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.
It is quickly recognized that in the previously treated cases, the formula N D → N◦A is provable, to the price set by the construction of D , which requires the use of some stability axiom at each of its steps; that amounts to say that some differences intervene at this level between formulas of which the behaviors at the lower levels were similar. By building tableaux and searching to minore the requirements for building them65one is led to the followingabstract Facts 5.3.1.1: The formulas N A(n) → N◦A for n > 0, N(A → N N A) → N◦A , N(A↔N A)→N◦A are formally provable inaincCn,ainiC, ainciC respectively66.
Fortunately, the proposal ofN◦Aas not valid has to appear at the beginning of the construction, followed immediately by the proposal of◦Aas valid; in the present cases, whereD isA(n), N(A→N N A), N(A↔N A) respectively, the stability axioms which are added are those which allow, starting from this last proposal, to end, by the lemma 4.5.2, on the proposal of ◦D as valid, so that, owing to the earlier proposal ofN D as valid, it becomes permitted to propose Das not valid, and we are then in the situation described by the figure 13, with D at the place indicated byA?.The closure follows.
5.3.2 Consistent certificates of consistency.
Now, as it seems that for proving the reverse implications, we need to prove
◦◦A ,we will try to take them as axioms. Then, to the axiom (ia), I will make a new axiom (ia’) correspond, which shall beN◦A→N D ,and start the study of what results from thesimultaneous adjunction of both axioms (ia) and (ia’).
The paraconsistent logic so obtained will here be labelled by appending the sign
“prime” to the label of its fragment obtained by removing just the axiom (ia’).
By the described simultaneous adjunctions,biCgives rise to the extension labelled biia’which might be called the “high (weak) concrete basic intuition-isticia’paraconsistent logic” (and so forth for other basic logics), andaCgives rise toaia’wich might be called the “high (weak) concrete fullia’paraconsistent logic” (and so forth for other full abstract logics)67.
Alas, as it can be seen by building suitable tableaux (which begin by applying the auxiliary special concrete rule to the proposal of ◦◦A as not valid, and continue by using as soon as possible of the derived rule 1 applied to (ia’), meant as added to the list of formulas proposed as valid, in order to introduce the formula N D corresponding to the suitable formula D of 5.3.1), the trick does not work:
65This leads to favour the use of a certificate of well behavior towards the recourse to (m0), in order to pass from the proposal as valid of a formula of the formN N Ato the proposal as valid of the formulaA .However, in virtue of what was established in the preceding subsection, the savings so realized are totally illusory when ET is a thesis, since the passage to the negation has to be taken in account in all systems then so extending the systems to which the facts to establish at present apply.
66The formula (A∧N A)→N◦Ais even provable inbiC.
67Wheniaisdc, this last logic is that which is labelledCilaby Carnielli and Marcos [2002].
Facts 5.3.2.1: The formula◦◦Ais provable inaincdc’n forn >0,inainiws’, and inaincigm’.
That is to say, exactly in the logics in which the implications in the sense which is opposite to (dc’n), (ws’), (gm’) are provable. But the link is more narrow than what precedes:
Theorem 5.3.2.2. The equivalences (dc’n)↔◦◦A forn >0,(ws’)↔◦◦A , (gm’)↔◦◦Aare provable since aincdcn,ainiws,aincigmrespectively.
Proof. The implications from the left says in other words just what the facts 5.3.2.1 are saying. The implications from the right are just instances of T5, in which ◦A is substituted for B and the suitable formula D for A . Then (ia’) comes as substituted forN B→N A ,and (ia), as substituted forA→B ,using the suitable value of (ia), and then, can be detached.
5.3.3 A glance at some high weak concrete full paraconsistent logics.
Let us recapitulate what we learned until now on the high weak concrete full paraconsistent logics of the kind in which we have here been interested.
For eachn >0 have been established the equalities and can be conjectured with good reasons the inequality which follow:
aidc’n6= ahdc’n= aωdc’n = adc’n.
The same holds for the equalities and the inequality which follow:
aigm’6= ahgm’= aωgm’= agm’.
We established also the equality and the last of the inequalities which follow, the first one of the last remaining conjectural:
aiws’6= ahws’= aωws’6= aws’.
