Download e-book for kindle: A New Perspective on Nonmonotonic Logics by Dov M. Gabbay, Karl Schlechta

By Dov M. Gabbay, Karl Schlechta

ISBN-10: 3319468154

ISBN-13: 9783319468150

ISBN-10: 3319468170

ISBN-13: 9783319468174

In this e-book the authors current new effects on interpolation for nonmonotonic logics, summary (function) independence, the Talmudic Kal Vachomer rule, and an equational resolution of contrary-to-duty duties. The bankruptcy on formal development is the conceptual center of the booklet, the place the authors mix the tips of various kinds of nonmonotonic logics and their research of 'natural' strategies right into a formal common sense, a unique preferential building that mixes formal readability with the intuitive benefits of Reiter defaults, defeasible inheritance, thought revision, and epistemic considerations.

It is appropriate for researchers within the sector of desktop technological know-how and mathematical logic.

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38 1 Introduction We start with probabilistic independence. 6 (1) Let I = ∅ be an arbitrary (index) set, for i ∈ I Ui = ∅ arbitrary sets. Let U := {Ui : i ∈ I}, and ⊆ U, an arbitrary set of sequences. If all Ui = {0, 1}, then U = P(I), and ⊆ P(I). Recall, that for X ⊆ I, U X := {Ui : i ∈ X}, etc. (2) Let P : P(U) → [0, 1] be a probability measure. ) (3) By abuse of language, for X ⊆ I, x ∈ U X, let P(x) := P({u ∈ U : u X = x}). Analogously, for X, Y ⊆ I, X ∩ Y = ∅, x ∈ U X, y ∈ U Y , let P(x, y) := P({u ∈ U : u X = x and u Y = y}).

10 (Loop1) AB1 B2 , . . , ABi−1 Bi , ABi Bi+1 , ABi+1 Bi+2 , . . , ABn−1 Bn , ABn B1 ⇒ AB1 Bn so we turn ABn B1 around to AB1 Bn . When we have to be more precise, we will denote this condition (Loop1n ) to fix the length. 3 There Is No Finite Characterization We turn to our main result. Consider the following simple, short, loop for illustration: ABC, ACD, ADE, AEF, AFG, AGB ⇒ ABG—so we can turn AGB around to ABG. Of course, this construction may be arbitrarily long. The idea is now to make ABG false, and, to make it coherent, to make one of the interior conditions false, too, say ADE.

We can combine f and f to a new function in , as long as they agree on Y . g. if X | Y | Z holds, and Z ⊆ Z, then X | Y | Z holds, too, for any suitable I and . The problem is to find a full characterization for this ternary relation. Our main result here is that there is no characterization of the relation, of any fixed finite size. Before we show the result, we describe the framework, so the reader can put the problem into context. But it is not necessary to know this context, in order to understand the result, which is very elementary.

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A New Perspective on Nonmonotonic Logics by Dov M. Gabbay, Karl Schlechta

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