Aristotle's Thesis between paraconsistency and modalization

If the arrow → stands for classical relevant implication, Aristotle's Thesis ¬(A→¬A) is inconsistent with the Law of Simplification (AB)→B accepted by relevantists, but yields an inconsistent non-trivial extension of the system of entailment E. Such paraconsistent extensions of relevant logics have been studied by R. Routley, C. Mortensen and R. Brady. After examining the semantics associated to such systems, it is stressed that there are nonclassical treatments of relevance which do not support Simplification. The paper aims at showing that Aristotle's Thesis may receive a sense if the arrow is defined as strict implication endowed with the proviso that the clauses of the conditional have the same modal status, i.e. the same position in the Aristotelian square. It is so grasped, in different form, the basic idea of relevant logic that the clauses of a true conditional should have something in common. It is proved that thanks to such definition of the arrow Aristotle's Thesis subjoined to the minimal normal system K yields a system equivalent to the deontic system KD.     

Combining linear-time temporal logic with constructiveness and paraconsistency

It is known that linear-time temporal logic (LTL), which is an extension of classical logic, is useful for expressing temporal reasoning as investigated in computer science. In this paper, two constructive and bounded versions of LTL, which are extensions of intuitionistic logic or Nelson's paraconsistent logic, are introduced as Gentzen-type sequent calculi. These logics, $\mathrm{IB}[l]$ and $\mathrm{PB}[l]$, are intended to provide a useful theoretical basis for representing not only temporal (linear-time), but also constructive, and paraconsistent (inconsistency-tolerant) reasoning. The time domain of the proposed logics is bounded by a fixed positive integer. Despite the restriction on the time domain, the logics can derive almost all the typical temporal axioms of LTL. As a merit of bounding time, faithful embeddings into intuitionistic logic and Nelson's paraconsistent logic are shown for $\mathrm{IB}[l]$ and $\mathrm{PB}[l]$, respectively. Completeness (with respect to Kripke semantics), cut–elimination, normalization (with respect to natural deduction), and decidability theorems for the newly defined logics are proved as the main results of this paper. Moreover, we present sound and complete display calculi for $\mathrm{IB}[l]$ and $\mathrm{PB}[l]$.In [P. Maier, Intuitionistic LTL and a new characterization of safety and liveness, in: Proceedings of Computer Science Logic 2004, in: Lecture Notes in Computer Science, vol. 3210, Springer-Verlag, Berlin, 2004, pp. 295–309] it has been emphasized that intuitionistic linear-time logic (ILTL) admits an elegant characterization of safety and liveness properties. The system ILTL, however, has been presented only in an algebraic setting. The present paper is the first semantical and proof-theoretical study of bounded constructive linear-time temporal logics containing either intuitionistic or strong negation.     

Special issue: Combining probability and logic

This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers.     

Anti-intuitionism and paraconsistency

This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of anti-intuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that anti-intuitionistic logics are paraconsistent, and in particular we develop a first anti-intuitionistic hierarchy starting with Johansson's dual calculus and ending up with Gödel's three-valued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these anti-intuitionistic logics with well-known paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) many-valued logics (Iⁿ)_nω we show that the anti-intuitionistic hierarchy (I^n*)_nω obtained from (Iⁿ)_nω does coincide with the hierarchy of the many-valued paraconsistent logics (Pⁿ)_nω. Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of self-duality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multiple-conclusion logics are used as an appropriate environment to deal with them.     

Relational semantics for full linear logic

Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In [4] it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [5] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form.In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [4], [5] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms.Traditionally, so-called phase semantics are used as models for (provability in) linear logic [8]. These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.     

Quasi-truth,paraconsistency, and the foundations of science

In order to develop an account of scientific rationality, two problems need to be addressed: (i) how to make sense of episodes of theory change in science where the lack of a cumulative development is found, and (ii) how to accommodate cases of scientific change where lack of consistency is involved. In this paper, we sketch a model of scientific rationality that accommodates both problems. We first provide a framework within which it is possible to make sense of scientific revolutions, but which still preserves some (partial) relations between old and new theories. The existence of these relations help to explain why the break between different theories is never too radical as to make it impossible for one to interpret the process in perfectly rational terms. We then defend the view that if scientific theories are taken to be quasi-true, and if the underlying logic is paraconsistent, it’s perfectly rational for scientists and mathematicians to entertain inconsistent theories without triviality. As a result, as opposed to what is demanded by traditional approaches to rationality, it’s not irrational to entertain inconsistent theories. Finally, we conclude the paper by arguing that the view advanced here provides a new way of thinking about the foundations of science. In particular, it extends in important respects both coherentist and foundationalist approaches to knowledge, without the troubles that plague traditional views of scientific rationality.     

