Abstract
This paper discusses logical accounts of the notions of consistency and negation, and in particular explores some potential means of defining consistency and negation when expressed in modal terms. Although this can be done with interesting consequences when starting from classical normal modal logics, some intriguing cases arise when starting from paraconsistent modalities and negations, as in the hierarchy of the so-called cathodic modal paraconsistent systems (cf. Bueno-Soler, Log Univers 4(1):137–160, 2010). The paper also takes some first steps in exploring the philosophical significance of such logical tools, comparing the notions of consistency and negation modally defined with the primitive notions of consistency and negation in the family of Logics of Formal Inconsistency (LFIs), suggesting some experiments on their expressive power.
A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines.
Ralph Waldo Emerson ( 1841 ).
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Notes
- 1.
It is commonly accepted that (PEx) and (PNC) are taken to be independent, in the sense that neither of them entails the other; see, e.g., Béziau and Franceschetto (2015) for an argument, where it is also argued that a legitimate paraconsistent logic must reject both principles. As we show in Sect. 15.4, however, such a conclusion should be taken cum grano salis.
- 2.
Notice that this definition implies that any classical contradiction is consistent; this is not any problem, however, since ‘consistent’ does not mean ‘true’.
- 3.
A modal system is classified as normal if it contains the Distribution Axiom (K) and the Necessitation Rule (Nec) among its axioms and rules, and as minimal if it has only (K) as a modal axiom and only (Nec) as a modal rule. In this sense, the above defined systems are the minimal normal modal extension of each propositional basis.
- 4.
The obvious example comes from some axiomatizations of geometry where ‘point’ is a primitive concept whose meaning is governed by geometric axioms.
- 5.
M. Fitting in (2017), proposes a formalization of the notion of evidence that defines an embedding of the logic BLE into the modal logic KX4 (a modification of S4). Formally, evidence can be understood as permitting contradictions. It is shown that BLE has both implicit and explicit evidence interpretations in a formal sense.
- 6.
- 7.
As an analogy, imaginary numbers do no exclude real numbers, but help to explain real numbers and expand the logical space where real numbers exist.
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Acknowledgements
Both authors acknowledge support from FAPESP Thematic Project LogCons 2010/51038-0, Brazil, and the second author thanks a research grant from The National Council for Scientific and Technological Development (CNPq), Brazil. We are indebted to David Gilbert, Raymundo Morado, and Peter Verdee for discussions that helped to improve this paper.
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Bueno-Soler, J., Carnielli, W. (2017). Experimenting with Consistency. In: Markin, V., Zaitsev, D. (eds) The Logical Legacy of Nikolai Vasiliev and Modern Logic. Synthese Library, vol 387. Springer, Cham. https://doi.org/10.1007/978-3-319-66162-9_15
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