Abstract
We study an array of logics defined on a small set of connectives (including an implication \(\rightarrow \) and a bottom particle \(\bot \)) by modularly considering subsets of a set of inference rules that we fix at the start of the game. We provide complete semantics based on possibly non-deterministic logical matrices and complexity upper bounds for the considered logics. As a consequence of the techniques applied, we also obtain completeness results for the negation-only fragments (obtained by defining the negation connective as \(\lnot p:=p\rightarrow \bot \), as usual) of all the above-mentioned logics, and analyze their possible paraconsistent character.
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Notes
- 1.
Note that in \(\textsf {CL}\) this is quite natural as all the other classical connectives are definable from \(\rightarrow \) and \(\bot \). In \(\textsf {IL}\) this is not the case, as for example neither conjunction nor disjunction can be defined from \(\rightarrow \) and \(\bot \). Still, for neatness of presentation, we commit this abuse of notation also here.
- 2.
The class of Hilbert algebras [11] is the algebraic semantics of the \(\{\rightarrow \}\)-fragment of intuitionistic logic. Hilbert algebras, also called (positive) implication(al) algebras [1, 15], correspond to \(\{\rightarrow \}\)-subreducts of Heyting algebras and thus have a definable natural order which has 1 as maximum element but need not have a minimum.
- 3.
Which is the \(\{\lnot \}\)-reduct of \(\mathbb {G}_{3}\), corresponding to three-valued Gödel logic, see [9].
- 4.
This easily follows from [5] and the fact that \(\textsf {MP}\) is weaker than \(\textsf {IL}\).
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Acknowledgements
The first two authors thank the support of the FEDER/FCT project PEst-OE/EEI/LA0008/2013 of Instituto de Telecomunicações. The first author also acknowledges the FCT postdoctoral grant SFRH/BPD/76513/2011. Finally, we would like to thank Fey Liang for pointing out a deadly mistake in an earlier version of the paper.
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Marcelino, S., Caleiro, C., Rivieccio, U. (2018). Plug and Play Negations. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_14
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