(i) (c) or (a1), (ii) (d) or (a2), (iii) (i) or (a3).
Henceforth any logic resulting by adjunction of one or several of the axioms (a0), (a1), (a2), (a3) to an extension ofbht? or ofbωdcnwill be labeled as the logic resulting of this extension by adjunction of the corresponding axiom(s) (n), (c), (d), (i); the same for the extensions of bit? not satisfying ET is allowed only in the case of (a2) and (d); this is due to the fact that each of (a0), (a1) and (a3) requires from the well-behaved formulas the satisfaction, respectively, of (m0), (m1) or (m3), among which each imposes a touch of classical well-behavior, what is the case neither of (m2) nor of any of the stability axioms;
these last remain neutral in their effect on the properties of well-behavior.
Remark 5.2.1.4. A queer point about the just contemplated concrete para-consistent logics is the following: it is perilous for paraconsistency that the re-placement theorem holds for the paraconsistent negation,i.e. thatN A↔N B would have to be provable ifA↔B is provable. On another side, one do not see why◦A ↔ ◦B would have to be provable in the same circumstance. And yet, ifA↔B is provable, (N A∧◦A)↔(N B∧◦B) has to be provable.
5.2.2 Some paraconsistent logics with classical strong negation.
Theorem 5.2.2.1. Sincebhdc, sincebhgm, and forn >1,sincebωdcn,the strong negation is classical.
Proof. We will proceed by a series of lemmas.
Fact 5.2.2.2: The schema ((A∨(N A∧◦A))↔(A∨N A)∧(A∨◦A)) is formally provable in Pil.
Corollary 5.2.2.3. The schema (A∨ ∼A)↔(A∨◦A) is formally provable in Uhland so in its extensionbhC.
Now, the theorem follows at once from the
Lemma 5.2.2.4. The schemesN A∨◦AandA∨◦Aare formally provable since bhdc, sincebhgm, and forn >1,sincebωdcn.
Proof. We check this by building a suitable closed tableau, which begins by the proposal, say, of A∨◦A as not valid, and continue by adding successively the separate proposals of A and ◦A as not valid. Note here that if a further proposal of a formula of the form B → C as not valid is exploited, then the proposal of the formula A as not valid will be removed, while it is needed for obtaining closure57. On the other hand, a proposal of a formula of the formN B as not valid can be exploited, provided that this should be done by using the classicalrule. Now, according to an auxiliary special concrete rule, the proposal as valid of the formulaAfollows, leading to closure. In this reasoning, the initial proposal of A as not valid can be replaced by a proposal of N A as not valid, since the special rule entails also the proposal of that formula as valid.
Corollary 5.2.2.5. ∼∼A→Aand DF are provable in all extensions ofbωdcn, forn >1,ofbhdcand ofbhgm.
Indeed, for seeing that, we have just to apply the transcription of the corol-lary 3.4.1.5 to the two last lemmas.
Corollary 5.2.2.6. For everyn >0, the logicsbωdcn and bdcn coincide, as coincide the logicsbhgmandbgm.
5.2.3 The power of the transmission of consistency by passing to negation.
In fact, we can say more:
57Here is the blocking point of the reasoning, in case where (ws) is used.
Theorem 5.2.3.1. The schema (m0) isprovable in all extensions ofahnt?. Proof58. Beginning by the cases of the logics to which the lemma 5.2.2.4 apply and which do not count (m0) amongst their axioms, we notice thatA→(m0) is a thesis of any such logic, as being an instance of an axiom of Pil,and recall that
◦A→(m0) also is a thesis, in virtue of 5.1.6 (i); whence (m0) also is provable.
Now we turn towards the case ofbhws, and proceed by a series of lemmas of independent interest.
Fact 5.2.3.2: The schemes N N A→◦AandN A∨◦Aare formally provable in bhws.
Proof. The formula N N A → (A → N N A) is an instance of an axiom of Pil, and so N N A → ◦A follows in this logic from the hypothesis ws, i.e.
