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Extending the standard format of adaptive logics to the prioritized case
Abstract
This paper introduces a new format for reasoning with prioritized standards of normality. It is applicable in a broad variety of contexts, e.g. dealing with (possibly conflicting) prioritized belief bases or combining different reasoning methods in a prioritized way. The format is a generalization of the standard format of adaptive logics (see [4]). Every logic that is formulated within it has a straightforward semantics in the style of Shoham's selection semantics (see [22]) and a dynamic proof theory. Furthermore, it can count on a rich meta-theory that inherits the attractive features of the standard format, such as soundness and completeness, reflexivity, idempo-tence, cautious monotonicity, and many other properties.
... The idea for Minimal Abnormality is to select all so-called minimally abnormal models from the LLL-models of a premise set , i.e., models that validate a minimal set of abnormalities. For Reliability LLL-models are selected that verify only abnormalities that are also verified by some of the minimally abnormal models (see [5,29,23]). ...
... As shown in [26,23], such logics have several interesting meta-theoretic properties. Given some additional restrictions, they are cautiously monotonic and cumulatively transitive, just like ALs in standard format. ...
... Given some additional restrictions, they are cautiously monotonic and cumulatively transitive, just like ALs in standard format. This also means that they are a fixed point, and have the reciprocity property -see again [26,23] for the details and related results. There it was also shown that these properties fail for SAL in the more general case. ...
The standard format for adaptive logics offers a generic and unifying formal framework for defeasible reasoning forms. One of its main distinguishing features is a dynamic proof theory by means of which it is able to explicate actual reasoning. In many applications it has proven very useful to superpose sequences of adaptive logics, such that each logic treats the consequence set of its predecessor as premise set. Although attempts have been made to define dynamic proof theories for some of the resulting logics, no generic proof theory is available yet. Moreover, the existing proof theories for concrete superpositions are sub-optimal in various respects: the derivability relations characterized by these proposals are often not adequate with respect to the consequence relation of the superposed adaptive logics and in some cases they even trivialize premise sets. An adequate and generic proof theory is needed in order to meet the requirement of explicating defeasible reasoning in terms of superpositions of adaptive logics. This paper presents two equivalent generic proof theories for superpositions of adaptive logics in standard format. By means of simple examples, the basic ideas behind these proof theories are illustrated and it is shown how the older proposals are inadequate.
... A simple way to achieve this is to use the set of abnormalities Ω defneg = {in A | A ∈ W}. Various techniques are available that allow to define adaptive logics in which both sets of abnormalities Ω defneg and Ω + • are considered (see e.g., [48,49]). The most straightforward way is to use an adaptive logic based on Ω • defneg = Ω defneg ∪Ω + • . ...
... Dynamic proof theories for such combinations are defined in [45,47]. Another option is e.g. to use lexicographic adaptive logics [49]. In such adaptive logics, both ¬in a and out d are derivable from Γ defneg , while neither out c nor out¬c are. ...
... The abnormalities are then presented by the set Ω = {• i A ∧ ¬A | i ≥ 1, A ∈ W }. Moreover, we can make use of socalled lexicographic ALs [49,50] that take care of the priorities in a natural way. Instead of 'hard-coding' priorities one may also consider priorities that arise in view of logical relationships among norms such as specificity cases in which more specific norms override conflicting norms (see e.g. ...
We translate unconstrained and constrained input/output logics as introduced by Makinson and van der Torre to modal logics, using adaptive logics for the constrained case. The resulting reformulation has some additional benefits. First, we obtain a proof-theoretic (dynamic) characterization of input/output logics. Second, we demonstrate that our framework naturally gives rise to useful variants and allows to express important notions that go beyond the expressive means of input/output logics, such as violations and sanctions.
... Similar cases are a specific variant of circumscription logic from [3], and the preferential semantics for open defaults from [1]. 6 In view of such examples, the requirement that the set of models of is ≺-smooth seems hardly justifiable, if we take the preference relation to be something more than merely a technical device-i.e. if we presuppose that ≺ represents a basic concept such as "is more normal then", or "is more plausible than". 7 It may well be that certain concrete constructions warrant smoothness-see e.g., [23] for a rather generic one-, but to assume this for preference relations in general is one bridge too far. Nevertheless, relatively little attention has been paid to preferential semantics that are well-behaved without the presupposition of the smoothness property. ...
