A difference between TL and the systems C is that in C functor `#' is defined through negation, `N', and conjunction, `∧', and then strong negation, `¬', is defined with those three functors, whereas in TL either `#' is taken as primitive, or else strong negation is taken as primitive (in which case ^{┌}#p^{┐} is defined as ^{┌}¬p∨¬Np^{┐}), or another primitive functor is introduced, one of strong afirmation, `H', such that ^{┌}¬p^{┐} is then defined as ^{┌}HNp^{┐} and ^{┌}#p^{┐} as (e.g.) ^{┌}H(p∨Np)^{┐} (strong affirmation distributes over conjunction and also over disjunction).
There are other differences between TL and systems C_{n}. In TL some schemata hold which do not hold in C, such as ^{┌}N(p∧Np)^{┐}, ^{┌}p⊃NNp^{┐}, ^{┌}p∨q≡N(Np∧Nq)^{┐}, ^{┌}p∧q≡N(Np∨Nq)^{┐}. Moreover, in TL there exist several primitive functors which do not exist in C, such as: (f1) an equivalential functor, `↔', for which the rule of inference p↔q, r ├ s holds, where ^{┌}r^{┐} differs from ^{┌}s^{┐} only by substituting to one or more occurrencess of ^{┌}p^{┐} as many occurrencess of ^{┌}q^{┐}; (f2) a functor of minimal affirmation, `Y', such that this schema holds: ^{┌}Yq⊃.q∧p↔q≡p^{┐} (or, abbreviating ^{┌}q∧p↔q^{┐} as ^{┌}q→p^{┐}: ^{┌}Yp⊃.q≡.p→q^{┐}); a strong conjunction, `●', such that ^{┌}p●q→.p∧q^{┐}, but not conversely, `●' being endowed with the properties of classical conjunction save idempotence (the schema ^{┌}p●p↔p^{┐} does not hold).
Such complicated pattern of relations has led us to find out a system with all the properties shared by TL and systems C_{m} (for m finite) and to see how it can be strengthened with properties of TL while remaining C-compatible.
Definition: a system S is C_{m}-compatible iff C_{m} is an extension of a system of which S is also an extension and, if the schemata which are theorematic in C_{m} but not in 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.
I shall show below that aystem CT -- to be sketched out in a moment -- is such that both TL and C_{m} (for m<∞) are extensions thereof. (I shall also show that, if we reinforce CT by adding certain functors and theorematic schemata of TL, we get systems some of which are C_{1}-compatible.) My hypothesis (which will not be proved here) is that system CT is the strongest system with such a feature.
System CT
Primitive symbols: ∨, ∧, N, #.
Definitions: ^{┌}Sp^{┐} abbreviates ^{┌}p∧Np^{┐}; ^{┌}¬p^{┐} abbr. ^{┌}#p∧Np^{┐}; ^{┌}p⊃q^{┐} abbr. ^{┌}¬p∨q^{┐}; ^{┌}p≡q^{┐} ^{┌}p⊃q∧.q⊃p^{┐}.
Inference rule: modus ponens (p, p⊃q ├ q)
Axiomatic schemata:
(Export)p∧q⊃r⊃.p⊃.q⊃r
(Transit)p⊃q∧(q⊃r)⊃.p⊃r
(L-simpl.)p∧q⊃p
(R-simpl.)p∧q⊃q
(L-addition)p⊃.p∨q
(R-addition)q⊃.p∨q
(Conv2Neg)NNp⊃p
(Chrisippus)#p∨Sp
(hered)#p∧#q⊃.#(p∨q)∧#(p∧q)∧#Np
(Clas-Clas)##p
(Conj2Disj)p⊃r∧(q⊃r)⊃.p∨q⊃r
(Conj2Conj)p⊃q∧(p⊃r)⊃.p⊃.q∧r
(Notice that this axiomatization is heavily indebted to the one proposed for CL by Prof. Hubert Hubien in his paper [H1].)
This system is stronger than C_{ω} but less strong than C_{m} (for m finite).
Now I am going into some crucial points concerning system CT.
It 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', `∨', `∧' and `⊃' are concerned. Oddly enough he has failed to offer any natural-language 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'.
