LLAL@GSIS (V)

Date & Venue

• Date: February 24, 2025.

• Venue: Room 207, Graduate School of Information Sciences, Tohoku University.  

Speakers

• Matteo Bizzarri (Scuola Normale Superiore)

• Yudai Kubono (Shizuoka University)

• Elise Perrotin (AIST)

• Satoshi Tojo (Asia University)

• Masanobu Toyooka (Hokkaido University) 

Program

10:30--11:30 Matteo Bizzarri "How to dissolve the Lottery Paradox as a lump of sugar in water: Wittgenstein's probability and supraclassical logics"

11:30--12:30 Masanobu Toyooka "Gentzenization of Wansing's expansions of Nelson's logics"

12:30--14:00 Lunch 

14:00--15:00 Yudai Kubono "A Representation of Explicit Knowledge and Epistemic Indistinguishability in a Logic of Awareness"

15:00--15:15 Break

15:15--16:15 Elise Perrotin "Towards lightweight epistemic-doxastic planning: on true beliefs and mere beliefs"

16:15--16:30 Break

16:30--17:30 Satoshi Tojo "Labeled Lambek Calculus for Belief Context"

Abstracts

Matteo Bizzarri: In the Tractatus, Wittgenstein introduced a method for calculating probability using truth tables—a technique that later influenced Carnap and Ramsey. However, his approach received little attention in the literature. Wittgenstein's method involves comparing two propositions: the first is considered only in cases where it is true, while the second is analyzed within this restricted domain. This bears a resemblance to Makinson's supraclassical logic, albeit through different means.

This paper aims to clarify Wittgenstein’s probabilistic method, examining its foundational principles and drawing connections to supraclassical logic. We argue that Wittgenstein anticipated certain modern developments in logic, formulating one of the earliest systems capable of integrating beliefs within a formal calculus. Finally, we show how his framework dissolves the Lottery Paradox, revealing that the paradox does not arise within this system.

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Yudai Kubono: In the logic of awareness proposed in (Fagin and Halpern, 1988), a notion of awareness is introduced and knowledge is classified into explicit and implicit knowledge to represent human knowledge. Focusing on epistemic indistinguishability dependent on awareness, we propose logical semantics for a representation of explicit knowledge. Using a concrete example, we demonstrate that the formalization allows us to intuitively describe explicit knowledge gained through logical inference. We also provide the axiomatic system ALP and prove its completeness.

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Elise Perrotin: Epistemic-doxastic planning allows us to plan while taking into account the agents' knowledge and beliefs. For example, a video game enemy might try a sneak attack if they believe the player has not seen them, but they will instead sound an alarm if they believe they have been seen. While epistemic planning has recently received some attention, adding beliefs into the mix means having to handle possible false beliefs and belief revision, which complicates things quite a bit. For this reason epistemic-doxastic planning is still relatively unexplored in the literature. In this talk I will introduce a recently proposed lightweight epistemic-doxastic logic which is based on the non-standard operators of "true belief about" and "mere belief about", and discuss its use as a tool for specifying actions and planning tasks which integrate both knowledge and beliefs of agents.

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Satoshi Tojo: Looking back at the history of epistemic logic, we first discuss how the scope of belief has been represented in categorial grammar. When we translate the grammar into the sequent calculus, we can employ the Box operator with axiom (K). However, we would like to implement more precise notions in the calculus, including the accessibility of Kripke semantics. In this talk, we propose to employ labeled Lambek calculus for belief context.

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Masanobu Toyooka: This presentation provides cut-free sequent calculi for Wansing's expansions of Nelson's logics. Gabbay (1982) proposed an expansion of the intuitionistic logic by the consistency operator M as a logic of non-monotonic reasoning. Its semantics is obtained by adding the satisfaction relation for M to the Kripke semantics for intuitionistic logic, where M has the same satisfaction relation as that for a diamond operator in the Kripke semantics for modal logics. Łukaszewicz (1990) pointed out some defects of this logic, and Wansing (1995) argued that they were overcome by changing the base logic from intuitionistic logic from Nelson's three-valued logic in the propositional setting. He semantically proposed two different logics having the different falsification clauses for M, with Nelson's three-valued logic being the base logic of both of the two logics. Skura (2017) provided the refutation system for one of its logics. Omori (2016) proposed the Hilbert system for a logic having the same falsification clause as Skura investigated, where the base logic is changed from Nelson's three-valued logic to Nelson's four-valued logic. As is seen from these preceding studies, the two different falsification clauses exist for M and the two different base logics, i.e., Nelson's three-valued and four-valued logics, exist. Thus, four different logics can be obtained. In this presentation, we newly propose cut-free sequent calculi for all of the four logics. Although the display calculus was proposed by Wansing (1999) for a similar logic, there have been no sequent calculi for the four logics. Moreover, the decidability of each logic is proved, and the Hilbert system for each logic is extracted.

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Acknowledgement

This workshop is partially supported by the Graduate School of Information Sciences, Tohoku University.