This paper is devoted to combining the negation of Classical Logic (**CL**) and the negation of Graham Priest's **LP** in a way that is faithful to central properties of the combined logics. We give a number of desiderata for a logic **L** which combines both negations. These desiderata include the following: (a) **L** should be truth functional, (b) **L** should be strictly non-explosive for the paraconsisent negation ˜ (i.e. if *A* and ˜*A* both have a non-trivial set of consequences, then this should also be the case for the set containing both) and (c) **L** should be a conservative extension of **CL** and of **LP**. The desiderata are motivated by a particular property-theoretic perspective on paraconsistency. Next we devise the logic **CLP**. We present an axiomatization of this logic and three semantical characterizations (a non-deterministic semantics, an in nitely valued set-theoretic semantics and an in nitely valued semantics with integer numbers as values). We prove that **CLP** is the only logic satisfying all postulated desiderata. The in nitely valued semantics of **CLP** can be seen as giving rise to an interpretation in which inconsistencies and inconsistent properties come in degrees: not every sentence which involves inconsistencies is equally inconsistent.

It is shown that a set of semi-recursive logics, including many fragments of **CL** (Classical Logic), can be embedded within **CL** in an interesting way. A logic belongs

to the set iff it has a certain type of semantics, called nice semantics. The set includes

many logics presented in the literature. The embedding reveals structural properties of the embedded logic. The embedding turns finite premise sets into finite premise sets. The partial decision methods for **CL** that are goal directed with respect to **CL** are turned into partial decision methods that are goal directed with respect to the embedded logics.

In the context of non-monotonic reasoning different kinds of consequence relations are defined for reasoning from (possibly) inconsistent information. Examples are consequence relations that are characterized in terms of maximal consistent subsets of the premise set. The *strong consequences* are those formulas that follow by Classical Logic from every maximal consistent subset. The *weak consequences* follow from some maximal consistent subset. The *free consequences* follow from the set of formulas that belong to every maximal consistent subset. In this paper the question is discussed which of these consequence relations should be applied in which reasoning context. First the concerns that are expressed in the literature with respect to the usefulness of the weak consequences are addressed. Then it is argued that making weak inferences is sensible for some application contexts, provided one has a (dynamic) proof theory for the corresponding consequence relation. Such a dynamic proof theory is what adaptive logics offer. Finally, all this is illustrated by means of a very simple adaptive logic reconstruction of the free, strong, and weak consequences

n this paper we present a logic that determines when implications in a classical logic context express a relevant connection between antecedent and consequent. In contrast with logics in the relevance logic literature, we leave classical negation intact - in the sense that the law of non-contradiction can be used to obtain relevantly implications, as long as there is a connection between antecedent and consequent. On the other hand, we give up the requirement that our theory of relevance can define a new standard of deduction. We present and argue for a list of requirements that such a logical theory of classical relevance needs to meet and go on to formulate a system that respects each of these requirements. The presented system is a monotonic and transitive logic that extends the relevance logic ℜ with a richer relevant implication that allows for Disjunctive Syllogism and similar rules. This is achieved by interpreting the logical symbols in the antecedents in a stronger way than the logical symbols in consequents. A proof theory and an algebraic semantics are formulated and interesting metatheorems (soundness, completeness and the fact that it satisfies the requirements for classical relevance) are proven. Finally we give a philosophical motivation for our non-standard relevant implication and the asymmetric interpretation of antecedents and consequents.

%B Studia Logica %V 82 %P 1–31 %G eng %0 Journal Article %J Foundations of science %D 2013 %T Non-monotonic set theory as a pragmatic foundation of mathematics %A Verdée, Peter %XIn this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödels incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive *dynamic* (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting *pragmatic* foundations.

In this article, I present a non-trivial but inconsistent set theory based on unrestricted comprehension. The theory is provably non-trivial and strong enough for most of the applications of regular mathematics. This is realized by distinguishing between strong and weak set membership and allowing for the derivation of strong membership from weak membership whenever this is not problematic (it does not lead to paradoxes). This idea of applying rules whenever unproblematic is formalized by means of an adaptive logic.

%B Logic Journal of the IGPL %V 21 %P 108-125 %G eng %R 10.1093/jigpal/jzs025 %0 Journal Article %J Synthese %D 2012 %T The Dynamics of Relevance: Adaptive Belief Revision %A Van De Putte, Frederik %A Verdée, Peter %XThis paper presents eight (previously unpublished) adaptive logics for belief revision, each of which define a belief revision operation in the sense of the AGM framework. All these revision operations are shown to satisfy the six basic AGM postulates for belief revision, and Parikhs axiom of Relevance. Using one of these logics as an example, we show how their proof theory gives a more dynamic flavor to belief revision than existing approaches. It is argued that this turns belief revision (that obeys Relevance) into a more natural undertaking, where analytic steps are performed only as soon as they turn out to be necessary in order to uphold certain beliefs.

%B Synthese %V 187 %P 1-42 %8 May %G eng %R 10.1007/s11229-012-0116-9 %0 Journal Article %J Logic Journal of the IGPL %D 2012 %T Modelling defeasible reasoning by means of adaptive logic games %A Verdée, Peter %XIn this article, I present a dynamic logic game for defeasible reasoning. I argue that, as far as defeasible reasoning is concerned, one should distinguish between practical and ideal rationality. Starting from the adaptive logic framework, I formalize both rationality notions by means of logic games. The presented adaptive logic games are based on (i) standard logic games on the one hand and (ii) dynamic proof procedures for adaptive logic on the other hand. The games are similar to standard logic games, but have the extra property that some moves are revisable. This is handled by means of a main control game, which starts different standard logic games. I argue that the adaptive logic games form intuitive reasoning models for rationality in defeasible reasoning contexts. Moreover, I will also demonstrate that the games give a good insight in the computational complexity of defeasible reasoning forms.