For each n > 0, the system adc’n, of which Carnielli and Marcos [2002]
showed that it is an extension ofbC, can be seen as a weakly concrete system which expresses what the strongly concrete “true” Cn expresses. It might be called the “weak Cn” and labelled wCn. Its collective well behavior and its strong negation are classical.
On another side, for eachn >0,the systemaidc’n hasintuitionistic collec-tive well behavior and strong negation. It might be called the “intuitionistic weak Cn” and labelled wCi
n.I did only skim its study in the lines which pre-cede. To be sure, this study should be much more difficult than that of the corresponding Cn, but not uninteresting owing to the close links of computer science and intuitionistic logic.
The systemsaigm’and agm’were here introduced much in order to learn a lesson from their comparison with the two other based on (dc) and (dc’), or on (ws) and (ws’), respectively, than in reason of their own interest, of which I do not have a clear idea. Technically, they are clearly more intricate to handle.
On another side, classically, A∧N A and A ↔ N A are equivalent formulas, which so express equivalent assertions; it can be meant that the same holds for their negations, but what paraconsistently ? Perhaps is it permitted to conceive
that the equivalence is maintained; in this case, this underlines that da Costa instinctively choose the simpler way for expressing a condition of consistency.
However, intuitionistically,A∧N Ais strictly stronger thanA↔N A ,but until now I did not have sufficient time to evaluate the consequences of this remark.
I am not in position to draw more conclusions on the systemaiws’,which Waragai and Shidori clearly had intended neither to invent nor to study. I have no more to say about the systemaws’; in some way, certain features presented by it could as well as PCL1 deceive some of the hopes they put on this last one.
The case ofaωws’is different, since it is a fragment of PCL1, what is not apparent at first. Then, this remains to clear up, and is what I will do in the next subsubsection.
5.3.4 More on PCL1 and da Costa’s hierarchy.
Theorem 5.3.4.1. PCL1 extendsaωws.
Corollary 5.3.4.2. aωC is shared by PCL1 and the wCn for all n > 0 as a common fragment68.
Theorem’s proof. Leaving aside in a first time all stability axioms (of which one is not of the same type as those which have been considered until now), let us start from the fragment of PCL1 which is axiomatized by the axioms for Pil and the PCL1 axiomsA10, this one to which the label (m0) was here allocated,A11 which isN A→N N N AandA12 which is, in abstract version,
◦A→((N A∨B)↔(A→B)),to which I will add, in view to obtain a weakly concrete system, (ws?) which is Ai→◦A ,where Ai is Waragai and Shidori’s shorthand forA↔N N A .
Note that a weakly concrete version of PCL1 requires the presence of (ws?), which is Ai ↔◦A , as an axiom, which implies (ws?). Note more that, after replacingAiby the formula it shortens, in (ws?),it becomes apparent that this last is equivalent to (N N A →A)→((A→ N N A)→ ◦A) in Pil, so that, in presence of (m0), we have (ws) at our disposal. Conversely, (ws) implies (ws?) in Pil.
We turn now toA12 which, taking in account the stability paraconsistent axioms, says, in virtue of what was said in subsubsection 5.2.5, that the collective well behavior is classical.
Indeed, reason in Pil: first, A12 implies ◦A → ((N A∨B) → (A →B)), from which cEF follows by taking in account the instanceN A→(N A∨B) of one of the axioms. Conversely, from◦A→(N A→(A→B)), we can deduce
◦A→((N A∨B)→(A→B)) by taking in account the axiomB →(A→B). Second,A12 implies◦A→((A→B)→(N A∨B)),of which the instance obtained by substitutingAforBallows at once to deduce cET. Conversely, the implication of ◦A → ((A → B) → (N A∨B)) by cET as a thesis of Pil can
68I do not know if this fragment is thegreatestcommon to these logics.
be obtained by substituting N Afor C in the thesis (A∨C)→ ((A →B) → (C∨B))69.
Hence, the systems obtained by adding to Pil, on one hand, A12, and on the other hand, both cEF and cET, are equivalent. Then, our first step will be to replaceA12 by the set of cEF and cET, in the set of axioms for our fragment.
