I will now examine the adjunction of a type of axioms, to which I assign the label (ia)48, and which are of the form ◦A↔D ,whereD is a schema of which Ais the unique prime component. Note that if we wish regardDas a shorthand of◦A ,this axiom becomes provable, being of the formD↔D .
The paraconsistent logic obtained by this adjunction will here be labelled by transforming the label of the logic to which it is added, replacing the “C” by what is effectively into the brackets of the axiom’s label.
SobiC gives rise to the extension labelled biia which might be called the
“low (weak) concrete basic intuitionisticia paraconsistent logic” (and so forth for other basic logics of consistency), and aωC gives rise to aωia which might be called the “low (weak) concrete fullia paraconsistent logic” (and so forth for
47Up to equivalence, since, in presence of ET and (m0), each of cEF and cRA can be deduced from the other.
48“Initials of the author”.
other full abstract logics of consistency)49.
I already noticed that if (ia) is accepted as an axiom, then, ◦A↔D must be a thesis of the paraconsistent logic, so thatD is a thesis of the aimed logic, which cannot be a thesis of the ground logic without making the paraconsistency disappear.
At the borderline of concrete considerations are the following abstract Facts 5.1.1: The following schemes are formally provable since biC:
U2 ◦A→N(A∧N A),
U3 ◦A→(A→N N A),
U4 ◦A→(A↔N A).
For the two first cases, proofs are brought by the tableau of the figure 13, which condenses in an only one four ordinary tableaux, and on which we can read some remarkable peculiarities of which it will be interesting to make use for the sequel.
Valid ? Not valid ?
• •
[[◦A]] •
• •
• A?
• •
• •
-[N(A∧N A)] ‡ [A→N N A]
[A∧N A] ‡
A ‡ [N N A]
N A ‡
- - - -[[A]]
Fig. 13
On the abstract side, the tableau condenses in an only one two tableaux of tbiC,which have the same left column except for the formula insimple square brackets, and also the same part of the right column above the first horizontal dashed lines, but with two different continuations under this line, which are distributed into two right columns separated by a vertical line made of signs
‡. In the tableau, the notation A?, which is the only explicited expression above the first horizontal dashed line, is meant to represent a shorthand for one of the two formulas in simple square brackets by which each the two separated
49I do not whish to rack my brain over the names to give to the low concreteia paraconsistent logics which are not full.
continuations of the right column begin. All in all, the inscription of the formula located just under the first horizontal dashed line corresponds, in each of the separated right continuations, just to the replacement of the shorthand by the shorten formula.
Then, on the side ofN(A∧N A),the formulaA∧N Ais proposed as valid, and so are next the formulasAandN A; on the side ofA→N N A ,the formula Ais proposed as valid and the formulaN N Ais proposed as not valid, and then, the formulaN Ais proposed as valid. As◦A is in the list of formulas proposed as valid, then, in both cases, the formulaAcan be added to the list of formulas proposed as not valid, what entails the closure of both tableaux which were condensed in one.
In the case whereA? would be meant to shorten the formula N(A↔N A), a splitting in two subtableaux intervenes after the inscription of the formula N A → A in the list of formulas proposed as valid, after what the formula A is proposed as valid in both newly introduced subtableaux, and then N A is proposed as valid in both also, by applying the derived rule 1 in virtue of the earlier introduction of the proposal of the formulaA→N Aas valid. Then, the closure intervenes as previously, in both subtableaux.
Now, let us delete from the condensed tableau the two formulas in double square brackets; what is obtained is a tableau which condenses in an only one two tableaux of tUil.Then, the later erasure of the formulas insimple square brackets (and the same operations on the evoked but not exhibited analogous tableau for the case where the formula N(A↔N A) is shorten intoA?) enters in a reasoning justifying the application of the following
Auxiliary special concrete rule 1 : In any extension of tbiC, if a formula of the form A? is proposed as not valid, where A? is a shorthand of one of the formulasN(A∧N A), A→N N A , N(A↔N A),then both formulasAandN A can simultaneously be proposed as valid; correlatively, the list of the formulas proposed as not valid is reduced to one unique formula of the form N B (B is respectively the formula A∧N A , N A , A ↔ N A), except in the case of an extension of tbhC, in which the list of formulas previously proposed as not valid remains unaltered.
Now, each of these formulas represents one of the meanings attributed to the notion of consistency of a formula by various authors: N(A∧N A) is the shorthand of ◦A in da Costa C1; in PCL1, this shorthand is equivalent to A → N N A50; N(A ↔ N A) is another possible interpretation that I add for extending the set of terms to compare.51
Owing to the facts 5.1.1, it is equivalent to adjoin, to any extension ofbiC: (a) the axiom (dc) ◦A↔N(A∧N A)
50More precisely, this shorthand isA↔N N A ,butN N A→Ais an axiom of the ground logic of PCL1 and takes no part in selecting the well-behaved formulas from the not necessarily well-behaved ones.
