Borderline vs. unknown: comparing three-valued representations of imperfect information

International audienceIn this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and González-Rodríguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility–necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions

Interpolation in Linear Logic and Related Systems

We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that they all enjoy a guarded form of Craig interpolation. We also exhibit continuum-many axiomatic extensions of each of these logics without the deductive interpolation property

Investigations in Belnap's Logic of Inconsistent and Unknown Information

Nuel Belnap schlug 1977 eine vierwertige Logik vor, die -- im Gegensatz zur klassischen Logik -- die Faehigkeit haben sollte, sowohl mit widerspruechlicher als auch mit fehlender Information umzugehen. Diese Logik hat jedoch den Nachteil, dass sie Saetze der Form 'wenn ..., dann ...' nicht ausdruecken kann. Ausgehend von dieser Beobachtung analysieren wir die beiden nichtklassischen Aspekte, Widerspruechlichkeit und fehlende Information, indem wir eine dreiwertige Logik entwickeln, die mit widerspruechlicher Information umgehen kann und eine Modallogik, die mit fehlender Information umgehen kann. Beide Logiken sind nicht monoton. Wir untersuchen Eigenschaften, wie z.B. Kompaktheit, Entscheidbarkeit, Deduktionstheoreme und Berechnungkomplexitaet dieser Logiken. Es stellt sich heraus, dass die dreiwertige Logik, nicht kompakt und ihre Folgerungsmenge im Allgemeinen nicht rekursiv aufzaehlbar ist. Beschraenkt man sich hingegen auf endliche Formelmengen, so ist die Folgerungsmenge rekursiv entscheidbar, liegt in der Klasse $\Sigma_2^P$ der polynomiellen Zeithierarchie und ist DIFFP-schwer. Wir geben ein auf semantischen Tableaux basierendes, korrektes und vollstaendiges Berechnungsverfahren fuer endliche Praemissenmengen an. Darueberhinaus untersuchen wir Abschwaechungen der Kompaktheitseigenschaft. Die nichtmonotone auf S5-Modellen basierende Modallogik stellt sich als nicht minder komplex heraus. Auch hier untersuchen wir eine sinnvolle Abschwaechung der Kompaktheitseigenschaft. Desweiteren studieren wir den Zusammenhang zu anderen nichtmonotonen Modallogiken wie Moores autoepistemischer Logik (AEL) und McDermotts NML-2. Wir zeigen, dass unsere Logik zwischen AEL und NML-2 liegt. Schliesslich koppeln wir die entworfene Modallogik mit der dreiwertigen Logik. Die dabei enstehende Logik MKT ist eine Erweiterung des nichtmonotonen Fragments von Belnaps Logik. Wir schliessen unsere Betrachtungen mit einem Vergleich von MKT und verschiedenen informationstheoretischen Logiken, wie z.B. Nelsons N und Heytings intuitionistischer Logik ab

Damned If You Do, Damned If You Don\u27t: A Logical Analysis of Moral Dilemmas

This project explores the logical structure of moral dilemmas. I introduce the notion of genuine contingent moral dilemmas, as well as basic topics in deontic logic. I then examine two formal arguments claiming that dilemmas are logically impossible. Each argument relies on certain principles of normative reasoning sometimes accepted as axioms of deontic logic. I argue that the principle of agglomeration and a statement of entailment of obligations are both not basic to ethical reasoning, concluding that dilemmas will be admissible under some logically consistent ethical theories. In the final chapter, I examine some consequences of admitting dilemmas into a theory, in particular how doing so complicates assignment of blame

Resolution Proof Technique in Linear Temporal Logic.

This dissertation presents a resolution proof technique for Propositional Linear Temporal Logic of discrete time with the Until operator. The presented proof technique stems from the resolution method developed by L. Farinas del Cerro and A. Cavalli. However, their method is incomplete, and their completeness proof, as originally reported, is incorrect. Unlike Farinas\u27s method, our proof technique incorporated the Until operator, which is a very powerful and useful in describing complex temporal relationships which are common in many areas of computer science. Our technique is also proved complete. The presented resolution method for linear temporal logic is similar to classical resolutions: the main goal is to show unsatisfiability of a set of temporal clauses by locating, either directly or indirectly, a state which contains unsatisfiability. Subsequent resolvents of a refutation are obtained by resolving out complementary literals referring to the same instant of time. In order to increase the efficiency of the resolution proof technique, we have developed a refinement of the presented basic method. This refinement is similar to the well-known Ordered Linear (OL) strategy for classical resolution. We also present the lifting of the basic resolution method to predicate linear temporal logic. Unlike First Order Logic, clauses of predicate linear temporal logic may contain variables which are quantified existentially, because skolemization is not valid here. We use standard unification with substitution on universally quantified variables. However, if a term substituted in place of a variable contains any flexible symbols, a constant or a function is flexible if it has different translation in different states, then all occurrences of the substituted variable must refer to the same instant of time (state). Otherwise, the unification may lead to incorrect results. Resolution in predicate linear temporal logic, though very useful from a practical standpoint, is incomplete, since no predicate temporal logic with arithmetic model of time is complete

Agential Obligation as Non-Agential Personal Obligation plus Agency

I explore various ways of integrating the framework for predeterminism, agency, and ability in[P.McNamara, Nordic J. Philos. Logic 5 (2)(2000) 135] with a framework for obligations. However,the agential obligation operator explored here is defined in terms of a non-agential yet personal obligation operator and a non-deontic (and non-normal) agency operator. This is contrary to the main current trend, which assumes statements of personal obligation always take agential complements. Instead, I take the basic form to be an agent’s being obligated to be such that p. I sketch some logics for agential obligation based on personal obligation and agency, first in a fairly familiar context that rules out conflicting personal obligations (and derivatively, conflicting agential obligations), and then in contexts that do allow for conflicts (of both sorts)