... The idea of a multi-selection semantics was obtained by a generalization of the Limit Variant from [18] -see Section 4. 23 The stronger Boutilier-variant which we present in Section 5 can also be characterized in terms of a multi-selection semantics. ...
... This section is of a rather technical nature. We will first provide a representation theorem for the class of all consequence operations that can be defined in terms of 23 In the introduction of [17], Schlechta also considers the generalization of semantics in terms of a single selection function f to semantics in terms of a set F of such functions, where semantic consequence is then defined as validity in at least one set f (M( )) for an f ∈ F . It can be easily verified that our approach covers Schlechta's idea as a special case. ...
This paper studies the properties of eight semantic consequence relations defined from a Tarski-logic L and a preference relation ≺ . They are equivalent to Shoham’s so-called preferential entailment for smooth model structures, but avoid certain problems of the latter in non-smooth configurations. Each of the logics can be characterized in terms of what we call multi-selection semantics. After discussing this type of semantics, we focus on some concrete proposals from the literature, checking a number of meta-theoretic properties and elaborating on their intuitive motivation. As it turns out, many of their meta-properties only hold in case ≺ is transitive. To tackle this problem, we propose slight modifications of each of the systems, showing the resulting logics to behave better at the intuitive level and in metatheoretic terms, for arbitrary ≺ .
... Our logic MP ⊏ allows us to solve the problem of prioritized conflicts within a new generic format of prioritized adaptive logics that was studied in [35]. As a result, many important meta-theorems, as well as a full-blown proof theory and semantics are immediately available. ...
... Note that the lower the index of an abnormality, the more we should avoid this abnormality, since it corresponds to an obligation of a higher level. We will use the generic format from [35] to deal with the priority order on the obligations resp. abnormalities. ...
... Let Φ ⊏ s (Γ) be the set of ⊏-minimal choice sets of Σ s (Γ). It is proven in [35] that at every stage s of a proof from Γ, Φ ⊏ s (Γ) is non-empty. Let us try to give an intuitive characterization of the above concepts. ...
In this article we present the logic MP⊏ that explicates reasoning on the basis of prioritized obligations. Although formal criteria to handle prioritized obligations have been formulated in the literature, little attention has been paid to the actual (non-monotonic) reasoning that makes use of these criteria. The dynamic proof theory of MP⊏ fills this lacuna. This article focuses on premise sets consisting of possibly conflicting prima facie obligations that have a modular order. MP⊏ allows to derive—inter alia—the actual, all-things-considered obligations from such premise sets. It is defined in the format of lexicographic adaptive logics from [34], whence a rich meta-theory is immediately available (e.g. soundness and completeness, idempotence, reflexivity, etc.). In addition, we establish some meta-theoretic results that are specific to the context of prioritized obligations. With the aid of concrete examples, we illustrate properties of MP⊏ which improve on other existing criteria for prioritized obligations.
... There are several ways to do so -see [Makinson, 2005] for a reader-friendly introduction to the field and a primer on some of the terminology used in this section. We will use the format of lexicographic adaptive logics developed in [Van De Putte and Straßer, 2012b] (see also [Van De Putte, 2012, ch. 5]). ...
... The definitions of final derivability are the same as for adaptive logics in the standard format (definitions 6.10 and 6.11 on page 156). Given a monotonic LLL with a sound and complete axiomatisation, Van De Putte and Straßer prove that the prioritised adaptive logic built on top of it is sound and complete as well [Van De Putte and Straßer, 2012b;Van De Putte, 2012]. ...
... This can be done in at least three clearly distinct ways -see [98] for a detailed study of these. Here we will only discuss one of these three, viz. the so-called lexicographic adaptive logics first presented in [102]; we moreover confine ourselves to the minimal abnormalityvariant of these systems. Although these logics can be fully characterized in terms of a dynamic proof theory, we focus on their semantics, which is a straightforward generalization of the AL m -semantics. ...
... The preference relation on abnormal parts of models yields a smooth preference relation on every set M LLL (Γ) [102]. Hence, just as for minimal abnormality, we can select the -minimal models of a premise set and define semantic consequence in terms of those models. ...
The article discusses the problem of choosing between inductive methods in
Rudolf Carnap’s inductive logic. It has been held that, if one has no background
knowledge, there is no way to justify the use of one inductive method instead
of another one. Even the non-inductive method, which gives no regard to
experience, cannot be considered worse than the inductivist methods giving
regard to experience. However, it can be shown that, by using Carnap’s own
success criterion for inductive methods, there is an inductivist method which is
to be preferred over the non-inductive method. The result will be compared to
Hans Reichenbach’s pragmatic justification of induction.
