Abstract
Some insights were gained from the study of inconsistency-adaptive logics. The aim of the present paper is to put some of these insights to work for the study of logics of formal inconsistency. The focus of attention is application contexts of the aforementioned logics and their theoretical properties in as far as they are relevant for applications. As the questions discussed are difficult but important, a serious attempt was made to make the paper concise but transparent.
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Notes
- 1.
Being merely an abbreviation in \(\mathbf {C}_{n}\) logics, the consistency operator adds nothing to the expressive power of the language. That the definiens, for example \(\lnot (A\wedge \lnot A)\) in \(\mathbf {C}_{1}\), expresses the consistency of A is somewhat awkward in the context of the \(\mathbf {C}_{n}\) logics.
- 2.
Some paraconsistent logicians defend a specific negation as the correct one. Priest [17], for example, seems to assign this role to the negation of \(\mathbf {LP}\). Other paraconsistent logicians, for example da Costa [12], consider a multiplicity of operators as paraconsistent negations, but sometimes impose certain conditions. Often a more general outlook is taken, as for example by Béziau [9].
- 3.
The insight was Suszko’s [18]. The resulting semantics may be ugly but is obviously adequate.
- 4.
The reference to \(\Gamma \) may be dropped for Tarski logics (reflexive, transitive, and monotonic logics).
- 5.
The syntactic justification refers to the complementing character of the non-paracomplete paraconsistent negation. \(A, \mathord {\sim }A \vdash _\mathbf {L} \lnot A\) by explosion for the classical \(\mathord {\sim }\) and \(\lnot A, \mathord {\sim }A \vdash _\mathbf {L} \lnot A\) by reflexivity. Both together entail \(\mathord {\sim }A \vdash _\mathbf {L} \lnot A\) in view of the is complementing character of \(\lnot \).
- 6.
Here too the reference to \(\Gamma \) may be dropped for Tarski logics.
- 7.
This is the reason why the converses of the inferences mentioned in Fact 15.11 do not hold for all consistency connectives.
- 8.
The situation may have been influenced by the somewhat odd behaviour of the (defined) consistency operator \(A^{(n)}\) in da Costa’s \(\mathbf {C}_{n}\) systems with \(n>1\). Another relevant consideration might have been that the consistent behaviour of a formula A on a premise set \(\Gamma \), viz. that the logic does not require \(\Gamma \) to entail A as well as \(\lnot A\), should not cause \(\mathord {\circ }A\) to be derivable from \(\Gamma \). However, this danger is nonexistent even in case \(\mathord {\circ }A\) is the suitable truth-function of A and \(\lnot A\).
- 9.
A non-monotonic logic may assign to \(\Gamma \) a selection of models that verify all members of \(\Gamma \). The lemma contains references to all \(\mathbf {L}\)-models.
- 10.
Adding or removing the trivial model—the model verifying all closed formulas—to the set of models defined by a semantics may require that the semantic clauses are adjusted. In view of the definition of the semantic consequence relation, it is obvious that such addition or removal does not affect the consequence relation.
- 11.
There is no reason to handle Modus Ponens and Modus Tollens on a par. The first states a property of the implication. The justification of Modus Tollens requires a reference to negation: if A is true, then B is true (by Modus Ponens); but \(\lnot B\) is true; so if inconsistencies are not true, then neither is A.
- 12.
This regularity requirement is stronger than the requirement for being a negation in the sense of Lemma 15.14. If S contains all \(\mathbf {CL}\)-models as well as the trivial model, \(\lnot \) is a negation but the regularity requirement is not fulfilled.
- 13.
Do not confuse the question whether a logic is regular with the question whether a specific semantics of this logics is deterministic. See Sect. 2 of another paper [7] in this volume for a method to turn an indeterministic semantics of a certain type into a deterministic one.
- 14.
There is no need to add “with respect to \(\mathbf {L}\)” as \( Dab \)-consequences of \(\Gamma \) will always be considered for a specific logic.
- 15.
The set of minimal \( Dab \)-consequences obviously depends on the logic. For some paraconsistent logics, like \(\mathbf {CLuN}\) mentioned in a subsequent section, \((p\wedge \lnot p)\vee (q\wedge \lnot q)\) is the only \( Dab \)-consequence of \(\Gamma _{1}\). Other paraconsistent logics assign infinitely many \( Dab \)-consequences to \(\Gamma _{1}\). Still, I cannot picture any formal paraconsistent logic for which \(\mathord {\circ }t\) has an effect on the minimal \( Dab \)-consequences of \(\Gamma _{1}\). This is weaker than what is claimed in the text, but I shall buy you a beer if you show my imagination lacking at this point and that is stronger than what is said in the text.
- 16.
To be more precise, this is the case for some (not necessarily all) sets \(\{A_1, \ldots , A_n\}\) such that \((A_1\wedge \lnot A_1)\vee \ldots \vee (A_n\wedge \lnot A_n)\) is a minimal \( Dab \)-consequence of the non-logical axioms of the theory.
- 17.
- 18.
Suitable are a classical conjunction or a gappy one. Glutty conjunctions have to be considered contextually because they allow for models that verify a conjunction and falsify one of the conjuncts. While such models are clearly abnormal with respect to \(\mathbf {CL}\) and many other logics, it depends on further properties whether a consistency operator should handle this. See for example [8] on gluts and gaps of all kinds.
- 19.
For present purposes, this may be identified with a compact Tarski logic.
- 20.
The set \(\Omega \) may comprise formulas of the form \(\exists (A\wedge \lnot A)\). If A is any formula, the form is unrestricted; if A is required to be an atomic formula, the form is restricted.
- 21.
The upper limit logic \(\mathbf {ULL}\) is obtained by extending \(\mathbf {LLL}\) with a rule that causes all abnormalities to entail triviality.
- 22.
If one of those symbols would be glutty or gappy, the abnormalities would need to contain members that describe the gluts or gaps in the existential quantifier and the conjunction in order to handle the situation in an adequate way. See [8] for more information.
- 23.
The absence of the restriction may cause the adaptive logic to be a flip-flop, which means that the adaptive consequence set reduces to the lower limit consequence set whenever the premise set is abnormal.
- 24.
This \(\Omega \) will also be adequate for some combinations of non-classical quantifiers, but that need not concern us in the present paper.
- 25.
If \(\Gamma \) has no \( Dab \)-consequences, \(\Phi (\Gamma )=\{\emptyset \}\); if \(\Gamma \) has no \(\mathbf {LLL}\)-models, \(\Phi (\Gamma )=\{\Omega \}\); \(\Phi (\Gamma )\ne \emptyset \) always holds.
- 26.
Where a logic \(\mathbf {L}\) is defined over \(\mathcal {L}\), a set \(\Gamma \) is \(\mathbf {L}\)-trivial iff \( Cn _{\mathbf {L}}(\Gamma ) = \mathcal {W}\).
- 27.
If consistency decisions do not trivialise the theory, consistency reclaims do not either; if consistency decisions trivialise the theory, consistency reclaims cannot make that situation worse.
- 28.
Actually for all formulas not in the set \(\{p,q\}\), but never mind.
- 29.
The reasons for the \(\mathcal {X}\) is as in (15.9); it would be tiresome to make this more precise here.
- 30.
If you frown here, realize that \(\lnot p\) is not a \(\mathbf {mbC}\)-consequence of \(\lnot (p\vee q)\).
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Batens, D. (2015). Some Adaptive Contributions to Logics of Formal Inconsistency. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_15
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