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A Reinterpretation of Beall’s ‘Off-Topic’ Semantics

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Abstract

Jc Beall’s off-topic interpretation of Weak Kleene logic offers a logic of ‘true-and-topic’ preservation. However, Nissim Francez has recently argued that being ‘off-topic’ is a relational and not an absolute semantic property; as such, it fails to satisfy the conditions of truth-functionality. For Francez, this means that it ‘cannot serve as an interpretation of a truth-value’. In this paper, I propose a two-layered reinterpretation of Beall’s off-topic semantics. This two-layered framework has two crucial features: a sentential topic-tagging device and a two-layered evaluation tool. I show that this framework results in a logic that preserves Beall’s key insight and addresses Francez’s concerns.

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Notes

  1. Beall contrasts his interpretation from the interpretation of Demolombe and Jones (1999), which, he claims, is more in-line with the relevant logic tradition of Angell (1962) and Parry (1933) rather than WK3 per se (Beall 2016, 140).

  2. Francez also offers a proof-theoretic counterpart for TVM, which will not be discussed in this paper.

  3. This is the right result since addition—the inference from a sentence, A to a disjunctive conclusion, \((A \vee B)\) is invalid in WK3.

  4. My thanks to this journal’s referee for pointing out this worry.

  5. My thanks to the referee for this clarification.

  6. My thanks to the referee for this important suggestion.

  7. This is a concern raised by this journal’s referee.

  8. As the referee correctly points out, since the proposed 2L framework does not provide a substantive theory of topics, it might be difficult to apply it to natural examples. In particular, the 2L requirement that tagged atomic sentences must be about exactly one topic is very restrictive. For example, suppose that topics are noun-phrases occurring in subject and object positions in atomic sentences headed by transitive verbs. Now consider the atomic sentence ’John loves Mary’. Arguably, this atomic sentence has both John and Mary as its topic. This means that, contra the proposed 2L semantics framework, ‘John loves Mary or Sue’ is an on-topic compound sentence, and the following inference is valid and topic-preserving: ‘John loves Mary or Sue’, ‘John does not love Mary’, so ‘John loves Sue’.

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Acknowledgements

My thanks go to Jc Beall, Franz Berto, Hazel T. Biana, Ben Blumson, Peter Eldridge-Smith, Brian Garrett, Greg Restall, Dave Ripley, Raymond R. Tan, and the anonymous referee of this journal for the comments and suggestions that greatly improved this paper.

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  1. Department of Philosophy, De La Salle University, Manila, Philippines

    Jeremiah Joven B. Joaquin

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Joaquin, J.J.B. A Reinterpretation of Beall’s ‘Off-Topic’ Semantics. J of Log Lang and Inf 31, 409–421 (2022). https://doi.org/10.1007/s10849-022-09353-2

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