Abstract
In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar many-valued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them discusses our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction.
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Notes
- 1.
We will sometimes omit the subscript u, when contexts disambiguates. Also, we may sometimes make reference of \(\models ^{\mathbf {L}}\) instead of \(\models ^{M_{\mathbf {L}}}\).
- 2.
Notice that this does not suggest that the infectious value does not belong to the set of designated values if and only if the logic is paracomplete, for there might well exist paracomplete logics which do not count with an infectious value at all, as in e.g. the well-known Strong Kleene logic \({\mathbf {K}}_{\mathbf{3}}\) (cf. [19]).
- 3.
Analogous to the previous footnote, notice that this does not suggest that the infectious value does belong to the set of designated values if and only if the logic is paraconsistent, for there might well exist paraconsistent logics which do not count with an infectious value at all, as in e.g. the Logic of Paradox due to Priest (cf. [27]).
- 4.
We should remark that providing a full overview of these motivations will require much more space than we have here. For that reason, we refrained from commenting on some of the motivations for infectious logics, e.g. (the first degree of) Parry systems (cf. [25]) and of Epstein’s Dependence and Dual Dependence systems (cf. [11]) discussed in e.g. [12,13,14, 23, 24], Deutsch’s logic from [9], Daniels’ logic from [7], and Priest’s logic \({\mathbf {FDE}}_{\varphi }\) from [28].
- 5.
As is noted in [12], while this framework is regarded as a ‘single address’ approach to Belnap computers, a ‘two address’ approach can also motivated, with the subtlety that it induces a weaker nine-valued logic.
- 6.
- 7.
- 8.
Unlike the previous interpretations of both the paracomplete and the paraconsistent infectious systems, the following account is our original thought. We would like to thank one of the reviewers for the suggestion to develop further the epistemic readings of infectious logics. For a full technical development of these ideas, see [30].
- 9.
We would like to thank one of the reviewers for the suggestion to restructure the presentation of plurivalent semantics.
- 10.
As an anonymous reviewer points out, since univalently designated values need not be identified with truth, preserving designated values from premises to conclusion, does not collapse with truth-preservation (namely, the preservation of the value \({\mathbf {t}}\)).
- 11.
Although for an alternative, see [30], where designated infectious values are understood as truth-value gluts, i.e. as both-true-and-false.
- 12.
Notice that we took the notational liberty of using e.g. \(\ddot{\delta }_{\wedge }\) as the paradigmatic case, but nothing really depends on this, and \(\dddot{\delta }_{\wedge }\) might be used as well, without any loss.
- 13.
For further arguments in favor of the traditional account of negation, see [8].
References
Armour-Garb, B., Priest, G.: Analetheism: a pyrrhic victory. Analysis 65(2), 167–173 (2005)
Beall, J.: Off-topic: a new interpretation of weak-kleene logic. Australas. J. Log. 13(6), 136–142 (2016)
Beall, J., Ripley, D.: Analetheism and dialetheism. Analysis 64(1), 30–35 (2004)
Belnap, N.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–55. Oriel Press, Newcastle upon Tyne (1977)
Bochvar, D.: On a three-valued calculus and its application in the analysis of the paradoxes of the extended functional calculus. Matamaticheskii Sbornik 4, 287–308 (1938)
Brady, R., Routley, R.: Don’t care was made to care. Australas. J. Philos. 51(3), 211–225 (1973)
Daniels, C.: A note on negation. Erkenntnis 32(3), 423–429 (1990)
De, M., Omori, H.: There is more to negation than modality. J. Philos. Log. (2017)
Deutsch, H.: Paraconsistent analytic implication. J. Philos. Log. 13(1), 1–11 (1984)
Dunn, M.: Intuitive semantics for first-degree entailments and ‘coupled trees’. Philos. Stud. 29(3), 149–168 (1976)
Epstein, R.: The semantic foundations of logic. In: Propositional Logics, 2nd edn., vol. 1. Oxford University Press, New York (1995)
Ferguson, T.M.: A computational interpretation of conceptivism. J. Appl. Non Class. Log. 24(4), 333–367 (2014)
Ferguson, T.M.: Logics of nonsense and Parry systems. J. Philos. Log. 44(1), 65–80 (2015)
Ferguson, T.M.: Faulty Belnap computers and subsystems of FDE. J. Log. Comput. 26(5), 1617–1636 (2016)
Fitting, M.: Kleene’s three valued logics, their children. Fundamenta Informaticae 20(1, 2, 3), 113–131 (1994)
Fitting, M.: Bilattices are nice things. In: Bolander, T., Hendricks, V., Pedersen, S.A. (eds.) Self-Reference, pp. 53–78. CSLI Publications (2006)
Frankowski, S.: Formalization of a plausible inference. Bull. Sect. Log. 33(1), 41–52 (2004)
Halldén, S.: The Logic of Nonsense. Uppsala Universitets Årsskrift (1949)
Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)
Malinowski, G.: Q-consequence operation. Rep. Math. Log. 24(1), 49–59 (1990)
McCarthy, J.: A basis for a mathematical theory of computation. In: Braffort, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, pp. 33–70. North-Holland Publishing Company, Amsterdam (1963)
Omori, H.: Halldén’s logic of nonsense and its expansions in view of logics of formal inconsistency. In: Proceedings of DEXA 2016, pp. 129–133. IEEE Computer Society (2016)
Paoli, F.: Regressive analytical entailments. Technical report number 33, Konstanzer Berichte zur Logik und Wissenschaftstheorie (1992)
Paoli, F.: Tautological entailments and their rivals. In: Béziau, J.Y., Carnielli, W., Gabbay, D. (eds.) Handbook of Paraconsistency, pp. 153–175. College Publications (2007)
Parry, W.T.: Ein axiomensystem für eine neue art von implikation (analytische implikation). Ergeb. Eines Math. Kolloqu. 4, 5–6 (1933)
Priest, G.: Hyper-contradictions. Logique et Analyse 27(107), 237–243 (1984)
Priest, G.: An Introduction to Non-classical logic: From If to Is, 2nd edn. Cambridge University Press, Cambridge (2008)
Priest, G.: The logic of the catuskoti. Comp. Philos. 1(2), 24–54 (2010)
Priest, G.: Plurivalent logics. Australas. J. Log. 11(1), 2–13 (2014)
Szmuc, D.: An epistemic interpretation of Paraconsistent Weak Kleene. Typescript
Szmuc, D.: Defining LFIs and LFUs in extensions of infectious logics. J. Appl. Non Class. Log. 26(4), 286–314 (2016)
Acknowledgments
We would like to thank the anonymous referees for their helpful (and enthusiastic!) comments that improved our paper. Hitoshi Omori is a Postdoctoral Research Fellow of Japan Society for the Promotion of Science (JSPS). Damian Szmuc is enjoying a PhD fellowship of the National Scientific and Technical Research Council of Argentina (CONICET) and his visits to Kyoto when this collaboration took place were partially supported by JSPS KAKENHI Grant Number JP16H03344.
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Omori, H., Szmuc, D. (2017). Conjunction and Disjunction in Infectious Logics. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_19
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