In Remark 45 we have mentioned the possibility of defining in mbC an inconsistency connective that is dual to its native consistency connective.
This could be done by setting•αdef==∼◦α, where∼αdef==α→(β∧(¬β∧ ◦β)) (for an arbitrary β) is a classical negation. Now, how could we enrich mbC so as to be able to define the inconsistency connective by using the paraconsistent negation instead of the classical ∼, that is, by setting•αdef==
¬◦α? This is exactly what will be done in this subsection by extending mbCinto the logicmCi. In fact,mCiwill reveal to be a logic that can be presented in terms of either ◦ or • as primitive connectives. Moreover,•α and ¬◦αwill be inter-translatable, and the same will happen with ◦αand
¬•α, as proven in Theorem 98.
From Theorem 49(i) we know that α∧ ¬α `mbC ¬◦α. The converse property (which does not hold in mbC) will be the first additional axiom we will add to mbC in upgrading this logic. On the other hand, we wish
that formulas of the form ¬◦α ‘behave classically’, and we wish to obtain in fact a logic that is controllably explosive in contact with formulas of the form¬n◦α, where¬0αdef==αand¬n+1αdef==¬¬nα. Any formula of the form
¬n◦αwould thus be assumed to ‘behave classically’, and {¬n◦α,¬n+1◦α}
would be an explosive theory. This desideratum leads us into considering the following (cf. [Marcos, 2005f]):
DEFINITION 75. The logic mCi is obtained from mbC (recall Defini-tion 42) by the addiDefini-tion of the following axiom schemas:
(ci) ¬◦α→(α∧ ¬α) (cc)n ◦ ¬n◦α (n≥0)
To the above axiomatization we add the definition by abbreviation of an inconsistency connective• by setting•αdef==¬◦α.
Notice that ¬◦α and (α∧ ¬α) are equivalent in mCi. Clearly every set{¬n◦α,¬n+1◦α} is explosive inmCi, in view of (bc1) and (cc)n. This expresses the ‘classical behavior’ of formulas of the form◦α(with respect to the paraconsistent negation). In other words, a formulaαin general needs the extra assumption◦αto ‘behave classically’, but the formula◦αand its iterated negations will always ‘behave classically’. In Theorem 78 below we will see that ¬•αis equivalent to ◦α, and in Definition 97, further on, we will introduce a new formulation ofmCi that introduces• as a primitive connective. Notice in that case how close is the bond that is established here in between inconsistency and contradictoriness by way of the paraconsistent negation.
We can immediately check that the equivalence in mCi between ¬◦α and (α∧ ¬α) is in fact logically weaker than the identification of ◦α and
¬(α∧ ¬α) assumed inC1 (recall also Theorem 49(iii)–(iv)) since the latter formula implies the former, inmCi, but the converse is not true.
THEOREM 76. This rule holds good inmCi:
(i)¬◦α`mCi(α∧ ¬α),
but the following rules do not hold:
(ii)¬(α∧ ¬α)`mCi◦α;
(iii)¬(¬α∧α)`mCi◦α.
Proof. Item (i) is obvious. In order to prove that (ii) and (iii) do not hold inmCi, observe thatmCiis sound for the truth-tables of LFI1(see Examples 17 and 18), where 0 is the only non-designated value. Then it is enough to check that (ii) and (iii) have counter-models in such a
truth-functional semantics.
It should be clear that, even though in mCi there is a formula in the classical languageF or(namely, the formula (α∧ ¬α)) that is equivalent to a formula that expresses inconsistency (the formula•α), there is no formula in the classical language that can express consistency inmCi. We also have the following:
THEOREM 77. (i)¬(α∧ ¬α) and¬(¬α∧α) are not top particles inmCi.
(ii)◦αand¬◦αare not bottom particles.
(iii) The schemas (α→ ¬¬α) and (¬¬α→α) are not provable inmCi.
Proof. Items (i), (ii) and the first part of item (iii) can be checked using again the truth-tables ofP1, enriched with the (definable) truth-table for◦ (Example 19), and using the fact thatmCiis sound for such a semantics.
For the second part of item (iii) one could use for instance the bivaluation
semantics of mCi(see Example 90).
It is straightforward to check the following properties ofmCi:
THEOREM 78. The following rules hold good inmCi:
(i)¬¬◦α`mCi◦α;
(ii)◦α`mCi¬¬◦α;
(iii)◦α,¬◦α`mCiβ;
(iv) (Γ, β`mCi◦α) and (∆, β`mCi¬◦α) implies (Γ,∆`mCi¬β).
Proof. For item (i), from ¬¬◦αand ◦α we obviously prove ◦α in mCi.
On the other hand, from¬¬◦αand¬◦αwe also prove◦αin mCi, because
◦¬◦αand (bc1) are axioms ofmCi. Using proof-by-cases we conclude that
¬¬◦α`mCi◦α. The other items are proven similarly. Notice in particular how items (i) and (ii) together show that¬•αa`mCi◦αholds good.
Item (ii) of Theorem 77 and item (iii) of Theorem 78 guarantee thatmCi is controllably explosive in contact with◦p0(recall Definition 9(iii)). In fact, the following relation between consistency and controllable explosion can be checked:
THEOREM 79. Let L be a non-trivial extension of mCi such that the implication (involving the axioms ofmCi) is deductive (recall Definition 6).
A schema σ(p0, . . . , pn) is provably consistent in L if, and only if, L is controllably explosive in contact withσ(p0, . . . , pn).