In CT we easily prove the four following results:
(R1) Chrisippus's principle -- namely ^{┌}#p∨Sp^{┐} -- is, in the presence of the other axiomatic schemata and inference rules, equivalent to ^{┌}N#p⊃Sp^{┐}, i.e. the assertion that whatever fails to stand by classical strictures is contradictory. Proof: first we prove ^{┌}p⊃p^{┐}, hence (by definition) ^{┌}p∨¬p^{┐} and ^{┌}p∨q⊃.q∨p^{┐}. By instantiation we have ^{┌}¬p∨¬¬p^{┐}, hence ^{┌}p⊃¬¬p^{┐}. We thus prove ^{┌}p⊃q⊃.¬q⊃¬p^{┐}. We also prove de Morgan: ^{┌}¬(p∨q)⊃.¬p∧¬q^{┐} and associativity: ^{┌}p∨q∨r⊃.p∨.q∨r^{┐}. Then we prove distributivity: ^{┌}p∨q∧r⊃.p∧r∨.q∧r^{┐} and ^{┌}p⊃.q⊃.p∧q^{┐}. Hence ^{┌}p∨q⊃.¬p⊃q^{┐}. Then from ^{┌}#p∨Sp^{┐} we prove ^{┌}¬#p⊃Sp^{┐}. But since ^{┌}##p^{┐} is theorematic, we have ^{┌}N#p⊃¬#p^{┐}; hence ^{┌}N#p⊃Sp^{┐}. Q.e.d. The converse proof is also straightforward: from ^{┌}N#p⊃Sp^{┐} we get ^{┌}¬N#p∨Sp^{┐}, hence ^{┌}#N#p∧NN#p∨Sp^{┐}, hence ^{┌}NN#p∨Sp^{┐}, hence ^{┌}#p∨Sp^{┐}. Q.e.d.
(R2) CT contains, among others, the following theorem-schemata: ^{┌}p⊃q∨p^{┐} (Funnel), ^{┌}p⊃q⊃p⊃p^{┐} (Peirce), ^{┌}p∧¬p⊃q^{┐} (Cornubia for strong negation). The proof is trivial: with ^{┌}p∧q⊃r⊃.p⊃.q⊃r^{┐} plus ^{┌}p∧q⊃p^{┐} and ^{┌}q∧p⊃p^{┐} prove, first, ^{┌}p⊃.q⊃q^{┐} and hence ^{┌}p⊃p^{┐}; then prove (by definition) ^{┌}p∨¬p^{┐}, hence (thanks to ^{┌}p⊃.p∨q^{┐} and ^{┌}q⊃.p∨q^{┐}) ^{┌}¬¬p∨¬p∨q^{┐}, i.e. e [prorsus] falso quodlibet, namely: ^{┌}¬p⊃.p⊃q^{┐}; whence Funnel follows thanks to ^{┌}p⊃r∧(q⊃r)⊃.p∨q⊃r^{┐} and exportation; now take a particular case of Funnel, namely ^{┌}p⊃q⊃q∨.p⊃q^{┐}. Whence conjunctive assertion (namely ^{┌}p⊃q∧p⊃q^{┐}) follows (again thanks to ^{┌}p⊃r∧(q⊃r)⊃.p∨q⊃r^{┐}). Peirce is proved as follows: by exportation we get (once we have proved ^{┌}p⊃p^{┐}) ^{┌}p⊃q⊃p⊃.p⊃q∨p⊃p^{┐} (again thanks to ^{┌}p⊃r∧(q⊃r)⊃.p∨q⊃r^{┐}) and by transitivity (and Funnel) ^{┌}p⊃q⊃p⊃p^{┐}. Cornubia follows from e prorsus falso quodlibet plus the lemma ^{┌}p⊃(p⊃q)⊃.p⊃q^{┐}, i.e. absorption, which can be easily proved from conjunctive assertion.
