Nonmonotonic Reasoning, Expectations Orderings, and Conceptual Spaces

In Gärdenfors and Makinson (Artif Intell 65(2):197–245, 1994) and Gärdenfors (Knowledge representation and reasoning under uncertainty, Springer-Verlag, 1992) it was shown that it is possible to model nonmonotonic inference using a classical consequence relation plus an expectation-based ordering of formulas. In this article, we argue that this framework can be significantly enriched by adopting a conceptual spaces-based analysis of the role of expectations in reasoning. In particular, we show that this can solve various epistemological issues that surround nonmonotonic and default logics. We propose some formal criteria for constructing and updating expectation orderings based on conceptual spaces, and we explain how to apply them to nonmonotonic reasoning about objects and properties.

Meta-inferences and Supervaluationism

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic NT. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented.

DDLV: A System for rational preferential reasoning for datalog

Datalog is a powerful language that can be used to represent explicit knowledge and compute inferences in knowledge bases. Datalog cannot, however, represent or reason about contradictory rules. This is a limitation as contradictions are often present in domains that contain exceptions. In this paper, we extend Datalog to represent contradictory and defeasible information. We define an approach to efficiently reason about contradictory information in Datalog and show that it satisfies the KLM requirements for a rational consequence relation. We introduce DDLV, a defeasible Datalog reasoning system that implements this approach. Finally, we evaluate the performance of DDLV.

Bilattice Logic for Rough Sets

Rough set theory is studied to manage uncertain and inconsistent information. Because Pawlak’s decision logic for rough sets is based on the classical two-valued logic, it is inconvenient for handling inconsistent information. We propose a bilattice logic as the deduction basis for the decision logic of rough sets to address inconsistent and ambiguous information. To enhance the decision logic to bilattice semantics, we introduce Variable Precision Rough Set (VPRS). As a deductive basis for bilattice decision logic, we define a consequence relation for Belnap’s four-valued semantics and provide a bilattice semantic tableau TB4 for a deduction system. We demonstrate the soundness and completeness of TB4 and enhance it with weak negation.

Consequences

The consequences of our actions seem to matter. But what is the nature of the consequence relation that a particular act bears to, well, its consequences? This essay considers a number of traditional approaches to understanding the consequence relation. While many traditional approaches treat the consequence relation as built upon a causal relation, I hold that there are good reasons to doubt that the consequence relation should be understood in terms of causal relations, even if supplemented with the identity relation. Instead, I argue for a contrastive approach that, while not entirely free of problems, does a better job than standard accounts at capturing the relationship between an act and its consequences.

Extensions of paraconsistent weak Kleene logic

Paraconsistent weak Kleene ($\textrm{PWK}$) logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} (\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

Sette’s calculus P1 and some hierarchies of paraconsistent systems

The necessary condition for a calculus to be paraconsistent is that its consequence relation is not explosive. This results in rejection of the principle of ex contradictione sequitur quodlibet. In 1973, Sette presented a calculus, denoted as $P^1$, which is paraconsistent only at the atomic level, i.e. $\alpha $ and ${\sim }\alpha $ yield any $\beta $ if, and only if the formula $\alpha $ is not a propositional variable. The calculus has been viewed as one of the noteworthy paraconsistent calculi since then. The objective of this paper is to propose a new semantics for Sette’s calculus and present some hierarchies of the paraconsistent calculi, which are based on $P^1$. We demonstrate that $P^1$ is sound and complete with respect to the semantics and so are all the calculi under consideration.

A Note on Fernández–Coniglio’s Hierarchy of Paraconsistent Systems

A logic is called explosive if its consequence relation validates the so-called principle of ex contradictione sequitur quodlibet. A logic is called paraconsistent so long as it is not explosive. Sette’s calculus P 1 is widely recognized as one of the most important paraconsistent calculi. It is not surprising then that the calculus was a starting point for many research studies on paraconsistency. Fernández–Coniglio’s hierarchy of paraconsistent systems is a good example of such an approach. The hierarchy is presented in Newton da Costa’s style. Therefore, the law of non-contradiction plays the main role in its negative axioms. The principle of ex contradictione sequitur quodlibet has been marginalized: it does not play any leading role in the hierarchy. The objective of this paper is to present an alternative axiomatization for the hierarchy. The main idea behind it is to focus explicitly on the (in)validity of the principle of ex contradictione sequitur quodlibet. This makes the hierarchy less complex and more transparent, especially from the viewpoint of paraconsistency.

Disjunctive Multiple-Conclusion Consequence Relations

The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r (dened on the power set of a set of all formulas of a given language) the least multiple-conclusion consequence relation containing r, is dened on the grounds of a natural Galois connection. It is shown that the very closure C is an isomorphism from the power set algebra of a simple binary relation to the Boolean algebra of all multiple-conclusion consequence relations.

Paraconsistency and Paracompleteness

A logic $\langle \mathcal{L},\vdash_{p}\rangle$ is said to be paraconsistent if, and only if $\{\alpha, \neg \alpha\} \nvdash_{p} \beta$, for some formulas $\alpha, \beta$. In other words, the necessary and sufficient (the latter is problematic) condition for a logic to be paraconsistent is that its consequence relation is not $\textit{explosive}$. The definition is very simple but also very broad, and this may create a risk that some logics, which have not too much in common with the $\textit{paraconsistency}$, are considered to be so. Nevertheless, the definition may still serve as a reasonable starting point for more thorough research. Paracomplete logic can be defined in many different ways among which the following one may be of some interest: A logic $\langle \mathcal{L},\vdash_{q}\rangle$ is said to be paracomplete if, and only if $\{\beta \rightarrow \alpha, \neg \beta \rightarrow \alpha\} \nvdash_{q} \alpha$, for some formulas $\alpha, \beta$. But again, just as in the case of paraconsistent logic, the definition is very general and may be seen to overlap with the logics that have nothing in common with the \textit{paracompleteness}. In the paper, we define some calculi of paraconsistent and paracomplete logics arranged in the form of hierarchies, determined by several criteria. We put central emphasis on logical axioms admitting only the rule of detachment as the sole rule of inference and on the so-called bi-valuation semantics. The hierarchies (no matter which one) are expected to shed some light on the aforementioned issue.