Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen [13]. Both proof-theoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay in [10], while the latter has been advocated by Y. Shoham in [38]. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams in [2]. The pr...

Conditional logics, introduced by Lewis and Stalnaker, have been utilized in artificial intelligence to capture a broad range of phenomena. In this paper we examine the complexity of several variants discussed in the literature. We show that, in general, deciding satisfiability is PSPACE-complete for formulas with arbitrary conditional nesting and NP-complete for formulas with bounded nesting of conditionals. However, we provide several exceptions to this rule. Of particular note are results showing that (a) when assuming uniformity (i.e., that all worlds agree on what worlds are possible), the decision problem becomes EXPTIME-complete even for formulas with bounded nesting, and (b) when assuming absoluteness (i.e., that all worlds agree on all conditional statements), the decision problem is NP-complete for formulas with arbitrary nesting. 1 INTRODUCTION The study of conditional statements of the form "If : : : then : : :" has a long history in philosophy [Sta68, Lew73, Che80, Vel8...

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Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.

DAVID LEWIS proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend LEWIS's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter. Canonicity for Intensional Logics Without Iterative Axioms Timothy J. Surendonk May 13, 1996 Keywords: Canonicity, strong completeness, non-normal modal logics, neighborhood frame semantics, non-iterative logics. Abstract DAVID LEWIS proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend LEWIS's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter. 1 Introduction In 1974 DAVID LEWIS [13] showed that every intension...

Modal logics see a wide variety of applications in artificial intelligence, e.g. in reasoning about knowledge, belief, uncertainty, agency, defaults, and relevance. From the perspective of applications, the attractivity of modal logics stems from a combination of expressive power and comparatively low computational complexity. Compared to the classical treatment of modal logics with relational semantics, the use of modal logics in AI has two characteristic traits: Firstly, a large and growing variety of logics is used, adapted to the concrete situation at hand, and secondly, these logics are often non-normal. Here, we present a shallow model construction that witnesses PSPACE bounds for a broad class of mostly non-normal modal logics. Our approach is uniform and generic: we present general criteria that uniformly apply to and are easily checked in large numbers of examples. Thus, we not only re-prove known complexity bounds for a wide variety of structurally different logics and obtain previously unknown PSPACE-bounds, e.g. for Elgesem’s logic of agency, but also lay the foundations upon which the complexity of newly emerging logics can be determined.

For every Kripke complete modal logic L we define its hybrid companion LH . For a reasonable class of logics, we present a satisfiability-preserving translation from LH to L. We prove that for this class of logics, complexity, (uniform) interpolation, finite axiomatization transfer from L to LH .

In this paper we show that the Hilbert system of agency and ability presented by Dag Elgesem is incomplete with respect to the intended semantics. We argue that completeness result may be easily regained. Finally, we shortly discuss some issues related to the philosophical intuition behind his approach. This is done by examining Elgesem’s modal logic of agency and ability using semantics with different flavours. 1

this paper. Elgesem's semantics for the modal logic of agency and ability is a structure f 1 , . . . , f n ,V (cf. [7, p. 20] and [6, p. 54]), where each f i , 1 n is a function as in the structure described above and i is an agent. Since there are no interactions among the agents and all functions f are independent from each other and obey the same conditions, we can restrict ourselves to the case of a single agent. Elgesem also considers some foundational aspects of the notions he deals with and introduces some additional functions in order to capture the idea of avoidability and accidence. However those functions do not play any relevant role in the characterisation of the modal operators E and C. The valuation function and the constraints on the model are given in terms of properties of f . The other functions are used to specify constraints on concrete instances of f . Finally V is a valuation function while v is an assignment. S1. w # E p iff w v(p); S2. w # E iff w E A; S3. w # E A B iff w E A or w # E B; S4. w # E EA iff w , w); S5. w # E CA iff f (#A# / 0

This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then introduce "ultrafilter semantics," a way of expressing all sets over a canonical frame in an `effable' way. This provides us with a conceptually easy way of dispatching some questions about intensional logics. In particular, we show that all non-iterative intensional logics are canonical and we go on to indicate how we can use ultrafilter semantics to demonstrate the canonicity of the Sahlqvist Logics. Neighborhoods, Ultrafilters, and Canonicity Timothy J. Surendonk September 20, 1996 Abstract This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then i...