(A → N N A) → ◦A . Now, a formal proof of N A∨◦A can be obtained by substitutingN AforAand◦AforB in the following
Fact 5.2.3.3: The schema (A∨N A)→ ((N A →B)→ (A∨B)) is formally provable in Pil.
The following step consists in demonstrating the strong peculiarity of frag-ments of PCL1 which follows.
Lemma 5.2.3.4. The schema T11 ◦N A is formally provable inahnws. Proof. By using now (n), i.e. ◦A → ◦N A , we obtain, in a way similar to what precedes, a formal proof of N N A → ◦N A , whence a formal proof of (N N A∨◦N A)→◦N A .But (N N A∨◦N A) is an instance of a schema which is formally provable according to the fact 5.2.3.2; then,◦N Ais formally provable.
Now, we reach the asserted theorem by using the schema (an) from the facts 4.4.1 which allows us to obtain a formal proof of (m0).
Corollary 5.2.3.5. The logicsahnt?and bωt?coincide59.
5.2.4 The power of the transmission of consistency by passing to conjunction.
In sum, taking in account that the formula (na) of theorem 5.1.3 is a thesis of all concerned systems, what we proved is that (n)↔(m0) is a thesis ofbht?.
In this respect, the situation of (m1) is similar: (c)↔(m1) is a thesis of bht?,taking this time in account the formula (ca) and the following
Lemma 5.2.4.1. The schema (m1) isprovable in all extensions ofahct?. Proof60. Assume (c). Then◦A→(◦B→(m1)) becomes provable, owing to the provability of the formula (ac) of theorem 4.4.1. Now note that N A → (m1) andN B→(m1) are provable in Pil,as thus such isN A→(◦B→(m1)) ; then
58In a manuscript for the first chapter of a projectedIntroduction to Paraconsistent Logic, J.Y. B´eziau made use of this result in order to give a very simple Hilbert’s style axiomatization of C1.
59Also with aωnt?,but this last coincide with the preceding ones, by using the formula (na) of theorem 5.1.3.
60Da Costa and Guillaume [1965].
from the provability of N A∨◦A follows that ◦B → (m1) is provable; finally, repeat that argumentation.
5.2.5 More on the aimed logics.
We are now in position to say more on the links between any concrete full paraconsistent logic and its aimed logic, which obey the following
Theorem 5.2.5.1. Letp∗ia be a concretefull paraconsistent logic admitting of a tableaux version tp∗ia, and letD be the formula defining the meaning of the consistency connective inp∗ia in terms of occurrences of some metavariable Aas prime components and of the usual (not paraconsistent) connectives.
Assume moreover that the logicp∗ia is a fragment of a not paraconsistent logic N∗lwhich provesD and admits of acomplete tableaux version tN∗l.
Then, N∗l is the aimed logic of p∗ia iff any ground rule-making axiom of which every instance is provable in N∗ladmits of some version of which every instance is provable in p∗ia.
Caution. The consistency connective, in principle, doesnot belong to the lan-guage which is proper to not paraconsistent logics as Pil, N∗land the aimed logic A∗lofp∗ia.Hence, any possible occurrence of a (sub)formuladominated by a consistency connective functions, and must be counted, as aprime component of any formula of one of these logics in which it appears.
Having said this, I can offer here, for the theorem 5.2.5.1, the followingsketch of a proof.
First of all, notice too that to say that a formulaF is a thesis of the aimed logic does not mean the same as to say that a formula is a thesis of the other logics which are here involved: we lack of any notion of formal proof for the aimed logic. Let us prefer to say that F is accepted in the aimed logic for expressing that F fulfils the criterion which is required for that, to wit, that
∧◦[F]→F is a thesis ofp∗ia,where∧◦[F] is meant to denote the conjunction of the certificates of consistency of the prime components ofF .
