Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency (LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8] and [9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8] and [9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbert?s program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B∧¬B and falsifies many arbitrarily complex instances of non-contradiction principle ¬(A∧¬A). Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACIarithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.     

Justification logics and hybrid logics

Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since justification logics are newer and less well known than hybrid logics, we sketch their background, and give pointers to their range of applicability. We conclude with suggestions for future research. Indeed, the main goal of this paper is to encourage others to continue the investigation begun here.     

Parallelism in the Early Moist Texts

Parallelism is present everywhere in the early Moist texts: at the syntactic level, at the semantic level, between sentences, between sets of sentences, between argumentative structures. The present article gives many examples of the phenomenon: parallelism of insistence, insistence from top to bottom, insistence from bottom to top, parallelism with symmetry, parallelism involving negation, subcontraries and negation at deeper levels, parallelism of the argumentative structures. Logic is particularly applied to the study of parallelism involving negation. From the point of view of argumentation, it is shown that many of those constructions have an important role in supporting arguments such as: arguments of generalization, a fortiori arguments, arguments of exemplarity, consequentialist arguments, arguments by comparison. This study draws the attention to the importance of argumentation in the study of Moism and gives a new light on the argument by parallelism (mou 侔) in the “Xiaoqu”: It is a natural extension of what we call “parallelism involving negation,” already very common in the early Moist texts.     

Don’t Ever Do That! Long-term Duties in <Emphasis Type="Italic">PD</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis></Subscript><Emphasis Type="Italic">L</Emphasis>

This paper studies long-term norms concerning actions. In Meyer’s Propositional Deontic Logic (PD _e L), only immediate duties can be expressed, however, often one has duties of longer durations such as: “Never do that”, or “Do this someday”. In this paper, we will investigate how to amend PD _e L so that such long-term duties can be expressed. This leads to the interesting and suprising consequence that the long-term prohibition and obligation are not interdefinable in our semantics, while there is a duality between these two notions. As a consequence, we have provided a new analysis of the long-term obligation by introducing a new atomic proposition I (indebtedness) to represent the condition that an agent has some unfulfilled obligation. Presented by Jacek Malinowski     

Applied Logic without Psychologism

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.     

Resource-origins of Nonmonotonicity

Formal nonmonotonic systems try to model the phenomenon that common sense reasoners are able to “jump” in their reasoning from assumptions Δ to conclusions C without their being any deductive chain from Δ to C. Such jumps are done by various mechanisms which are strongly dependent on context and knowledge of how the actual world functions. Our aim is to motivate these jump rules as inference rules designed to optimise survival in an environment with scant resources of effort and time. We begin with a general discussion and quickly move to Section 3 where we introduce five resource principles. We show that these principles lead to some well known nonmonotonic systems such as Nute’s defeasible logic. We also give several examples of practical reasoning situations to illustrate our principles. Edited by Hannes Leitgeb     

Methodological naturalism and epistemic internalism

Epistemic naturalism holds that the results or methodologies from the cognitive sciences are relevant to epistemology, and some have maintained that scientific methods are more compatible with externalist theories of justification than with internalist theories. But practically all discussions about naturalized epistemology are framed exclusively in terms of cognitive psychology, which is only one of the cognitive sciences. The question addressed in this essay is whether a commitment to naturalism really does favor externalism over internalism, and we offer reasons for thinking that naturalism in epistemology is compatible with both internalist and externalist conceptions of justification. We also argue that there are some distinctively internalist aims that are currently being studied scientifically and these notions, and others, should be studied by scientific methods. This essay is dedicated to Deborah Mayo, who has long advocated using error statistical techniques to analyze and resolve epistemological puzzles in the philosophy of science. This essay follows the same spirit by advocating that computational concepts and techniques be applied within the heart of traditional, analytic epistemology.