Proof. If`L◦σ(α0, . . . , αn) then, by axiom (bc1), Γ, σ(α0, . . . , αn),¬σ(α0, . . . , αn)`Lβ for any choice of Γ andβ.
Conversely, assume that Γ, σ(α0, . . . , αn),¬σ(α0, . . . , αn)`Lβ for any Γ and β. Since, from (ci), we have that¬◦σ(α0, . . . , αn)`L(σ(α0, . . . , αn)∧
¬σ(α0, . . . , αn)), then it follows that ¬◦σ(α0, . . . , αn) is a bottom particle.
As in the proof of Theorem 46(i) (using here the fact that the original implication of mCi is still deductive in L), we get `L ¬¬◦σ(α0, . . . , αn).
By Theorem 78(i), we conclude that `L◦σ(α0, . . . , αn).
Complementing the versions of contraposition mentioned in Theorem 48, we have:
THEOREM 80. Here are some restricted forms of contraposition introduced bymCi:
(i) (α→ ◦β)`mCi(¬◦β → ¬α);
(ii) (α→ ¬◦β)`mCi(◦β → ¬α);
(iii) (¬α→ ◦β)`mCi(¬◦β →α);
(iv) (¬α→ ¬◦β)`mCi(◦β→α).
Proof. Item (i). By axiom (cc)0,◦◦βis a theorem ofmCi. The result now follows from Theorem 48(iii). The other items are proven similarly.
On the other hand, properties such as (◦α→β)`mCi(¬β→ ¬◦α) do not hold; this can easily be checked after Corollary 93, to be established below.
Notice how the above theorem opens yet another way for the internalization of classical inferences, as discussed in Subsection 3.6.
Recall now the replacement property (RP) discussed in Remark 51. We had already proven in Theorem 52 that (RP) cannot hold in certain para-consistent extensions of mbC. On what concerns its possible validity in paraconsistent extensions ofmCi, we can now prove that:
THEOREM 81.
(i) The replacement property (RP) is not enjoyed bymCi.
The replacement property (RP) cannot hold in any paraconsistent extension ofmCiin which:
(ii)¬(¬α∧ ¬β)`mbC(α∨β) holds; or (iii) (¬α∨ ¬β)`mbC¬(α∧β) holds.
Proof. (i) Consider again the first set of truth-tables (with the same set of designated values) used in the proof of Theorem 50.
(ii) Consider the supplementing negationoα= (¬α∧ ◦α) formCiproposed in Remark 43. By Theorem 78 this last formula is equivalent to (¬α∧¬¬◦α).
In Theorem 94, this negation will be shown to behave classically inside this logic. But then,¬oα`α∨¬◦α, by hypothesis, and so¬oα`α, using axiom (ci), proof-by-cases and conjunction elimination. The rest of the proof now follows exactly like in Theorem 52(ii).
Finally, for item (iii), recall that, from (Ax10), (¬α∨ ¬¬α) is a theorem of mCi. But then, by hypothesis,¬(α∧ ¬α) would also be a theorem. From Theorem 49(ii) and replacement it follows that ¬¬◦α is provable, and by Theorem 78(i) this results in◦αbeing provable. Thus, the resulting logic
would be explosive.
In the case of the logicmbC, we have called the reader’s attention to the fact that the validity of (RP) required the validity of rules (EC) and (EO) (see the end of Subsection 3.2). Interestingly, now in mCi we can check that (EC) is enough:
THEOREM 82. In extensions of mCithe validity of:
(EC)∀α∀β((αaβ) implies (¬αa¬β))
guarantees the validity of:
(EO) ∀α∀β((αaβ) implies (◦αa◦β)).
Proof. Suppose (αa` β). By (EC) we have that (¬αa` ¬β), and from these two equivalences we conclude that (α∧ ¬α)a`(β∧ ¬β). But from Theorems 49(ii) and 76(i) we have that ¬◦γ a`mCi (γ∧ ¬γ), so we have that¬◦αa` ¬◦β. By Theorem 80(iv) we conclude then that◦αa` ◦β.
Suppose now we considered the addition tomCiof a stronger rule than (EC), in order to ensure replacement:
THEOREM 83. Consider the following rule:
(RC) ∀α∀β((αβ) implies (¬β¬α)).
Then, the least extension LofmCithat satisfies (RC) and proof-by-cases collapses into classical logic.
Proof. From the axioms ofmCiwe first obtain¬◦α`Lα, and¬◦α`L¬α.
By (RC) and Theorem 78(i) we then get ¬α`L ◦αand ¬¬α`L ◦α. But then, using proof-by-cases, we conclude that`L◦α, that is, all formulas are consistent. The result now follows, as usual, from Remark 29.
Notice that our paraconsistent formulations of the normal modal logics from Example 34 donotextend the logicmCi(contrast this with Remark 53 about mbC). As we said at the beginning of this subsection, an inconsis-tency connective• can often be defined from a consistency connective◦ by taking ∼◦, where ∼is a classical negation. The definition of an inconsis-tency connective by taking¬◦is an innovation of mCiovermbC, and it is typical in fact of mostLFIs from the literature, as the ones we will be study-ing in the rest of this chapter. The reader should not think though that the latter class of C-systems has any intrinsic advantage over the former. This far, it only seems to have more often met the intuitions of the working para-consistentists, for some reason or another —or maybe by pure coincidence.
At any rate, the distinction between the two classes is only made clear in a framework such as the one set in the present study, where consistency and inconsistency are taken as (primitive or defined) connectives in their own right.