Online research in philosophy

This paper spells out a dynamic proof format for the pure logic of relevant implication. (A proof is dynamic if a formula derived at some stage need not be derived at a later stage.) The paper illustrates three interesting points. (i) A set of properties that characterizes an inference relation on the (very natural) dynamic proof interpretation, need not characterize the same inference relation (or even any inference relation) on the usual set-theoretical interpretation. (ii) A proof format may display an internal dynamics (defeasible conclusions) in the absence of an external dynamics (non-monotonicity). (iii) A monotonic logic may have a non-monotonic characterization.

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

This paper analyses the logical structure of the balancing of conflicting normative arguments, and asks whether non-monotonic logic is adequate to represent this type of legal or practical reasoning. Norm conflicts are often regarded as a field of application for non-monotonic logics. This paper argues, however, that the balancing of normative arguments consists of an act of judgement, not a logical inference, and that models of deductive as well as of defeasible reasoning do not give an adequate account of its structure. Moreover, it argues that as far as the argumentation consists in logical inferences, deductive logic suffices for reconstructing the argumentation from the internal point of view of someone making normative judgements.

In this paper, I discuss the analysis of logic in the pragmatic approach recently proposed by Brandom. I consider different consequence relations, formalized by classical, intuitionistic and linear logic, and I will argue that the formal theory developed by Brandom, even if provides powerful foundational insights on the relationship between logic and discursive practices, cannot account for important reasoning patterns represented by non-monotonic or resource-sensitive inferences. Then, I will present an incompatibility semantics in the framework of linear logic which allow to refine Brandom’s concept of defeasible inference and to account for those non-monotonic and relevant inferences that are expressible in linear logic. Moreover, I will suggest an interpretation of discursive practices based on an abstract notion of agreement on what counts as a reason which is deeply connected with linear logic semantics.

Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance.

Any inferential system in which the addition of new premises can lead to the retraction of previous conclusions is a non-monotonic logic. Classical conditional probability provides the oldest and most widely respected example of non-monotonic inference. This paper presents a semantic theory for a unified approach to qualitative and quantitative non-monotonic logic. The qualitative logic is unlike most other non- monotonic logics developed for AI systems. It is closely related to classical (i.e., Bayesian) probability theory. The semantic theory for qualitative non-monotonic entailments extends in a straightforward way to a semantic theory for quantitative partial entailment relations, and these relations turn out to be the classical probability functions.

One of the most important developments over the last twenty years both in logic and in Artificial Intelligence is the emergence of so-called non-monotonic logics. These logics were initially developed by McCarthy [10], McDermott & Doyle [13], and Reiter [17]. Part of the original motivation was to provide a formal framework within which to model cognitive phenomena such as defeasible inference and defeasible knowledge representation, i.e., to provide a formal account of the fact that reasoners can reach conclusions tentatively, reserving the right to retract them in the light of further information.

Non-monotonic inference is inference that is defeasible: in contrast with deductive inference, the conclusions drawn may be withdrawn in the light of further information, even though all the original premises are retained. Much of our everyday reasoning is like this, and a non-monotonic approach has applications to a number of technical problems in artificial intelligence. Work on formalizing non-monotonic inference has progressed rapidly since its beginnings in the 1970s, and a number of mature theories now exist – the most important being default logic, autoepistemic logic, and circumscription.

The emergence, over the last twenty years or so, of so-called “non-monotonic” logics represents one of the most significant developments both in logic and artificial intelligence. These logics were devised in order to represent defeasible reasoning, i.e., that kind of inference in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further evidence.