Lou Goble

A logic for deontic dilemmas

Abstract The possibility of deontic dilemmas poses a significant problem for deontic logic. Here I review some proposals to resolve this problem, and then offer a new account. This is a simple modification of standard deontic logic that enables the system to accommodate deontic dilemmas without inconsistency and without deontic explosion, while at the same time accounting for the range of genuinely valid inferences.

A Proposal for Dealing with Deontic Dilemmas

In this paper I propose a simple modification of standard deontic logic that will enable the system to accommodate deontic dilemmas without inconsistency and without deontic explosion, while at the same time preserving the range of genuinely valid inferences. The proposal applies both to monadic deontic logic and to a dyadic logic of conditional obligation. In the Appendix these systems are proved to be sound and complete with respect to an appropriate semantics and also to be decidable.

Opacity and the Ought-to-Be

In this regard, deontic logic would seem to be the nicest member of the modal logic family. This is how it should be. One expects no problems interpreting quantification into deontic contexts. Unlike the logics for alethic necessity or belief, with deontic logic there is not even a temptation to think that the values of the individual variables are intensions, or individual concepts, or possibilia (permissibilia ?). Statements of obligation, permission, prohibition, etc. seema to be directly about persons or things regardless of how they are conceived or described. The deontic operators thus certainly seem to be extensional; nevertheless, when principles of extensionality are assumed for standard systems of deontic logic, paradox appears.1

An Incomplete Relevant Modal Logic

The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley–Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that characterizes G. (iii) The class of frames for G characterizes the classically based logic T.

Neighborhoods for entailment

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the neighborhood semantics meets the Routley–Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable.

Combinatory Logic and the Semantics of Substructural Logics

The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B∘T, then LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself.