Extensions in graph normal form

Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.publishedVersio

There are only two paradoxes

Using a graph representation of classical logic, the paper shows that the liar or Yablo pattern occurs in every semantic paradox. The core graph theoretic result generalizes theorem of Richardson, showing solvability of finite graphs without odd cycles, to arbitrary graphs which are proven solvable when no odd cycles nor patterns generalizing Yablo's occur. This follows from an earlier result by a new compactness-like theorem, holding for infinitary logic and utilizing the graph representation.Comment: 7 pages, submitted to a journa

Paraconsistent resolution

Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance

Singular and plural non-deterministic parameters

The article defines algebraic semantics of singular (call-time-choice) and plural (run-time-choice) nondeterministic parameter passing and presents a specification language in which operations with both kinds of parameters can be defined simultaneously. Sound and complete calculi for both semantics are introduced. We study the relations between the two semantics and point out that axioms for operations with plural arguments may be considered as axiom schemata for operations with singular arguments

Complete axiomatizations of finite syntactic epistemic states

Abstract. An agent who bases his actions upon explicit logical formulae has at any given point in time a finite set of formulae he has computed. Closure or consistency conditions on this set cannot in general be assumed – reasoning takes time and real agents frequently have contradictory beliefs. This paper discusses a formal model of knowledge as explicitly computed sets of formulae. It is assumed that agents represent their knowledge syntactically, and that they can only know finitely many formulae at a given time. Existing syntactic characterizations of knowledge seem to be too general to have any interesting properties, but we extend the meta language to include an operator expressing that an agent knows at most a particular finite set of formulae. The specific problem we consider is the axiomatization of this logic. A sound system is presented. Strong completeness is impossible, so instead we characterize the theories for which we can get completeness. Proving that a theory actually fits this characterization, including proving weak completeness of the system, turns out to be non-trivial. One of the main results is a collection of algebraic conditions on sets of epistemic states described by a theory, which are sufficient for completeness. The paper is a contribution to a general abstract theory of resource bounded agents. Interesting results, e.g. complex algebraic conditions for completeness, are obtained from very simple assumptions, i.e. epistemic states as arbitrary finite sets and operators for knowing at least and at most.

Kernels of digraphs with finitely many ends

According to Richardson’s theorem, every digraph without directed odd cycles that is either (a) locally finite or (b) rayless has a kernel (an independent subset with an incoming edge from every vertex in ). We generalize this theorem showing that a digraph without directed odd cycles has a kernel when (a) for each vertex, there is a finite set separating it from all rays, or (b) each ray contains at most finitely many vertices dominating it (having an infinite fan to the ray) and the digraph has finitely many ends. The restriction to finitely many ends in (b) can be weakened, admitting infinitely many ends with a specific structure, but the possibility of dropping it remains a conjecture

Strongly complete axiomatizations of ”knowing at most” in standard syntactic assignments

Abstract. Standard syntactic assignments (SSAs) model knowledge directly rather than as truth in all possible worlds as in modal epistemic logic, by assigning arbitrary truth values to atomic epistemic formulae. It is a very general approach to epistemic logic, but has no interesting logical properties — partly because the standard logical language is too weak to express properties of such structures. In this paper we extend the logical language with a new operator used to represent the proposition that an agent “knows at most ” a given finite set of formulae and study the problem of strongly complete axiomatization of SSAs in this language. Since the logic is not semantically compact, a strongly complete finitary axiomatization is impossible. Instead we present, first, a strongly complete infinitary system, and, second, a strongly complete finitary system for a slightly weaker variant of the language.