The second step70will be to replace cET by ET, by making use of A11, of the instances (N A→N N N A)→◦N Aof (ws) and ◦N A→(N A∨N N A) of cET, and of (m0). This step is reversible relatively to Pilsince the implication of cET by ET is an instance of one of its axioms.
Now it is made clear that our fragment extendsbiC. Hence we can prove U3 in it (see 5.1.1), and as (m0) is taken as an axiom, we can prove◦A→Ai, from what it follows that (ws?) implies (ws?); moreover, since (ws) implies (ws?),as noticed above, we can replace, in the axiomatization of our fragment, (ws?) by (ws) without making the desired (ws?) cease being a thesis of our fragment.
The fourth step consists in noticing that the fragment extracted from our fragment by omitting A11 is bωws. Hence, according to 5.2.3.5 and 5.2.3.4,
◦N A is a thesis of it. But ◦N A → (N A → N N N A) is an instance of U3, so provable; then A11 is superfluous and can be suppressed, what reduces our fragment to justbωws.
Now, PCL1 axioms which are labeledA13 and A14 are those which were here labeled (a1) and (a2) respectively, and according to the corollaries 5.1.6 and 5.1.7, it comes to the same to add them or to add (c) and (d). Hence, in order to prove that PCL1 extendsaωws, hence alsoaωC, as the wCn are doing for n >0, it remains just to examine what happens with the axiom of PCL1 labeledA15.
This axiom is equivalent to the conjunction of the formulas ◦A → (m3) and ◦A → ((A∧N B) → N(A → B)), whether it is given under the form
◦A → (N(A →B)↔ (A∧N B)) or by separately postulating on a side that
◦A → ((A∧N B) → N(A → B)) is an axiom and on another side that the sequent◦A`N(A→B)→(A∧N B) holds.
Now, it follows immediately from the formula (ia) of theorem 5.1.3 that (◦A → (m3)) → (i) is provable, so that the stability axiom (i) is provable in PCL1, what ends a proof of my claim thataωws is a fragment of PCL1.
This time, I donot see how we could prove more; neither ◦A →(m3) nor
◦A→((A∧N B)→N(A→B)) seem deducible from the axiom (i).
Scholium 5.3.4.3. I have still to complete a proof that I was right by replacing the formulaA↔N N A shorten byAiby the formulaA→N N Ain my weakly
69In the same way, one proves that to adjoin the sole (A→B)↔(N A∨B) to Pilsuffices for axiomatizing Fcl.The same is true for (A∨B)↔(N A→B) ; the proof results from the previous one by permuting the rˆoles ofAandN A .
70Inspired by what Waragai and Shidori are doing.
concrete setting of PCL171.
Now, previously, (ws?) was shown provable in bhws; now it remains to show that the formula N Ai ↔ N◦A , that I will label (ws’?), is provable in the weakly concrete version of PCL1. I will content myself by showing that it is provable inahws’.
First, from (ws?) we can deduce the conjunct (ws’?), i.e. N◦A →N Ai, by having recourse to the instance of T5 obtained by substitutingAiforAand
◦AforB ,noticing that the permission required is then the provable◦◦A(here is where we use (ws’).)
Second, for proving the reverse implication, we can start a tableau in tahC with the intention to check the sequent N(A ↔ N N A) ` N◦A . Then the formula◦Ais quickly proposed as valid; as the only prime component occurring in the checked sequence is A , we can, by iterated uses of stability axioms, assume that for all formulas of the form N B proposed as valid, the formula
◦B is proposed as valid. Then, the tableau can be built just by applying the classical rules for drawing the consequences of the proposals of formulas of the formN B as valid or not valid. As the formulaN(A↔N N A) was proposed as valid, there is a splitting in two subtableaux after the proposal of the formula A↔N N Aas not valid; in one of them, the formulaA→N N Ais proposed as not valid, now just under the conditions under which we checked the formula U3; so this subtableau is closed, as was the case in figure 13. In the other subtableau, the formula N N A→A is proposed as not valid; after application of the rule for drawing the consequences of such a proposal, there remains no new application of this rule to do, and we are led to closure as if we were building a classical tableau for checking this last classical thesis. This ends the proof.