%B Logic Journal of the IGPL %V 20 %P 417–437 %G eng %R http://dx.doi.org/10.1093/jigpal/jzq060 %0 Journal Article %J Logic Journal of IGPL %D 2012 %T A proof procedure for adaptive logics %A Verdée, Peter %XIn this article, I present a procedure that generates proofs for finally derivable adaptive logic consequences. A proof procedure for the inconsistency adaptive logic CLuNr is already presented in [7]. In this article a procedure for CLuNm is presented and the results for both logics are generalized to all adaptive logics, on the presupposition that there exists a proof procedure for the lower limit logic. The generated proofs are so called goal-directed proofs, i.e. proofs that (i) start with the formula (the goal) of which one wants to know whether it is a consequence of a certain premise set and (ii) only consist of lines that may potentially be useful for proving or disproving the goal. The goal-directed proofs form good explications of actual problem-solving reasoning processes.

%B Logic Journal of IGPL %V 21 %P 743-766 %G eng %R 10.1093/jigpal/jzs046 %0 Conference Paper %B WoLLIC 2011 Proceedings LNAI Series %D 2011 %T Strong paraconsistency by separating composition and decomposition in classical logic %A Verdée, Peter %E Goebel, R %XIn this paper I elaborate a proof system that is able to prove all classical first order logic consequences of consistent premise sets, without proving trivial consequences of inconsistent premises (as in A, ¬A\,\unmatched{22a2}\,B). Essentially this result is obtained by formally distinguishing consequences that are the result of merely decomposing the premises into their subformulas from consequences that may be the result of also composing ‘new’, more complex formulas. I require that, whenever ‘new’ formulas are derived, they are to be preceded by a special +-symbol and these +-preceded formulas are not to be decomposed. By doing this, the proofs are separated into a decomposition phase followed by a composition phase. The proofs are recursive, axiomatizable and, as they do not trivialize inconsistent premise sets, they define a very strong non-transitive paraconsistent logic, for which I also provide an adequate semantics.

%B WoLLIC 2011 Proceedings LNAI Series %I Springer %@ 364220919X %G eng %U http://dx.doi.org/10.1007/978-3-642-20920-8\_26 %0 Journal Article %J Synthese %D 2009 %T Adaptive Logics using the Minimal Abnormality strategy are \$\textbackslashPi\^ 1\_1\$-complex %A Verdée, Peter %XIn this article complexity results for adaptive logics using the minimal abnormality strategy are presented. It is proven here that the consequence set of some recursive premise sets is Pi(1)(1)-complete. So, the complexity results in ( Horsten and Welch, Synthese 158: 41- 60, 2007) are mistaken for adaptive logics using the minimal abnormality strategy.

%B Synthese %V 167 %P 93–104 %G eng %U http://dx.doi.org/10.1007/s11229-007-9291-5 %0 Journal Article %J Logique et Analyse %D 2009 %T On the Transparency of Defeasible Logics: Equivalent Premise Sets, Equivalence of Their Extensions, and Maximality of the Lower Limit %A Batens, Diderik %A Straßer, Christian %A Verdée, Peter %XFor Tarski logics, there are simple criteria that enable one to conclude that two premise sets are equivalent. We shall show that the very same criteria hold for adaptive logics, which is a major advantage in comparison to other approaches to defeasible reasoning forms.

A related property of Tarski logics is that the extensions of equivalent premise sets with the same set of formulas are equivalent premise sets. This does not hold for adaptive logics. However a very similar criterion does.

We also shall show that every monotonic logic weaker than an adaptive logic is weaker than the lower limit logic of the adaptive logic or identical to it. This highlights the role of the lower limit for settling the adaptive equivalence of extensions of equivalent premise sets.

This paper answers the philosophical contentions defended in Horsten and Welch (2007, Synthese, 158, 41-60). It contains a description of the standard format of adaptive logics, analyses the notion of dynamic proof required by those logics, discusses the means to turn such proofs into demonstrations, and argues that, notwithstanding their formal complexity, adaptive logics are important because they explicate an abundance of reasoning forms that occur frequently, both in scientific contexts and in common sense contexts.

%B Synthese %V 166 %P 113–131 %G eng %U http://dx.doi.org/10.1007/s11229-007-9268-4 %0 Generic %D 2008 %T Logische Bewijsdynamieken voor de Formele Explicatie van Wetenschappelijke Probleemoplossingsprocessen %A Verdée, Peter %I Ghent University %8 April 22 %9 phd %1Diderik Batens

%0 Journal Article %J Fuzzy Sets and Systems %D 2008 %T Modeling sorites reasoning with adaptive fuzzy logic %A van der Waart van Gulik, Stephan %A Verdée, Peter %XWe present and discuss a new solution for reasoning with sorites series and their related paradoxes.We argue that a suitable logic for sorites series should be able to apply specific classical logic rules like modus ponens until and unless it becomes apparent that these applications generate unacceptable results. When the latter happens, the logic should be able to retract those applications of classical logic rules that are problematic. The formal core of our solution consists of several adaptive logics based on a Łukasiewicz fuzzy logic extended with the Baaz △-operator and a non-singleton interval of designated values. The natural dynamics characteristic of adaptive logics allows these logics to perform necessary retractions in an intuitive and elegant manner. © 2008 Elsevier B.V. All rights reserved.

%B Fuzzy Sets and Systems %V 159 %P 1869–1884 %G eng %R 10.1016/j.fss.2008.01.001