Abstract
I argue that a recent philosophical interpretation by Jc Beall of the middle value of Weak Kleene (three-valued) logic (known also as Bochvar’s logic) as ‘being off-topic’ is untenable. My main claim is that “being off-topic” is a relation, not a property, and as such cannot serve as an interpretation of a truth-value.
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Notes
I use 0.5 as the third truth-value just because Beall does. It should not suggest that the truth-values are ordered in any way.
As the issue touches also on natural language (NL), I will use ‘sentence’ and ‘formula’ synonymously.
I confine the discussion here to propositional logics only.
I ignore here the more delicate issue of distinguishing truth-values from arbitrary algebraic values, since the issue of interpreting the truth-table is orthogonal to this distinction.
For the argument here, I can safely ignore relational assignments.
Standardly, this dependence is not transparent in propositional logics, usually seen as assigning arbitrary truth-values to atomic sentences, in all possible ways. In a setting in which atomic sentences are viewed as regimentation of sentences in a more expressive language, having a richer space of semantic properties, this dependence can be made more transparent.
I relate here to binary operations only just for convenience. The same holds for applying an nary operation for any n.
Actually, I believe that for logical object languages a theory of topics is better formulated in a first-order setting, offering more natural candidates for counting as topics than propositional languages do.
I do not commit myself here to a specific view of how many topics can a sentence have. I just assume that this number is at least one.
I very much share Beall’s rejection of Addition—Gentzen’s disjunction introduction—but for reasons differing from his altogether.
This can easily be extended to the case where a sentence has more than one topic.
The distinction is not always sharp. I simplify here and take it as such for explanatory purposes, as this is not the topic of the paper.
Again, there are exceptions. For example, \({\textsf {Mary\ is\ a\ sister\ and\ John\ is\ brother\ of\ Bill}}\). I again simplify the linguistic issues for explanatory reasons.
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Appendix
Appendix
Below are the truth-tables of \(WK_{3}\).
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Francez, N. On Beall’s New Interpretation of \(WK_{3}\). J of Log Lang and Inf 28, 1–7 (2019). https://doi.org/10.1007/s10849-018-9274-6
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DOI: https://doi.org/10.1007/s10849-018-9274-6