G. Mints. A Short Introduction to Modal Logic. CSLI Lecture Notes 30, CSLI Publications, Stanford, 1992. 15  Home/Search   Document Not in Database   Summary   Related Articles   Check

....sequent calculus based on the idea of residuation and Galois connection. 3. 1 Tableau calculi Tableau calculi incorporating the accessibility relation of possible worlds models were first introduced by Kripke [27] and were later linearized by various authors, notably Fitting [13, 14] and Mints [33]. The basic declarative unit of these calculi is not just a formula A, but rather a formula plus label (#, A) In general, the label # is a non empty finite sequence of positive integers. A simplification is possible for S5. Since S5 is characterized by the class of all frames with a universal ....

G. Mints. A Short Introduction to Modal Logic. CSLI Publications, 1992.

....of providing a cut free formulation for S5 can be a solved with the help of hypersequents. In a sense, the idea goes back at least to [Kr59a] where a semantic tableaux for S5 is presented. This tableaux system can easily be presented in the form of a Gentzen type calculus (see, e.g. Mi74] and [Mi92]) which in turn can be viewed as a hypersequential calculus. Following Kripke, Mints calculus uses formulae labeled with worlds, and one of these worlds is designated. The hypersequential form is therefore only implicit here. It is made explicit in [Po83] where hypersequents and nothing beyond ....

....of (MS) is due to the fact that in an S5 model (M; v) a formula of the form A is either true in all elements of M , or false in all them. Xi The completeness of GS5 and the cut elimination theorem are proved simultaneously in the next theorem. Its proof was essentially given in [Kr59a] see also [Mi92]) Since it was not stated as such there, we give below a simplified version. Theorem 3. If a hypersequent G is valid then it has a cut free proof. Proof: Call a hypersequent H saturated if the following conditions are satisfied. 1) If Gamma ) Delta is a component of G and A B 2 Delta then ....

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Mints G, A Short Introduction to Modal Logic, CSLI Lecture Notes, No. 30, 1992.

....reflexive, i.e. R(x; x) for all worlds x. ffl System S4. In this logic, the accessibility relation R should be reflexive and transitive (R is a preorder) i.e. 8w:R(w; w) and 8w; w 0 ; w 00 : R(w; w 0 ) R(w 0 ; w 00 ) R(w; w 00 ) For a quick introduction to modal logic, see [20]. System S4 gives a logical explanation of computational stages. Each world in the semantic model corresponds to a computational stage. A term of type 2 corresponds to a code that can be executed in a future stage, i.e. an accessible world. Terms of a modal type (which corresponds to ....

Grigori Mints. A short introduction to modal logic. Center for the study of language and information, 1992.

....On leave from University of Karlsruhe, Institute for Logic, Complexity and Deduction Systems, D 76128 Karlsruhe, Germany. Our object language uses labelled formulae like oe : A, where oe is a label and A is a formula, with intuitive reading the possible world oe satisfies the formula A ; see [6, 18, 11] for details. Thus, 1 : 2p says that the possible world 1 satisfies the formula 2p. Our box rule then reduces the formula 1 : 2p to the labelled formula 1: x) p which contains the universal variable x in its label and has an intuitive reading the possible world 1: x) satisfies the formula p . ....

G. Mints. A Short Introduction to Modal Logic. CSLI, Stanford, 1992.

.... A ) A External Structural Rules G GjH GjSjSjH GjSjH GjS 1 jS 2 jH GjS 2 jS 1 jH (External weakening, contraction and permutation, respectively) 27 Hypersequents were first introduced by Pottinger in [Po83] and independently in [Av87] Related structures were used before by Mints (see [Mi92]) Internal Structural Rules Gj Gamma 1 ; A; B; Gamma 2 ) DeltajH Gj Gamma 1 ; B; A; Gamma 2 ) DeltajH Gj Gamma ) Delta 1 ; A; B; Delta 2 jH Gj Gamma ) Delta 1 ; B; A; Delta 2 jH Gj Gamma; A ) DeltajH Gj Gamma; A; A ) DeltajH Gj Gamma ) Delta; AjH Gj Gamma ) Delta; A; AjH ....

Mints G., A Short Introduction to Modal Logic, CSLI Lecture Notes, No. 30, 1992. smallskip

....results briefly, to connect the intuitions behind new proof systems with the intuitions behind old ones, and to highlight the distinctive complexities that remain in modal deduction and how those relate to standard characterizations of the complexity of modal deduction. The reader may consult (Mints, 1992) for a more thorough introduction to modern modal proof theory and (Gallier, 1986) for an introduction to the connections between proof theory and automated deduction. I begin in section 2.1 with an informal example intended to motivate modal scoping mechanisms and to introduce a key theme: how ....

Mints, G. (1992). A Short Introduction to Modal Logic. Number 30 in CSLI Lecture Notes. CSLI.

.... A P style implementation compares favourably with existing fast implementations of modal tableau systems like LWB [12] Our object language uses labelled formulae like oe : A, where oe is a label and A is a formula, with intuitive reading the possible world oe satisfies the formula A ; see [6, 17, 10] for details. Thus, 1 : 2p says that the possible world 1 satisfies the formula 2p. Our box rule then reduces the formula 1 : 2p to the labelled formula 1: x) p which contains the universal variable x in its label and has an intuitive reading the possible world 1: x) satisfies the formula p . ....

Grigori Mints. A Short Introduction to Modal Logic. CSLI, Stanford, 1992.

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G. Mints. A Short Introduction to Modal Logic. CSLI Lecture Notes 30, CSLI Publications, Stanford, 1992. 15

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G. E. Mints, A Short Introduction to Modal Logic, Center for the Study of Language and Information (CSLI), Stanford University, Stanford, California, 1992.

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G. Mints. A Short Introduction to Modal Logic. CSLI, 1992.

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G. Mints. A Short Introduction to Modal Logic. CSLI Publications, 1992.

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G. Mints, A short introduction to modal logic, CSLI (Center for the Study of Language and Information), Stanford University, Stanford, CA, 1992.

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Grigori Mints. A Short Introduction to Modal Logic. CSLI, Stanford, 1992.

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Mints G, A Short Introduction to Modal Logic, CSLI Lecture Notes, No. 30, 1992.

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Mints, G.: 1992, A Short Introduction to Modal Logic, CSLI Lecture Notes 30, Stanford.

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