(R3) The fragment of CT expresible only with `⊃', `∧', `∨', `¬' is exactly CL. Proof: take any standard presentation of CL and show the equivalence between its set of axioms and that of CT when symbol `N' is omitted. In fact Hubien's axiomatization is but a variant of the well-known axiomatization of Hilbert & Ackermann, which is clearly equivalentent to our positive system of axioms plus classical negation endowed with ^{┌}p⊃q⊃.¬q⊃¬p^{┐}, ^{┌}p⊃¬¬p^{┐} and ^{┌}¬¬p⊃p^{┐}. The three are provable in CT (since ^{┌}p⊃q^{┐} abbreviates ^{┌}¬p∨q^{┐}). Therefore, CT contains CL. The converse can also be proved quite easily, since in CL ^{┌}¬p∨q≡.p⊃q^{┐}. Thus replace the set pf Hilbert & Ackermann's primitive symbols {¬, ⊃, ∨, ∧, ≡} with {¬, ∨} and define `⊃' and `≡'. The three nonpositive axioms then become redundant or idle.
(R4) CT is stronger than C_{ω} (since C_{ω} lacks Peirce and Funnel). Proof: C_{ω} is positive (intuitionistic) logic enlarged with a very weak negation satisfying just ^{┌}NNp⊃p^{┐} and ^{┌}p∨Np^{┐}. CT is of course stronger, since it includes the whole classical positive calculus. (See (R2) above.)
We 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 two-valued but not truth-functional: we can easily devise such a semantic for CT failing to satisfy any one of the following schemata; devising it is left as an exercise to the reader):
(2negation)p⊃NNp
(DeMorgan-1)p∨q⊃N(Np∧Nq)
(DeMorgan-2)p∧q⊃N(Np∨Nq)
(DeMorgan-3)N(p∧q)⊃.Np∨Nq
(DeMorgan-4)N(p∨q)⊃.Np∧Nq
Let LTL, or lean transitive logic, be the fragment of TL expressible with symbols occurring in the CT language. LTL is the result of adding to CT all those five axioms plus the principle of contradiction or Ænesidemus, namely: ^{┌}½≡N½^{┐}, where `½' is a sentential constant with whatever meaning. We can call JTL (jejune transitive logic) the result of adding to CT the just mentioned principles except Ænesidemus. CL is JTL plus the axiomatic schema: ^{┌}#p^{┐}. TL is of course a conservative extension of CL, but it cannot be classically «strengthened» (once Ænesidemus has been added, no classical meaning can be given to `N').
LTL and even JTL are not C_{m}-compatible (for m finite). Proof: with DeMorgan-1 or ^{┌}p∨q⊃N(Np∧Nq)^{┐} we prove a variant of noncontradiction (^{┌}N(Np∧NNp)^{┐}) from the principle of excluded middle, which is theorematic in CT. Thus with the help of 2negation or ^{┌}p⊃NNp^{┐} (the converse of which is theorematic in CT) we prove the general principle of noncontradiction, which of course is incompatible with da Costa's systems (it collapses them into CL).
In fact starting with CT we obtain C_{1} by adding BF -- or the Back-to-the-Fold principle --, which is the converse-Chrisippus principle, namely: ^{┌}NSp⊃#p^{┐} -- what is not contradictory is classical. (Again the proof is trivial but tedious, and is thus left to the reader. Hint: in C_{1} prove all axioms of CT (defining ^{┌}#p^{┐} as ^{┌}NSp^{┐} and thus getting BF quite cheap); then in CT plus BF prove any set of axioms of C_{1}.) In order to obtain C_{2} instead, we add ^{┌}NSp∧NSSp⊃#p^{┐}. In general C_{m} is CT plus ^{┌}NSp∧NSSp∧...∧NS_{m}p⊃#p^{┐}, where `S_{m}' stands for a string of m occurrences of `S'.
(Da Costa's original axiomatization was of course different: with ∧, ⊃, ∨, N as primitive, if we define `S' in such a way that ^{┌}Sp^{┐} abbr ^{┌}p∧Np^{┐} and we have Modus Ponens as the only inference rule, the axioms are: ^{┌}p⊃.q⊃p^{┐}; ^{┌}p⊃q⊃.p⊃(q⊃r)⊃.p⊃r^{┐}; ^{┌}p⊃.q⊃.p∧q^{┐}; ^{┌}p∧q⊃p^{┐}; ^{┌}p∧q⊃q^{┐}; ^{┌}p⊃r⊃.q⊃r⊃.p∨q⊃r^{┐}; ^{┌}p⊃.p∨q^{┐}; ^{┌}q⊃.p∨q^{┐}; ^{┌}NNp⊃p^{┐}; ^{┌}p∨Np^{┐}; ^{┌}NSp⊃.q⊃p⊃.q⊃Np⊃Nq^{┐}; ^{┌}NSp∧NSq⊃.NS(p⊃q)∧NS(p∧q)∧NS(p∨q)∧NSNp^{┐}.)