51Before all in the technicalities to come.
or the axiom (dc) N(A∧N A)→◦A52; (b) the axiom (ws) ◦A↔(A→N N A) or the axiom (ws) (A→N N A)→◦A; (c) the axiom (gm) ◦A↔N(A↔N A) or the axiom (gm) N(A↔N A)→◦A .
The usefulness of the auxiliary special concrete rule 1 consists in opening the way towards a unified treatment of some properties which are common to concrete logics in which one of the formulas (dc), (ws), (gm) is accepted as a thesis. In contexts which are related to such properties, the logics concerned will be indicated by labels in which “ia” will be replaced by “t?”.
On another side, there exists other meanings attributed to the possession of consistency which are not in the situation of the facts 5.1.1:
Fact 5.1.2: Forn >0 the formula◦A→A(n)is a thesis ofbωCand ofaincC. Indeed, to build a tableau for checking this forn >1 introduces proposals of formulas of the formN N Bas valid (forn= 1,the formula to prove is nothing else than U2, which is already provable, according to 5.1.1, in the fragmentbiC of bωC). Now, as for finding a rule to apply for going farther, the one which will appear below as slightly the more efficient is that which is associated to (m0)53 and allows to propose the formula B as valid – in the case of aincC, the same outcome requires the proposal as valid of the formula ◦B ,which can be deduced from that of◦AwhenBis built by using negation and conjunction, from prime components of which all are occurrences ofA .This leads to a cascade of proposals of formulas of the formN N(Ai∧N Ai) as valid, which ends when i becomes equal to 0, with the proposals of the formulas A andN A as valid, leading to closure, in reason of the initial proposal of◦Aas valid.
Hence, forn >1,I will take the formula (dcn) A(n)→◦A
as an axiom for the weakly concrete fragments of Cn extendingbωCoraincC. Anauxiliary special concrete rule1nis alsoat our disposal in the case where A? is a shorthand for A(n) with n >1 ; it says the same thing that the other special concrete rule,but only for the extensions of tbωCor of taincC.
The use as an axiom of (dc), (ws) or (gm), resp. of (dcn) for n > 1, allows, when◦Ais proposed as not valid, to employ the derived rule 2, after an imagined proposal as valid of that of these axioms corresponding to the context, for performing the first step which was explained on the tableau of figure 13 after the supposed proposal ofA? as not valid; to imagine to perform later the next steps boils down to apply the auxiliary special concrete rule 1,resp. 1n,directly in virtue of the earlier proposal of◦Aas not valid in place of that ofA?.
In such contexts, in which some properties, common to the logics of which the label is one of those that the labelt? symbolizes, are also partaken by the
52Thus, mybdcis Carnielli and Marcos’Cil(cf [2002]).
53The reader can also consult the proof given in 1964 by da Costa and Guillaume [1965].
logics accepting as a thesis one of the formulas (dcn) for n >1, the label t?n
will symbolize one of the labels symbolized byt?or bydcn forn >1.
As first applications of the preceding remarks, it can very easily be built closed tableaux for proving the propositions which follow54.
Theorem 5.1.3. The following schemes are formally provable sincebit? : (na) (m0)→(◦A→◦N A),
(ca) (m1)→
◦A→(◦B →◦(A∧B)) ,
(da) (m2)→
◦A→(◦B →◦(A∨B)) , (ia) (m3)→(◦B→◦(A→B)).
Corollary 5.1.4. The following schemes are formally provable since bht? : (ne) (◦A→◦N A)↔(◦A→(m0)),
(ce)
◦A→(◦B →◦(A∧B))
↔
◦A→(◦B→(m1)) ,
(ie)
◦A→(◦B →◦(A→B))
↔
◦A→(◦B→(m3)) . Corollary 5.1.5. The schema
(de)
◦A→(◦B →◦(A∨B))
↔
◦A→(◦B→(m2)) is formally provable since bit?.
Corollary 5.1.6. To any extension ofbht?,it amounts to the same to adjoin independently as axiom:
(i) (n) or (a0) ◦A→(m0),
(ii) (c) or (a1) ◦A→(◦B→(m1)), (iii) (i) or (a3) ◦A→(◦B→(m3)).
Corollary 5.1.7. To any extension ofbit?,it amounts to the same to adjoin independently as axiom:
(d) or (a2) ◦A→(◦B→(m2)).
To the preceding results are to be added those which follow, which can be etablished inductively by reasoning on building more complexes tableaux.
Theorem 5.1.8. “Guillaume’s thesis”55◦A→◦N Ais formally provable in all extensions ofbωt?n56.
Theorem 5.1.9. Forn > 1, the schemes (ca), (da), (ia), (ce), (de), (ie) are formally provable inbωdcn.
Corollary 5.1.10. To any extension ofbωdcn with n >1, it amounts to the same to adjoinindependently as axiom:
54The theorem 5.1.3 dates from [1965]. Waragai and Shidori recently discovered the impli-cations from the right to the left in the schemes (ce) and (de).
55According to Carnielli and Marcos [2002].
56It will be seen below thatahnt?coincide withbωt?.
(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.