclusion?
Now, with our overhauled implementation, we have a different situation: any asserted implication in a system can be inferred from any implication (can be inferred
provided it is asserted, of course). As regards implications, our ways are classical. That
does not mean that an asserted implication can be inferred from nothing or from
anything. The Ackermann property keeps sway. And proof theory becomes so much
simple!
Inference, so implemented, is still not necessarily the same as the consequence
operation. A theory may contain theorems less true than true implications are; yet an
implication, if true, does not imply those theorems. On the other hand, a true implication
is not implied by all sentences; so, in particular it is not the case that for any two
R!theorems,
p,
q, pq. But we can (and must) also recognize a different inference
relation which coincides with the consequence operation in the usual, classical sense;
let us say, *. From Sp it follows that S*p, but not conversely. (There are other links
Rv$between * and : if S*p, there is some truth,
q such that SB{q}p.)
While failing to be identical to the consequence operation (*) outright, as now
conceived is much much closer to it, due to the particular status of implications. But it
is not close enough yet. Even with Factor and PII implication is still too distant from our
methodological maxim: Remain as close to the classical model as is compatible with
carrying out your program of a logic of degrees of truth
. We have made implications
classical in some sense by rendering all true implications equally true. But what about
false implications? We'll advance in our classicalizing enterprise by rendering allK+
o.,,55implications which are not true enough to be assertible completely false. Which means
Rthat we countenance the principle of implicative funnel:
pqrV.pq: an implication
is either true or else so false that it implies everything. From a prooftheoretical
viewpoint that means that we split our proofs in two branches: one wherein we suppose
that pq, the other wherein we suppose that pqr; if the same conclusion follows from
both branches, we'll assert it.
But there is a similar consideration as regards another classical principle, compatible with our overhaul of relevant implication as a connective expressing that the
degree of falseness of the apodosis is at most as high as that of the protasis. I am
Rreferring to the principle of linearization,
pqV.qp. Same procedure: we split our
proofs, and look out for the outcome.
^
5." Proving (and Deriving) what in E has to be Taken as
[
Given#
Two striking results follow. One, Adjunction may cease being a primitive rule.
We can countenance this rule as the only inference rule in our Hilbertstyle system: for
Rp1n,
pNqV.pqV.8V.pq,
pN, 8,
p q. When 1=n, it is MP. The rationale
for the rule is that either p implies pandq or else q implies pandq.
Second striking result: a number of interpolation principles may become axiomatic " in alternative presentations of the system " through which some principles acRcepted, so to say, blindly and without justification in system E can be provided with reaRusonable warrant. For instance, E countenances distribution:
pVqUr.pUrV.qUr. Why
is it true? Within the framework of the system we are now considering, its proof is obtained from linearization (and Factor): since the disjunction between p and q either
implies p el else implies q, it is immediate that the conjunction between such disjunction
and r implies either pandr or qandr. More importantly, such (widely challenged, and
Ryet to my mind correct) principles of system E as conjoined assertion (
pqUpq) and
Rcontraction (
p(pq).pq) now become provable: the former is proved from
Rimplicative funnel: we have as a particular case of implicative funnel that
pqqV.Rpq: each of the disjuncts implies the principle of conjoined assertion. Another formula
Rwhich E countenances as an axiom (by force, so to speak "using A&B's own words)
Ris
prU(qr).pVqr. It seems very clear to me that the axiom is not obvious. A
R~natural link is missing, which is provided in our system by
pVqpV.pVqq. Another
Rgprinciple which is not altogether obvious is the principle of conjoined apodoses:
pqURP (pr).p.qUr, which can be proved, too, as a theorem within our system. The most
R9!controversial principles of E thus become theorems and are endowed with enhanced evidence (although to be sure our new axioms are probably as controversial as those of
R
#E).
Rn$In the same way, some suppression principles implicitly accepted in E (as R.
RY%Sylvan has pointed out) through which exported syllogism could be justified (but in E it
is not justified, just taken as an additional primitive and underivable evidence) now can
be taken as axiomatic, thus rendering exported syllogism a proved theorem. For
R(instance an Adjunction principle for implications:
pq.rs.pqU.rs. The
general Adjunction principle is wrong, but we are by now aware that implications are
R)special. That gives us a rationale for those suppressive principles of system E.)o.,,55ԌRWhat constitutes E's weirdness in that connection is that it countenances suppression in exported form but not in imported form; else, Factor would become provable.
RIn particular from
pqU(rr)s the protasis's second conjunct cannot be suppressed.
R3Moreover, some other anomalies of E are cured. For instance in E there is an
Rasymmetry between disjunction and conjunction:
p(qUr)I.pqU.pr (`I' is mutual
Rimplication), but not
p(qVr)I.pqV.pr. Likewise, in E we have
pVqrI.prU.qr
Rbut not
pUqrI.prV.qr. All those equivalences obtain in the system we are
sketching.
^:6." Conclusion
From an orthodox relevantist viewpoint " if there is such a thing " all this enterprise is pointless, for we are doomed to countenance irrelevances. That seems to me
the usual all or nothing
, a bad maxim we had better shy away from. Our middle
course offers a view of reasoning (if you like, of some idealized sort of reasoning) which
Rr
is in between that of classical logic and that of orthodox relevantism. I call it relevantoid
R]logic. It eschews VEQ (
p.qp); it has the Ackermann property. It has (at least
RHwithin the domain of the sentential calculus) the entailment property (
pN, 8,
p
q
R1iff
rr
pNU8Upq). It avoids the Cornubia rule for nonstrong negation. It also
Ravoids unqualified exportation (
pUqr.p.qr). It also avoids the validity of the
RDugundji formulae (for any finite n):
pNIpV.pIpV.8V.p-NIp.
The system just sketched is close enough to the most widely publicized relevant
RKsystem of entailment logic, E, with which, despite the chasm our strengthening has
opened, sufficient closeness remains to allow bridges to be built. (On the other hand,
our system is much closer to classical logic than in fact almost any other nonclassical
logic; to be more specific: we are very close to accepting what the classicist accepts,
but at the same time we are far apart from the classicist attitude as rejection is
concerned: we refrain from rejecting true contradictions, while the classicist wrongly
equates rejecting something with asserting a negation thereof.)
I think this system is a better logic of reasoning. Reasoning as thus implemented is of course somehow artificial. I do not deny that more natural systems can be
found. But naturalness has its price, too. I wonder if some part of the task can be afforded by a pragmatic rounding out of purely inferential logic. But those are matters for
a further inquiry.