Abstract
In this paper I describe a construction which can be applied to any many-valued logic to give a plurivalent logic, that is, a logic in which formulas may take more than one value. Various results are established concerning the relationship between the many-valued logic and the corresponding plurivalent logic; and a detailed analysis is provided of the relationship between the two for a small family of many-valued logics related to the logic of First Degree Entailment.
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Notes
- 1.
For a detailed description of Imaginary Logic, see Priest (2000).
- 2.
What follows is published as ‘Plurivalent Logic’, Australasian Journal of Logic, 12 (2014), pp. 2–13. It is reprinted with their permission.
- 3.
- 4.
See Priest (2008a), ch. 7.
- 5.
An interesting alternative is to define ‘\(\vartriangleright \) designates A’ as: for all v such that \(A\vartriangleright v\), v ∈ D. I defer discussion of this possibility to a brief appendix to the paper.
- 6.
These can be found in Czelakowski (2001), esp. Prop 0.3.3.
- 7.
- 8.
- 9.
Many thanks to Thomas Ferguson for drawing my attention to this.
- 10.
See Priest (1984).
- 11.
- 12.
It is worth noting that Shramko and Wansing (2011), chs. 3 and 4, show how, given any univalent logic M, to construct a logic whose values are the values of \(\ddot {M}\). Given their approach, it is natural to think of the empty set not simply as an absence of values, but as a positive value in its own right. This motivates different possible definitions for the truth functions in the logic, producing somewhat different results. In particular, with these definitions, the empty set does not generate a failure of ∨-introduction. In this context, it is worth noting that applying plurivalence to classical logic does not produce FDE.
- 13.
This is done for classical logic and positive plurivalence in Priest (1984).
- 14.
For the limit: anything valid in the limit logic is valid in each finite “approximation”. Conversely, anything invalid in it is invalid in some approximation, since only finitely many values are employed in the counter-model.
- 15.
- 16.
Many thanks go to Lloyd Humberstone for very helpful comments on an earlier draft of this paper.
References
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Appendix
Appendix
This appendix concerns a variation of the definition of designation for plurivalent logics. Specifically, we replace the definition of designation of Sect. 12.2 by: ‘\(\vartriangleright \) designates A’ as: for all v such that \(A\vartriangleright v\), v ∈ D. This changes nothing till Sect. 12.5, except the definition of the designated values in \(\dot {M}\), where the modification required is obvious. For Sect. 12.5, let M n be the semantics obtained from M by adding the value n, if necessary. Then line (6) holds provided we replace M b with M n. The homomorphism that establishes this is defined as follows:
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if e ∈ X then θ(X) = e
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else: if n ∈ X then θ(X) = n
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else: if t ∈ X and f ∈ X then θ(X) = n
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else:
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if X = {t, b} then θ(X) = t
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if X = {t, f} then θ(X) = f
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if X = {x} then θ(X) = x
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One may check that this is a homomorphism, and that it is onto.
Things are quite different with respect to the general plurivalence of Sect. 12.6, however; and matters do not modify in such a straightforward way. In particular, the alignment between \(\models _{p}^{M^{e}}\) and \(\models _{g}^{M}\) disappears in both directions. Thus, for M take classical logic (∅). Then, as one may check, \(p\wedge q\models _{p}^{e}p\), \(p\wedge q\nvDash _{g}^{\emptyset }p\) (let \(\vartriangleright \) relate q to just f, and p to nothing); and \(p\nvDash _{p}^{e}p\vee q\), but \(p\models _{g}^{\emptyset }p\vee q\) (for a counter-model, p must relate to just t; all the values of p ∨ q are then designated—even when q relates to nothing!). What one can say to characterise general plurivalence in this case is still an open question.
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Priest, G. (2017). Plurivalent Logics. 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_12
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