How has belief modality contributed to formal semantics?

How has belief modality contributed to formal semantics?∗ Satoshi Tojo JAIST Abstract. Looking back the history of formal treatment of linguistics, we cannot disregard the contri- bution of possible world semantics. Intensional logic of Montague semantics, DRT (Discourse Repre- sentation Theory), mental space, and situation theory are closely related to or compared with the notion of possible world. All these theories have commonly clarified the structure of belief context or uncertain knowledge, employing hypothesized worlds. In this talk, I firstly brief the pedigree of these theories. Next, I will introduce the recent development of modal logic for the representation of (i) knowledge and belief and (ii) time, in which belief modality is precisely discussed together with the accessibility among possible worlds. I will refer to BDI (belief-desire-intention) logic, CTL (computational tree logic), and sphere-based model in belief revision. Finally, I will discuss how these theories could be applied to the further development of analyses of natural language. 1. Introduction Let me introduce two examples how modality is used in natural language semantics. Firstly, when non-native English learners first learn two meanings of each of ‘must’ and ‘may’, that is epistemic/deontic readings, they are baffled: what the two meanings have in common? must may epistemic have firm knowledge have vague knowledge deontic force obligation permit tolerantly The possible world semantics may (or must?) convince some of them, that is, must for all possible worlds, denoted by □ may some possible worlds exist, denoted by ♦ Remember in First-Order Logic, ∀ (for all) and ∃ (some · · · exist) are interchangeable each other; i. e., ∀x p(x) is equal to ¬∃¬p(x) (there exists no x such that ¬p(x)). In the similar way, ‘□’ can be replaced by ‘¬♦¬’ in modal logic. In Japanese, the deontic reading of ‘must’ is translated as –nakere–ba–naranai. Here, the first ‘–nakere–’ is the negation, and the second ‘–ba–’ suggests a case, and the third ‘–naranai’ is again the negation. Thus, Japanese represent deontic ‘□’ by ‘¬♦¬’. Secondly, we can introduce a hypothetical belief situation either in the past tense or in the conjunc- tive mood in Indo-European languages, and also we may mention an imaginary current situation in the conditionalmood,1 as ,,Wenn ich mehr Zeit h¨utte, so w¨urde ich Ihnen einen l¨angeren Brief schreiben.“ ⟨⟨S’il venait demain, tu le verrais.⟩⟩ “Se facesse bel tempo domani, che faresti?” In English both the conjunctive and the conditional moods have been reduced to the subjunctive mood. “If I were a bird, I would fly to you.” ∗Copyright 2007 by Satoshi Tojo 1Konjunctive in German. 42 Why do we employ the past tense when we introduce a counterfactual situation? One explanation is that we consider retreating once to a past (by the conjunctive mood) to annihilate the current situation, and restarting from the past to the different ‘now’ (by the conditional mood) as in the following figure. ↗◦ new now −→ ◦ −→ ◦ ↘ ◦ −→ ◦ −→ now Actually, in English and German which do not possess the …

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