Abstract
Game semantics provides an alternative view on basic logical concepts. Provability games, i.e., games for the validity of a formula provide a link between proof systems and semantics. We present a new type of provability games, namely the Mezhirov game, for minimal propositional logic and two modal systems: the logic of functional frames and the logic of serial frames, i.e., \(\mathbf {KD}\) and prove their adequacy. The games are finite resulting in a finite search for winning strategies.
Research supported by FWF project W1255-N23. The author is grateful to the anonymous referees for comments which led to improvements in this paper.
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Notes
- 1.
\(\mathbf {MPC}\) is alternatively known as \(\mathbf {J}\) after Ingebrigt Johansson who proposed it in [12].
- 2.
To distinguish between the two, sometimes f is used as falsum constant for minimal logic. We, however, will stick to \(\bot \). Whenever we compare the intuitionistic \(\bot \) with the minimal one, we shall use the lower indexes i and m respectively.
- 3.
This version was originally proposed by Johansson. One can even trace it back to the famous paper by Kolmogorov.
- 4.
We shall generally omit the lower index if it does not lead to confusion.
- 5.
A directed path in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices with the additional restriction that the edges should be all directed in the same direction. The length of a path is the number of edges it contains.
- 6.
By boxed formula we understand a formula the main operator of which is \(\Box \), i.e., formula of the form \(\Box \psi \).
- 7.
To be more formal and precise, we can formulate this as follows: “for every game position \(C_{k}\) and every \((\Box \psi )^{i}\in \mathcal {P}_{k}\) s.t. \((\Box \psi )^{i}\ne \varphi ^{0}\): \(\mathbf {P}\) can unmark \((\Box \psi )^{i}\) at the game step \(k+1\) iff \(C_{k+1}\ne C_{l}\) where \(l\le k\)”.
- 8.
This condition indicates that \(x_{i}\) is not the final world and if \(x_{j}\) is the final world the accessibility relation between \(x_{i}\) and \(x_{j}\) is not symmetric.
- 9.
The same argument is applicable to show that \(\mathbf {O}\) does not need to be able to unmark formulae.
- 10.
Note that we are proving that \(\mathbf {P}\) can always keep playing rationally meaning that if \(\mathbf {P}\) was not playing rationally, then \(\mathbf {P}\) could have lost the opportunity to do so.
- 11.
Move (b) can be made iff it forces \(\mathbf {P}\) to have a mistake and \(\mathbf {O}\) to have none.
- 12.
We prove that it is always possible in what follows.
- 13.
The full formal proof is subject to a separate presentation due to its technicality.
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Pavlova, A. (2021). Provability Games for Non-classical Logics. In: Silva, A., Wassermann, R., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2021. Lecture Notes in Computer Science(), vol 13038. Springer, Cham. https://doi.org/10.1007/978-3-030-88853-4_25
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