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#XpiP;rEXP#hh#X| p.78X#v*YoESUKӊ=jCB C1Compatible Transitive Extensions of System CT
by Lorenzo Pe9a`r"#Hă
y]dddy
*
[C1ĩCompatible Transitive Extensions of System CT
RO!Qvv Logique et Analyse
N 161162163 (1998)
%pp. 135143
Pn$~Lorenzo Pe9a
R CSIC Institute of Philosophy
'Madrid
*
RK*
Da Costa's paraconsistent systems of the series Cm (for finite m) (see [C1], [C2],
R4and esp. [C3], pp. 237ff.) share important features with transitive logic, TL (which has
been gone into in [P1] and [P2]), namely, they all coincide in that: (c1) they possess a
strong negation, `', a conditional, `D', a conjunction, `U', and a disjunction, `V', with
Rrespect to which they are conservative extensions of CL or Classical Logic; (c2) they
Rpossess a nonstrong negation, `N' (notations are different for systems C) which does
not possess all properties of classical negation, but for which the following schemata are
theorematic (I use the letters `p', `q', etc as schematic letters; my notational conventions
are basically Church's: associativity leftwards; a dot stands for a left parenthesis with
Rits mate as far to the right as posible):
pVNp,
NNpDp,
pDNpDNp; (c3) they
possess a monadic functor, `#', for which the following schemata are theorematic:
RR
N#pD.pUNp,
#pU#qD.#(pVq)U#(pUq)U#(pDq)U#Np; (c4) they are almost unique
anong paraconsistent logics in their having the three aforementioned features. (In
R$general systems with features of that sort have been called `extensional paraconsistent
R
logics' by Diderik Batens, who has also proposed systems bearing a kinship of sorts to
those " even though they lack strong negation and a classicality operator, they can be
easily extended in that way; see [B1].)
R>A difference between TL and the systems C is that in C functor `#' is defined
through negation, `N', and conjunction, `U', and then strong negation, `', is defined with
R!those three functors, whereas in TL either `#' is taken as primitive, or else strong
R!negation is taken as primitive (in which case
#p is defined as
pVNp), or another
R"primitive functor is introduced, one of strong afirmation, `H', such that
p is then
R#defined as
HNp and
#p as (e.g.)
H(pVNp) (strong affirmation distributes over
conjunction and also over disjunction).
R&There are other differences between TL and systems Cn. In TL some schemata
R'hold which do not hold in C, such as
N(pUNp),
pDNNp,
pVqN(NpUNq),
R'
pUqN(NpVNq). Moreover, in TL there exist several primitive functors which do not
R(exist in C, such as: (f1) an equivalential functor, `', for which the rule of inference
R)pq, r s holds, where
r differs from
s only by substituting to one or more
R*occurrencess of
p as many occurrencess of
q; (f2) a functor of minimal affirmation,
R+`Y', such that this schema holds:
YqD.qUpqp (or, abbreviating
qUpq as
qp:
Rz,
YpD.q.pq); a strong conjunction, `', such that
pq.pUq, but not conversely, `'z,=o.o.o.being endowed with the properties of classical conjunction save idempotence (the
Rschema
ppp does not hold).
Such complicated pattern of relations has led us to find out a system with all the
R1properties shared by TL and systems Cm (for m finite) and to see how it can be
Rstrengthened with properties of TL while remaining Cĩcompatible.
R}Definition: a system S is Cmĩcompatible iff Cm is an extension of a system of
Rfwhich S is also an extension and, if the schemata which are theorematic in Cm but not
ROin S are added to S, the result is a paraconsistent system (for negation `N'), i.e. it does
not have the inference rule: p, Np q.
R I shall show below that aystem CT " to be sketched out in a moment " is
R
such that both TL and Cm (for m<) are extensions thereof. (I shall also show that, if
Rkwe reinforce CT by adding certain functors and theorematic schemata of TL, we get
RVsystems some of which are C1ĩcompatible.) My hypothesis (which will not be proved
R?
here) is that system CT is the strongest system with such a feature.
R&System CTă
Primitive symbols: V, U, N, #.
R^Definitions:
Sp abbreviates
pUNp;
p abbr.
#pUNp;
pDq abbr.
pVq;
pq
RG
pDqU.qDp.