On negation: Pure local rules

This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multiple-conclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextual sub-classical rules involving this logical constant—among others, well-known forms of proof by cases, consequentia mirabilis and reductio ad absurdum. Finer definitions of paraconsistency and the dual paracompleteness can thus be formulated, allowing for pseudo-scotus and ex contradictione to be differentiated and for a comprehensive version of the Principle of Non-Triviality to be presented. A final proposal is made to the effect that—pure positive rules involving negation being often fallible—a characterization of what most negations in the literature have in common should rather involve, in fact, a reduced set of negative rules.     

Interpretations of intuitionist logic in non-normal modal logics

Historically, it was the interpretations of intuitionist logic in the modal logic S4 that inspired the standard Kripke semantics for intuitionist logic. The inspiration of this paper is the interpretation of intuitionist logic in the non-normal modal logic S3: an S3 model structure can be 'looked at' as an intuitionist model structure and the semantics for S3 can be 'cashed in' to obtain a non-normal semantics for intuitionist propositional logic. This non-normal semantics is then extended to intuitionist quantificational logic.     

Mathematical modal logic: A view of its evolution

This is a survey of the origins of mathematical interpretations of modal logics, and their development over the last century or so. It focuses on the interconnections between algebraic semantics using Boolean algebras with operators and relational semantics using structures often called Kripke models. It reviews the ideas of a number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. It concludes with an account of several applications of modal model theory to mathematics and theoretical computer science.     

Logic programming as classical inference

We propose a denotational semantics for logic programming based on a classical notion of logical consequence which is apt to capture the main proposed semantics of logic programs. In other words, we show that any of those semantics can be viewed as a relation of the form $T{\models }^{\star }X$ where T is a theory which naturally represents the logic program under consideration together with a set of formulas playing the role of “hypotheses”, in a way which is dictated by that semantics, ${\models }^{\star }$ is a notion of logical consequence which is classical because negation, disjunction and existential quantification receive their classical meaning, and X represents what can be inferred from the logic program, or an intended interpretation of that logic program (such as an answer-set, its well-founded model, etc.). The logical setting we propose extends the language of classical modal logic as it deals with modal operators indexed by ordinals. We make use of two kinds of basic modal formulas: ${\square }_{\alpha }\phi$ which intuitively means that the logical program can generate φ by stage α of the generation process, and ${◇}_{\alpha }{\square }_{\beta }\phi$ with $\alpha >\beta$, which intuitively means that φ can be used as a hypothesis from stage β of the generation process onwards, possibly expecting to confirm φ by stage α (so expecting ${\square }_{\alpha }\phi$ to be generated). This allows us to capture Rondogiannis and Wadge's version of the well-founded semantics [27] where a member of the well-founded model is a closed atom which receives an ordinal truth value of ${\mathtt{true}}_{\alpha }$ or ${\mathtt{false}}_{\alpha }$ for some ordinal α: in our framework, this corresponds to having $T{\models }^{\star }{\square }_{\alpha }\phi$ or $T{\models }^{\star }{\square }_{\alpha }\neg \phi$, respectively, with T being the natural representation of the logic program under consideration and the right set of “hypotheses” as dictated by the well-founded semantics. The framework we present goes much beyond the proposed traditional semantics for logic programming, as it can for instance let us investigate under which conditions a set of hypotheses can be minimal, with each hypothesis being activated as late as possible and confirmed as soon as possible, setting the theoretical foundation to sophisticated ways of making local use of hypotheses in knowledge-based systems, while still being theoretically grounded in a classical notion of logical consequence.     

From reasonable preferences,via argumentation,to logic

This article demonstrates that typical restrictions which are imposed in dialogical logic in order to recover first-order logical consequence from a fragment of natural language argumentation are also forthcoming from preference profiles of boundedly rational players, provided that these players instantiate a specific player type and compute partial strategies. We present two structural rules, which are formulated similarly to closure rules for tableaux proofs that restrict players' strategies to a mapping between games in extensive forms (i.e., game trees) and proof trees. Both rules are motivated from players' preferences and limitations; they can therefore be viewed as being player-self-imposable. First-order logical consequence is thus shown to result from playing a specific type of argumentation game. The alignment of such games with the normative model of the Pragma-dialectical theory of argumentation is positively evaluated. But explicit rules to guarantee that the argumentation game instantiates first-order logical consequence have now become gratuitous, since their normative content arises directly from players' preferences and limitations. A similar naturalization for non-classical logics is discussed.