By Lev Beklemishev (auth.), Nick Bezhanishvili, Sebastian Löbner, Kerstin Schwabe, Luca Spada (eds.)

Edited in collaboration with FoLLI, the organization of good judgment, Language and data, this ebook constitutes the refereed lawsuits of the eighth overseas Tbilisi Symposium on common sense, Language, and Computation, TbiLLC 2009, held in Bakuriani, Georgia, in September 2009. The 20 revised complete papers incorporated within the ebook have been conscientiously reviewed and chosen from various shows given on the symposium. the focal point of the papers is at the following issues: normal language syntax, semantics, and pragmatics; confident, modal and algebraic common sense; linguistic typology and semantic universals; logics for synthetic intelligence; info retrieval, question resolution platforms; common sense, video games, and formal pragmatics; language evolution and learnability; computational social selection; historic linguistics, heritage of logic.

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**Example text**

Another example is a variant of modal logic K given in a language with a cover nabla modality ∇. The ﬁnitary cover modality ∇ operates on ﬁnite sets of formulas, so whenever Γ is a ﬁnite set of formulas, ∇Γ is a formula. Its semantics in a state s of a Kripke model S = (S, R, ) is that Γ and the future of s R[s] = {t | sRt} cover each other, formally: S, s ∇Γ iﬀ (∀t ∈ R[s]) (∃γ ∈ Γ ) S, t and (∀γ ∈ Γ ) (∃t ∈ R[s]) S, t γ γ. (1) Using the semantics it is easy to check that the following mutual deﬁnition between ∇ and the two standard modalities and ♦ hold: ∇Γ ↔ Γ ∧ ♦Γ φ ↔ ∇{φ} ∨ ∇∅ ♦φ ↔ ∇{φ, } (2) 34 M.

Now let X⊕ , τ⊕ be the Fs -sum Fs Xc . Lemma 4. X⊕ , τ⊕ is a spectral space. Proof. First we show that X⊕ , τ⊕ is T0 . Let x, y ∈ X⊕ be two distinct points. Let also π(x) = c and π(y) = d, where π : X⊕ Ws is the canonical map, sending x ∈ Xc to c. If c = d, then as Rs is irreﬂexive transitive, we have cR s d or dR s c. Without loss of generality we may assume that cR s d. Then d ∈ {c}∪Rs (c). Therefore, x ∈ π −1 ({c}∪Rs (c)) and y ∈ π −1 ({c}∪Rs (c)). Thus, π −1 ({c}∪Rs (c)) is an open subset of X⊕ separating x from y.

Modal case. Notre Dame Journal of Formal Logic (2011) (to appear) 5. : Modal Logic. Cambridge University Press, Cambridge (2001) 6. : Weak transitivity—a restitution. In: Logical Investigations, No. 8 (Moscow, 2001), Nauka, Moscow, pp. 244–255 (2001) (in Russian) 7. : Intuitionistic logic and modality via topology. Ann. Pure Appl. Logic 127(1-3), 155–170 (2004) 8. : Logics containing K4. II. J. Symb. Logic 50(3), 619–651 (1985) 9. : Modal deﬁnability in topology. Master’s Thesis, University of Amsterdam (2001) 10.