As N∗l is assumed to extend p∗ia, the formula ∧◦[F] → F will also be a thesis in it, as are also the instances of (Ia) corresponding to the prime compo-nents ofF and the corresponding instances ofD ,so that, bymodus ponens,F itself is a thesis of N∗l: then,every formula accepted by A∗lis a thesis of N∗l. Then, to prove 5.2.5.1 is reduced to prove the converse, which ensues, by taking Γ void and ∆ reduced toF , as an easy corollary from the following Theorem 5.2.5.2. Assume that the logicsp∗ia and N∗lfulfil the assumptions of the theorem 5.2.5.1. For every sequent Γ `∆ of N∗l, let ◦[Γ∆] be the set of the certificates of consistency of all the prime components of the formulas occurring in this sequent.
Then, the sequent◦[Γ∆],Γ `∆ holds inp∗ia iff the sequent Γ`∆ holds in N∗l.
First, let usprove that the condition isnecessary61. Indeed, as N∗lextends p∗ia,if the sequent◦[Γ∆],Γ`∆ holds in the second, then it holds in the first, in which all formulas of◦[Γ∆] are provable.
Now, I claim that the condition issufficient. I sketch aproof.
Suppose that the sequent Γ`∆ holds in N∗l. We proceed step by step.
First, since the tableaux version of N∗l is complete, we can build a closed tableau of tN∗l,started by cheking this sequent.
Second, by applying systematically the methods said “of justification” of the not basic intuitionistic rules described in section 3, modified just by agreeing to put the added proposals as valid of instances of rule-making axioms at the first places of the left column at the root of the tableau (and if needed of a variableX not occuring in Γ∪∆ at the first place of the right column at the root), we can transform this first tableau in a kept closed tableau of tPil, which could have been built in conformity with a correct application of the rules as started by checking the sequentR[Σ(Γ∆)],Γ`(X),∆,whereR[Σ(Γ∆)] denotes a set of instances of not basic intuitionistic rule-making axioms, built out of subformulas of formulas of Γ∪∆, and where (X) denotes the set reduced to X if it was made recourse to an instance of EF, or the void set otherwise.
Third, we form a set cR[Σ(Γ∆)] from the set R[Σ(Γ∆)] by applying, to each formula of this last, the transformation which consists in restablishing the prefixation(s) of the implication(s) by a certificate of consistency, required by the version, which holds inp∗ia,of the ground rule-making axiom of which this formula is an instance. In the same time, we collect the reintroduced certificates, which at the end form the set◦Σ(Γ∆),and add them, as also the formulas of cR[Σ(Γ∆)],at the beginning of the list of the proposals of formulas as valid, in the left column at the root of the tableau.
We cannot come against an obstruction here: in virtue of the assumptions made in the statement of the theorem 5.2.5.1, we can always find, amongst the rule-making axioms holding in p∗ia, some more or less controlled version of each ground rule-making axiom holding in N∗l.
Thus we have transformed the second tableau in a kept closed tableau of tPil, which could have been built in conformity with a correct application of the rules as started by checking the sequent◦Σ(Γ∆), cR[Σ(Γ∆)],Γ`(X), ∆. Indeed, the proposal as valid of every formula belonging toR[Σ(Γ∆)] can now be seen as resulting from an application of the auxiliary derived rule 1 in virtue of the proposals as valid of the corresponding formulas ofcR[Σ(Γ∆)] and◦Σ(Γ∆). Fourth, we can now transform the just obtained third tableau in a still closed tableau, this time of tp∗ia,and which could have been built, in confor-mity with a correct application of the rules, as started by checking the sequent
◦Σ(Γ∆),Γ`∆.This can be made by erasing the formulas ofcR[Σ(Γ∆)],those ofR[Σ(Γ∆)] and those of their subformulas viewed, in conformity with the
ex-61After Newton da Costa [1963].
planations of section 3, as transiently written in order to justify the applications of the corresponding rules by their imagined presence.
As it was seen in section 3, none of these processes did affect the pairs of occurrences of formulas proposed both as valid and as not valid in one and the same branch of the initial tableau, which so transmits its closure to its successive transforms, until the last, of which the closure proves that the sequent
◦Σ(Γ∆),Γ`∆ holds inp∗ia.