RInference rule: modus ponens (p, pDq q)
R/ *!Axiomatic schemata:YoY#Ѓ
Rf(Export)pUqDrD.pD.qDr
R(Transit)pDqU(qDr)D.pDr
R((L-simpl.)pUqDp
R(R-simpl.)pUqDq
R(L-addition)pD.pVq
RK(R-addition)qD.pVq/
R(Conv2Neg)NNpDp
R
(Chrisippus)#pVSp
Rn!(hered)#pU#qD.#(pVq)U#(pUq)U#Np
R"(Clas-Clas)##p
R0$(Conj2Disj)pDrU(qDr)D.pVqDr
R%(Conj2Conj)pDqU(pDr)D.pD.qUr
R&(Notice that this axiomatization is heavily indebted to the one proposed for CL
by Prof. Hubert Hubien in his paper [H1].)
R<)This system is stronger than C3 but less strong than Cm (for m finite).
R*Now I am going into some crucial points concerning system CT.*o.,,55ԌIt seems to be in order to propose readings of our symbols. Not that da Costa
has always bothered to provide us with such readings; more often than not he hasn't,
except in so far as `N', `V', `U' and `D' are concerned. Oddly enough he has failed to
Roffer any naturallanguage reading for either `(m)' (our `#') or `', or strong negation. #p
can be read as It is a classical matter whether or not p
, where a classical matter is
a disjunction between two entirely opposite situations each of which either completely
holds or else does not hold at all. Likewise p
can be read as It is not the case that
p at all
; Hp
as It is fully the case that
. `' is read `if, and only if'.
RIn CT we easily prove the four following results:
R (R1) Chrisippus's principle " namely
#pVSp " is, in the presence of the other
R
axiomatic schemata and inference rules, equivalent to
N#pDSp, i.e. the assertion that
whatever fails to stand by classical strictures is contradictory. Proof: first we prove
R
pDp, hence (by definition)
pVp and
pVqD.qVp. By instantiation we have
RԚ
pVp, hence
pDp. We thus prove
pDqD.qDp. We also prove de Morgan:
R
(pVq)D.pUq and associativity:
pVqVrD.pV.qVr. Then we prove distributivity:
R
pVqUrD.pUrV.qUr and
pD.qD.pUq. Hence
pVqD.pDq. Then from
#pVSp we
R|prove
#pDSp. But since
##p is theorematic, we have
N#pD#p; hence
Re
N#pDSp. Q.e.d. The converse proof is also straightforward: from
N#pDSp we get
RN
N#pVSp, hence
#N#pUNN#pVSp, hence
NN#pVSp, hence
#pVSp. Q.e.d.
R(R2) CT contains, among others, the following theoremschemata:
pDqVp
R(Funnel),
pDqDpDp (Peirce),
pUpDq (Cornubia for strong negation). The proof is
Rtrivial: with
pUqDrD.pD.qDr plus
pUqDp and
qUpDp prove, first,
pD.qDq and
Rlhence
pDp; then prove (by definition)
pVp, hence (thanks to
pD.pVq and
RU
qD.pVq)
pVpVq, i.e. e [prorsus] falso quodlibet, namely:
pD.pDq; whence
R>Funnel follows thanks to
pDrU(qDr)D.pVqDr and exportation; now take a particular
R'case of Funnel, namely
pDqDqV.pDq. Whence conjunctive assertion (namely
R
pDqUpDq) follows (again thanks to
pDrU(qDr)D.pVqDr). Peirce is proved as follows:
Rby exportation we get (once we have proved
pDp)
pDqDpD.pDqVpDp (again thanks
Rto
pDrU(qDr)D.pVqDr) and by transitivity (and Funnel)
pDqDpDp. Cornubia follows
Rfrom e prorsus falso quodlibet plus the lemma
pD(pDq)D.pDq, i.e. absorption, which
can be easily proved from conjunctive assertion.
R(R3) The fragment of CT expresible only with `D', `U', `V', `' is exactly CL.
RProof: take any standard presentation of CL and show the equivalence between its set
Rof axioms and that of CT when symbol `N' is omitted. In fact Hubien's axiomatization is
but a variant of the wellknown axiomatization of Hilbert & Ackermann, which is clearly
equivalentent to our positive system of axioms plus classical negation endowed with
R"
pDqD.qDp,
pDp and
pDp. The three are provable in CT (since
pDq
R#abbreviates
pVq). Therefore, CT contains CL. The converse can also be proved quite
R|$easily, since in CL
pVq.pDq. Thus replace the set pf Hilbert & Ackermann's
primitive symbols {, D, V, U, } with {, V} and define `D' and `'. The three nonpositive
axioms then become redundant or idle.