The main idea behind adding BF -- and of course, philosophically, da Costa's 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 not-p, 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, if contradictions can be true, one of those true contradictions may well be that p-and-not-p 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 not-p, his interlocutors are likely to rejoin: `Then you do not accept the principle of non-contradiction!'. 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 is 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 2negation, DeMorgan-2, DeMorgan-3 and DeMorgan-4 -- in enforcing an important constraint, viz. the confinement of contradictions (not to multiply contradictions beyond absolute necessity): thanks to the noninconsistency assumption -- or equivalently to the BF principle -- (plus the non-endorsement 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 `α', a given theory contains both `α' and `Nα', it is bound to also contain infinitely many different contradictions (^{┌}α∧Nα∧N(α∧Nα)^{┐}, ^{┌}N(α∨Nα)^{┐}, 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 Critical Notice: «Graham Priest's «Dialectheism» -- Is It Althogether True?», SORITES # 7 (November 1996) (ISSN 1135-1349), pp. 28-56). 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. First-level contradictions are bad enough as they are, but adding second-level 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 a case for the claim that climbing up to the C systems is not a whimsical choice.
Let me explain. System CT is classical logic plus a very weak negation endowed with only two principles: converse double negation and excluded middle. CT does not 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 schema ^{┌}Sp⊃q^{┐} (or ^{┌}#p^{┐}). C_{1} is instead obtained by adding ^{┌}SSp⊃q^{┐} (or ^{┌}#Sp^{┐}); C_{2} is obtained by adding ^{┌}SSSp⊃q^{┐} (or ^{┌}#SSp^{┐}); and so on. The idea behind the strengthening into classical logic is that all contradictions -- even first degree contradictions -- are bad and unacceptable; the one behind C_{n} 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 deeper-level contradictions seems baffling: you can admit «p and not p» but not «p and not-p and not (p and not p) and ...». 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 will espouse ^{┌}S...Sp^{┐} for any finite sequence of `S' and reject any of ^{┌}#p^{┐}, ^{┌}#Sp^{┐}, etc. But 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 «p» such that you do not accept ^{┌}Sp^{┐}, there is no valid ground on which you will base a 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 low-level contradictions and no higher-level 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.
Anyway, are there extensions of CT which are C_{m}-compatible (in the technical sense of the term we are using)? There probably are. If to CT we add one among 2negation, DeMorgan-2, DeMorgan-3, DeMorgan-4, the result can probably be shown to remain C_{m}-compatible. 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 add e.g. DeMorgan-2, then in a contradictorial theory wherein, for some particular constant ☼, we have `☼∧N☼', DeMorgan-2 will yield `N(N☼∨NN☼)', 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 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 not-p' ought to behave classically: if admitted as true, its negation must be thoroughly rejected as purely and unmixedly false.
Not only is it possible to obtain C_{m}-compatible systems by strengthening CT with 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 time, -- again with the exception of DeMorgan-1 -- without the resulting system losing its C_{m}-compatibility.
While a study of da Costa's preferred semantical account of his systems through the method of two-valueed non-truthfunctional 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 (^{┌}p⊃NNp^{┐}) requires that, for every valuation v, v(p)=v(NNp).
Our examination of a different way of setting up systems C_{m} shows that da Costa's whole logical enterprise -- as carried out in the construction of the C systems -- 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 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 framework of da Costa's original presentation of his systems. We now see that TL and the 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 well-behavedness), which we have written as `#', enjoying the expected properties (hereditariness, Chrisippus, and the classicality of classicality-judgments -- or what, from a gradualistic viewpoint [not da Costa's] can be termed the two-valuedness of two-valuedness-attributions, i.e. whether a situation is classical or not is a classical matter).
References