It is known that fot every n > 0, the formula A(n) → A(n+1) is provable in wCn72, but not the formula A(n+1) → A(n). I will show that a similar phenomenon arises between aωws’ and wCi
n : (A → N N A) → A(n), where A(n)is da Costa’s formula, is provable inbωws, since◦A→A(n)is so provable according to 5.1.2, and since (ws) is precisely (A→N N A)→◦A; but on the other side:
Theorem 5.3.4.4. Forno n >0 is the formulaA(n)→(A→N N A) provable in aωws’.
Proof. We begin with the casen= 1 :
Lemma 5.3.4.5. The formula U5 N(A∧N A)→(A→N N A) is not provable in aωws’
– just because a proof of U5 would allow immediately to build a proof ofN(A∧ N A)→◦A ,contrary to the theorem 5.2.6.1. Now,
Lemma 5.3.4.6. The formulaN(A∧N A)→A(n) is provable inbωws
71Note that, in the strongly concrete version of PCL1, such a proof is by no means necessary, since, after what is meant by the formula N N A→Awas postulated, the meaning of the formulaA→N N Abecomes equivalent to that of the formulaA↔N N A .
72In fact, the proof rests only on wCin’s axioms.
– just becauseN(A∧N A),alias A1,is precisely just the conjunct which lacks to the conjunction of the Ai for i going from 2 to n , all provable in bωws according to 5.2.6.3, for formingA(n).
Hence, inaωws’, a proof ofA(n)→(A→N N A) would allow immediately to build a proof of the forbiddenN(A∧N A)→(A→N N A).
Theorem 5.3.4.7. Forno n >0 is the formula (A→N N A)→A(n) provable in wCn,nora fortiori in wCin.
Proof. We begin with the casen= 1 :
Lemma 5.3.4.8. The formula U6 (A→N N A)→N(A∧N A) is not provable in wC1,nor,a fortiori, in wCi1.
Proof of the lemma. I will have recourse to the matrice M2 long ago devised by da Costa (cf. [1964]), which has three elements 0, β and 1, from which the two last are distinguished. The values indicated in its table for a→b are 1 everywhere, except for the cases b = 0 and a = 1 or a = β , where it is 0. In its table for a∧b the values indicated are all 0 when a = 0 or b = 0, and 1 in all other cases. The operationnintended to interpret the negation is defined by the equalitiesn1 = 0,n0 = 1,nβ =β .Then all axioms of wC1 are satisfied, but n(β∧nβ) =n(β∧β) =n1 = 0 and β →nnβ = 1 hold, so that (β→nnβ)→n(β∧nβ) = 0 holds, and thus U6 is not satisfied73.
Now, as noticed above, A(n) → N(A∧N A) is provable in Pil, and it is known for a long time that wC1extends wCn,for everyn >0.Hence, in wCn, a proof of (A →N N A) →A(n) would allow immediately to build a proof of U6, which in its turn allows to build a proof in C1 of the forbidden U674.
References
[1952] S.C. Kleene,Introduction to Metamathematics. Van Nostrand, New-York, 1952.
[1957] S. Kanger,Provability in Logic. Acta Universitatis Stockholmien-sis, Stockholm Studies in Philosophy 1. Almqvist and Wiksell, Stockholm.
[1959] E.W. Beth,The Foundations of Mathematics. A Study in the Philosophy of Science. North Holland, Amsterdam, 1959.
[1962] E.W. Beth, Formal Methods. An Introduction to Symbolic Logic and to the Effective Operations in Arithmetic and Logic. Reidel, Dordrecht, 1962.
73By luck, since this matrice was by no means devised with the aim to prove the lemma !
74The matrice used in the proof of theorem 5.2.6.1 provides also a proof of non provability of the formulaN(A↔N A)→(A→N N A) inaωws’,and the just described matriceM2, a proof of non provability of the formula (A→N N A)→N(A↔N A) inaωgm’.Yet these matrices assign the same operation for interpreting the schemesN(A∧N A) andN(A↔N A), what blocks the above reasonings in what regards the links ofaωgm’with wC1,and beyond with the wCn’s. I am compeled to leave that question open.