R'(R4) CT is stronger than C3 (since C3 lacks Peirce and Funnel). Proof: C3 is
R(positive (intuitionistic) logic enlarged with a very weak negation satisfying just
NNpDR)p and
pVNp. CT is of course stronger, since it includes the whole classical positive
calculus. (See (R2) above.)n*o.,,55ԌRWe can strengthen CT by adding one or several among the following principles.
(That by so doing we obtain proper strengthenings can be shown through a da Costa's
valuation semantics, which is twovalued but not truthfunctional: we can easily devise
Rsuch a semantic for CT failing to satisfy any one of the following schemata; devising it
is left as an exercise to the reader):
R(2negation)pDNNp
Rh(DeMorgan-1)pVqDN(NpUNq)
R(DeMorgan-2)pUqDN(NpVNq)
R* (DeMorgan-3)N(pUq)D.NpVNq
R
(DeMorgan-4)N(pVq)D.NpUNq
RLet LTL, or lean transitive logic, be the fragment of TL expressible with symbols
Roccurring in the CT language. LTL is the result of adding to CT all those five axioms
R
plus the principle of contradiction or $nesidemus, namely:
N, where `' is a
Rsentential constant with whatever meaning. We can call JTL (jejune transitive logic) the
Rresult of adding to CT the just mentioned principles except $nesidemus. CL is JTL plus
Rthe axiomatic schema:
#p. TL is of course a conservative extension of CL, but it
cannot be classically strengthened
(once $nesidemus has been added, no classical
meaning can be given to `N').
RLTL and even JTL are not Cmĩcompatible (for m finite). Proof: with DeMorgan-1
Ror
pVqDN(NpUNq) we prove a variant of noncontradiction (
N(NpUNNp)) from the
Rprinciple of excluded middle, which is theorematic in CT. Thus with the help of
Ru2negation or
pDNNp (the converse of which is theorematic in CT) we prove the
general principle of noncontradiction, which of course is incompatible with da Costa's
RIsystems (it collapses them into CL).
RIn fact starting with CT we obtain C1 by adding BF " or the BacktotheFold
Rprinciple ", which is the converseChrisippus principle, namely:
NSpD#p " what is
not contradictory is classical. (Again the proof is trivial but tedious, and is thus left to
Rgthe reader. Hint: in C1 prove all axioms of CT (defining
#p as
NSp and thus getting
RRBF quite cheap); then in CT plus BF prove any set of axioms of C1.) In order to obtain
R=C2 instead, we add
NSpUNSSpD#p. In general Cm is CT plus
NSpUNSSpU8UNSmpDR(#p, where `Sm' stands for a string of m occurrences of `S'.
(Da Costa's original axiomatization was of course different: with U, D, V, N as
Rp!primitive, if we define `S' in such a way that
Sp abbr
pUNp and we have Modus
RY"Ponens as the only inference rule, the axioms are:
pD.qDp;
ĚpDqD.pD(qDr)D.pDr;
RB#
pD.qD.pUq;
pUqDp;
pUqDq;
ĚpDrD.qDrD.pVqDr;
pD.pVq;
qD.pVq;
NNpDp;
R+$
pVNp;
ĚNSpD.qDpD.qDNpDNq;
NSpUNSqD.NS(pDq)UNS(pUq)UNS(pVq)UNSNp.)
R%The main idea behind adding BF " and of course, philosophically, da Costa's
Ru&chief motivation " is the noninconsistency assumption, namely that denying a contradiction entails accepting that the situation therein involved is a classical one. In other
words, if and when it is not the case that both p and notp, then p is a classical
situation. Whatever is noncontradictory is classical.
The noninconsistency assumption has of course been questioned by many other
paraconsistent logicians " including the present author ", who have argued that, ifx+o.,,55contradictions can be true, one of those true contradictions may well be that pandnotp
both obtains and does not obtain.
Yet da Costa's approach enjoys two significant characteristics, or perhaps
advantages. The first one is that, when somebody claims, for a certain particular
situation, p, to accept both p and notp, his interlocutors are likely to rejoin: `Then you
do not accept the principle of noncontradiction!'. Needless to say, other paraconsistent
schools regard such a rejoinder as stemming from a classicist confusion " mistaking
`not to accept s' for `to accept not-s', or `to accept r', if s=Nr. Even so, da Costa's point
Ris not entirely devoid of prima facie plausibility. That constitutes the first advantage of
the approach implemented in the C systems.
