Abstract
Recently, in an ongoing debate about informal provability, non-deterministic logics of informal provability BAT and CABAT were developed to model the notion. CABAT logic is defined as an extension of BAT logics and itself does not have independent and decent semantics. The aim of the paper is to show that, semantically speaking, both logics are rather complex and they can be characterized by neither finitely many valued deterministic semantics nor possible word semantics including neighbourhood semantics.
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Notes
For clarity, note that it’s possible to add the reflection principle for a formal provability predicate of a certain formal theory \(\mathbf {T}\). What we meant here is that it is impossible to have a provability predicate that shares the standard conditions put on formal provability and the reflection schema at the same time. In the first scenario the new predicate enriched by the reflection schema is no longer a formal provability predicate of the enriched theory.
To be fair if we add to the premise set a formula \(\mathtt {B}\lnot \lambda \rightarrow \lnot \lambda \) then the consequence does hold. But we are not interested here in an arbitraty instance of the reflection schema but in the translation of the standard proof of Löb’s theorem. Thus, in general the reflection schema is not assumed.
The etymology of the name CABAT is a bit more complex. “C” in the name stands for the closure condition and “A” for algebraic conditions that turned out to be redundant.
For the first interpretation, it’s impossible since the status of a mathematical claim may change. For instance, currently the Goldbach hypothesis is undecidable but it may turn out later that we will be able to prove or disprove it. Thus, a conjuction and disjuction of two independent sentences may turn out to be provable, refutable or remain undecided.
To be fair, it is also a perfectly viable motivation to approach informal provability from supervaluationistic perspective. This approach and the comparison between the two approaches, however, is beyond the scope of this paper.
On the other hand, it is an interesting system to study since it is a starting point and the system on top of which CABAT is defined.
Since BAT does not have any tautologies, we have to stick to its consequence relation.
From the fact that CABAT understood as the set of tautologies cannot have a finite many-valued characterization, it follows that CABAT understood as a consequence relation cannot have deterministic n valued semantics as well.
A non-normal Kripke semantics may contain non-normal worlds, resulting in the weakening of (Nec) [if \(\vdash \varphi \) then \(\vdash \Box \varphi \)] Chellas (1980).
For more information on this semantics see Chellas (1980).
Let F be a family of frames. We say that \(\varphi \) is a global semantic consequence of \(\Gamma \) (\(\Gamma \models \varphi \)) iff for all frames \(G\in F\), if \(G \models \Gamma \) then \(G\models \varphi \). For more details about this distinction see Chapter 1 of Blackburn (2001).
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Acknowledgements
I Would like to thank to Rafal Urbaniak for his time and comments on the previous versions of this paper.
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This work has been supported by FWO [Research Fuoundation Flanders].
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Pawlowski, P. Non-deterministic Logic of Informal Provability has no Finite Characterization. J of Log Lang and Inf 30, 805–817 (2021). https://doi.org/10.1007/s10849-021-09344-9
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DOI: https://doi.org/10.1007/s10849-021-09344-9