Fifth, asp∗ia is assumed to befull, the sequent◦[Γ∆]`◦Σ(Γ∆) holds in it, and from the two lastly established facts, the desired conclusion follows, Q.E.D.
Now, suppose thatp∗ia is widest. Then, the same holds for N∗l, and the only open possibilities are that N∗lis either Filor Fcl.We are led this way to the two following results.
Theorem 5.2.5.3. Filis the aimed logic of ait?n (or more genrally of faiia if the formulaDin (ia) is a thesis of Fil).
Theorem 5.2.5.4. Fclis the aimed logic ofaht? and ofaωdcn forn >1. 5.2.6 The main original features of PCL1.
As it was already shown by the lemma 5.2.3.4, by comparison with da Costa’s initial views, what happens through the development of PCL1 makes abig dif-ference: all formulas of the formN Aare well-behaved – and this is inescapable;
moreover, as we will see, the strong negation is not classical, what is not in itself a sin, but also not what happens to da Costa’s systems Cn,even just to C1.Indeed:
Theorem 5.2.6.1. Neither of the formulas DF,A∨◦A , N(A∧N A)→◦A is formally provable inaωws62.
Proof. Let us consider the linear Heyting algebra with three elements 0, a , 1, ordered as in this enumeration, with 1 as only distinguished element. Keep the heytingian interpretations of the conjunction, disjunction, implication: so, all theses of Pil are satisfied by the matrix so obtained. As interpretation of the paraconsistent negation, let us take the unary operation ndefined by the equalities n0 = 1, na = 1, n1 = 0. According to ws, compute ◦0 = 0 → nn0 = 1 ; ◦1 = 1 →nn1 = 1 ; ◦a = a →nna = 0 : then, the equation
◦x∧nx∧x= 0 holds, from what it results that cEF is satisfied; and ET also, since the equation x∨nx= 1 holds. But a∨(a→ 0) = a∨◦a =a∨0 = a hold; hence DF and A∨◦A are not satisfied. Also, n(a∧na) = 1 holds; so n(a∧na)→◦a= 0 holds, and thusN(A∧N A)→◦Ais not satisfied63.
62This results extends to the logicaωws’introduced in the next subsection.
63Note that∼0 =n0∧◦0 = 1,∼1 =n1∧◦1 = 0,∼a=na∧◦a= 0 hold, so that on this matrix (of which I owe the idea to some tentative of Waragai and Shidori), one can see how the interpretation of the strong negation coincide with the heytingian negation, and how its restriction coincide with the restriction of the interpretation of the paraconsistent negation on the two well-behaved elements 0 and 1.
Moreover, it is very easy to verify that all stability axioms are satisfied by this matrix, so that it shows that the logicawsis astrictextension of the logic aωws64. Hence, the fragments of the first strictly extend the corresponding fragments of the other; so, for example, bωC is a strict fragment of bC (and also ofbcC, since the first does not satisfy DF which is an axiom of the second).
The following theorem, wich will be useful in the sequel, is as striking as the previous one:
Theorem 5.2.6.2. The schema nEF N N A→ (N A→B) is for-mallyprovable in all extensions ofbωws.
Proof. From cEF it follows that the sequent ◦A , N A , A`B holds; from 5.2.3.2, that the sequent N N A ` ◦A holds; from (m0), that the sequent N N A ` A holds. Hence, the sequent N N A , N A ` B holds; this entails the theorem by repeated applications of the deduction theorem.
Corollary 5.2.6.3. Da Costa’s formulaAn+1 is a thesis of every extension of bωws for everyn >1.
Proof. An+1 is defined as a shorthand ofN(An∧N An),for everyn >0.Try to propose it as not valid; then, bothAnandN An have to be proposed as valid.
But by assuming ngreater than 1, one assumes thatAn is itself a shorthand a formula of the form N B , where B is An−1∧N An−1; thus, the formulas just proposed as valid areN BandN N B ,and according to nEF, the formulaAn+1 would have to be proposed as valid, a contradiction.