Moreover " and this constitutes the second advantage of da Costa's preferred
approach ", the noninconsistency assumption succeeds " in the absence of
R2negation, DeMorgan-2, DeMorgan-3 and DeMorgan-4 " in enforcing an important
Rconstraint, viz. the confinement of contradictions (not to multiply contradictions beyond
absolute necessity): thanks to the noninconsistency assumption " or equivalently to the
RBF principle " (plus the nonendorsement of involutivity and De Morgan), a given
contradiction not only fails to render the theory deliquescent (trivial) but also fails to
trigger an infinite chain of further contradictions, whereas, upon other paraconsistent
underlying logics (such as a relevant or a transitive logic), once, for a certain constant
`', a given theory contains both `' and `N', it is bound to also contain infinitely many
R$different contradictions (
UNUN(UN),
N(VN), etc).
Now, da Costa's is not the only paraconsistent approach to have implemented
the confinement constraint. In fact the late Richard Sylvan's approach often (although
perhaps not always) leaned towards some sort of containment policy; but especially
Graham Priest's approach is arguably a containment view (see the present writer's
R'Critical Notice: Graham Priest's Dialectheism " Is It Althogether True?
, SORITES
# 7 (November 1996) (ISSN 1135-1349), pp. 2856). Admittedly, those other approaches
put the containment constraint to serve different purposes.
Both advantages (if they are such) are of course closely related. Probably what
is implicitly assumed by those interlocutors who equate asserting a contradiction with
denying (and in fact rejecting) the principle of noncontradiction is that nobody is so
unreasonable as to both swallow a contradiction and yet also espouse the very same
denial of that contradiction. Contradictions are assumed to be bad and even irrational.
Firstlevel contradictions are bad enough as they are, but adding secondlevel
contradictions and so on is still more irrational. Now, all approaches implementing the
confinement constraint somehow or other assume as much " namely that contradictions are bad and thus not to be endorsed except as an extreme measure, when nothing
else works to solve a difficulty, and even so perhaps only temporarily.
Whatever our final views on such a debate (and my own opinion is that, infinite
chains being harmless, no serious mishap ensues from advocating both noncontradiction and also certain contradictions), the present discussion (or digression) makes out
R'a case for the claim that climbing up to the C systems is not a whimsical choice.
R)Let me explain. System CT is classical logic plus a very weak negation endowed
R)with only two principles: converse double negation and excluded middle. CT does not
R*prejudge any additional principles as regards negation. In fact CT can be strengthened
into classical logic (thus collapsing `N' into a notational variation of `') by adding the+o.,,55Rschema
SpDq (or
#p). C1 is instead obtained by adding
SSpDq (or
#Sp); C2 is
Robtained by adding
SSSpDq (or
#SSp); and so on. The idea behind the strengthening into classical logic is that all contradictions " even first degree contradictions " are
Rbad and unacceptable; the one behind Cn is that contradictions of (n+1)th degree are
bad; and one (perhaps the) reason that can be adduced for that is that an infinite chain
of deeperlevel contradictions seems baffling: you can admit p and not p
but not p
and notp and not (p and not p) and 8
. At some level or other you are bound to stop,
or else nobody will really understand what your point amounts to. Perhaps that level is
not the first level, but it must be some finite level or other.
On the other hand, instead of climbing to any of those systems, you can choose
to accept contradictions of any level of complexity. Then, for some particular p
, you
Ry
will espouse
S8Sp for any finite sequence of `S' and reject any of
#p,
#Sp, etc.
RbBut then by the same token you will also accept
NSp,
NSSp, etc, that is to say all
corresponding instances of the principle of noncontradiction. Now, for all other formulae
R4
p
such that you do not accept
Sp, there is no valid ground on which you will base
Ra rejection of
NSp. Thus, for every p
you will then accept
NSp. Which means that
then you accept the principle of noncontradiction.
Those are two legitimate, plausible options: either (1) only some lowlevel
contradictions and no higherlevel contradiction, and no general principle of noncontradiction; or else (2) contradictions of every level plus the principle of noncontradiction.
(The classicist's choice is the former, with admission of contradictions of 0 level only,
i.e. no contradictions at all.)
Even though I personally happen to think that the latter choice is better, more
elegant, I nevertheless acknowledge the rationality and the motivation of da Costa's own
choice.
RAnyway, are there extensions of CT which are Cmĩcompatible (in the technical
Rsense of the term we are using)? There probably are. If to CT we add one among
Ro2negation, DeMorgan-2, DeMorgan-3, DeMorgan-4, the result can probably be shown
RZto remain Cmĩcompatible. But then why has da Costa kept clear of them all, thus
impoverishing his weak negation beyond necessity? The probable reason is that, if you
R,add e.g. DeMorgan-2, then in a contradictorial theory wherein, for some particular
Rconstant , we have `UN', DeMorgan-2 will yield `N(NVNN)', i.e. a negation of an
instance of excluded middle. And da Costa tries to confine (unavoidable) contradictions
to atomic sentences, as far as possible. In some of his systems every nonatomic
formula must be classical; that's not always the case as regards the modelizations of
R!his main systems of the C series, though; but even so, he clearly leans towards taking
(most) nonatomic formulae to stand by classical standards " at least once disjunction
has been entered " and thus to thinking that, whatever the behaviour of p " classical
or not ", `p or notp' ought to behave classically: if admitted as true, its negation must
be thoroughly rejected as purely and unmixedly false.
R&Not only is it possible to obtain Cmĩcompatible systems by strengthening CT with
R'at least one among double negation and the DeMorgan principles (except DeMorgan-1),
but, what is more, apparently all those principles can be added together, at the same
R)time, " again with the exception of DeMorgan-1 " without the resulting system losing
Rj*its Cmĩcompatibility.j*o.,,55ԌWhile a study of da Costa's preferred semantical account of his systems through
the method of twovalueed nontruthfunctional valuations invented by da Costa (and
developed by I. Arruda, E. Alves and others) needn't concern us here, adapting the
technique to the envisaged enrichments is rather straightforward. Thus double negation
R(
pDNNp) requires that, for every valuation v, v(p)=v(NNp).
ROur examination of a different way of setting up systems Cm shows that da
Costa's whole logical enterprise " as carried out in the construction of the C systems
R" must not be reduced to espousing BF; that you can perfectly well reject BF while
keeping many of the programmatic points implemented in the C systems; that your path
and da Costa's can bifurcate without your being bound to part company with his
R orientation right from the start. Even without BF a lot of the significance and usefulness
of [something close to] the C systems remains.
Thus, our main result has been to clarify the true relations between Transitive
Logic and da Costa's C systems, a clarification which was hard to attain within the
R
framework of da Costa's original presentation of his systems. We now see that TL and
Rthe C systems are built up on an underlying common ground, system CT, i.e. classical
logic plus: (a1) a weak nonclassical negation enjoying at least converse double negation
and excluded middle; and (a2) a symbol for classicality (or classical wellbehavedness),
which we have written as `#', enjoying the expected properties (hereditariness,
Chrisippus, and the classicality of classicalityjudgments " or what, from a gradualistic
R(viewpoint [not da Costa's] can be termed the twovaluedness of twovaluednessRattributions, i.e. whether a situation is classical or not is a classical matter).
[pp%Referencesă
R
[B1] Diderik Batens, Paraconsistent Extensional Propositional Logics
, Logique et
RAnalyse N 9091 (1980), pp. 195234.#
[C1] Newton C.A. da Costa, Calculs de pr)dicats pour les syst/mes formels
R6inconsistants
, Comptes Rendus de l'Acad)mie des Sciences de Paris, T.
257 (1963), pp. 37903792.#
R
[C2] Newton C.A. da Costa, On the Theory of Inconsistent Formal Systems
, Notre
RkDame Journal of Formal Logic, XV/4 (1974), pp. 497510.#
R
[C3] Newton C.A. da Costa, Ensaio sobre os fundamentos da l;gica. SMo Paulo:
Hucitec, 1980.#
R!
[H1] Hubert Hubien, A New Basis for Classical Propositional Calculus
, Logique et
R"Analyse 79 (1977), pp. 2257.#
Rb#
[M1] C. Mortensen, Every Quotient Algebra for C1 is Trivial
, Notre Dame Journal of
RM$Formal Logic, vol XXI (1980), pp. 694700.#
[P1] Lorenzo Pe9a, Contradictions and Paradigms: A Paraconsistent Approach
, in
P&Cultural Relativism and Philosophy: North and Latin American PerspecR'tives, ed, by Marcelo Dascal. Leiden & New York: E.J. Brill, 1991, pp. 2956.#
R(
[P2] Lorenzo Pe9a, Rudimentos de l;gica matemtica. Madrid: Servicio de Publicaciones del CSIC, 1991. Pp. vi+324.#