... This can be done in at least three clearly distinct ways -see [98] for a detailed study of these. Here we will only discuss one of these three, viz. the so-called lexicographic adaptive logics first presented in [102]; we moreover confine ourselves to the minimal abnormalityvariant of these systems. Although these logics can be fully characterized in terms of a dynamic proof theory, we focus on their semantics, which is a straightforward generalization of the AL m -semantics. ...
... The preference relation on abnormal parts of models yields a smooth preference relation on every set M LLL (Γ) [102]. Hence, just as for minimal abnormality, we can select the -minimal models of a premise set and define semantic consequence in terms of those models. ...
Adaptive Logics (ALs) are a viable and useful formal tool to handle various issues in deontic logic. In this paper, we motivate, explain, illustrate, and discuss the use of ALs in deontic logic. Published work on deontic ALs focusses mainly on conflicttolerant deontic logics (logics that can accommodate conflicting obligations) and-to a lesser extent-on problems concerning factual and deontic detachment. So does the present paper. Near the end of the paper, however, we also indicate some of the possibilities that the adaptive logic framework creates for tackling other types of problems within deontic logic.
... This format is a generalization of the Standard Format of adaptive logic defined in Batens (2007). It is developed in Putte and Straßer (2012), where also a proof theory and the metatheory are given for every logic defined within that format. The format extends the Standard Format in the sense that it allows for prioritization within the logic itself (Standard Format adaptive logics have to be combined in order to enable prioritization). ...
This paper is devoted to a consequence relation combining the negation of Classical Logic () and a paraconsistent negation based on Graham Priest’s Logic of Paradox (). We give a number of natural desiderata for a logic that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. We devise the logic by means of an axiomatization and three equivalent semantical characterizations (a non-deterministic semantics, an infinite-valued set-theoretic semantics and an infinite-valued semantics with integer numbers as values). By showing that this logic is maximally paraconsistent, we prove that is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations.
... That is, the inferential closure of CL m • differs from that of | ∼ e and | ∼ n for a large range of premise sets T . For the proof theory of | ∼ p , priorities can be introduced along the lines of van de Putte and Straßer [42]. ...
Some scientific theories are inconsistent, yet non-trivial and meaningful. How is that possible? The present paper aims to show that we can analyse the inferential use of such theories in terms of consistent compositions of the applications of universal axioms. This technique will be represented by a preferred models semantics, which allows us to accept the instances of universal axioms selectively. For such a semantics to be developed, the framework of partial structures by da Costa and French will be extended by a few elements of the Sneed formalism, also known as the structuralist approach to science.
... Second, one may consider more complex comparisons of two ALs, where they use both a different underlying monotonic core and a different set of abnormalities. Third, one may try to generalize these results to more generic frameworks which have the standard format as a special case; examples are the format from [24] which does not assume supraclassicality of L, the format of lexicographic ALs from [31] in which abnormalities can have various priority degrees, and the format of [28,Chapter 5] which generalizes the notion of a strategy using so-called threshold functions. Our current results will be useful for all three types of investigation. ...
Adaptive logics (ALs) in standard format are defined in terms of a monotonic core logic L, a distinct set of ‘abnormal’ formulas Ω and a strategy, which can be either reliability or minimal abnormality. In this
article we we ask under which conditions the consequence relation of two ALs that use the same strategy are identical, and
when one is a proper subrelation of the other. This results in a number of sufficient (and sometimes necessary) conditions
on L and Ω which apply to all ALs in standard format. In addition, we translate our results to the closely related family of default
assumption consequence relations.
... So, as the addition is avoidable, it better is avoided-the formulation of a logic should refrain from taking a philosophical stance. Finally, the checked symbols led to confusion, for example to the mistaken claim that adaptive logics are in a sense incomplete because not all semantic consequences would be derivable from premise sets in which occur checked symbols [62,63]. 26 All that we really need in the standard format is a classical disjunction, to which I refer by∨. ...
This paper contains a concise introduction to a few central features of inconsistency-adaptive logics. The focus is on the aim of the program, on logics that may be useful with respect to applications, and on insights that are central for judging the importance of the research goals and the adequacy of results. Given the nature of adaptive logics, the paper may be read as a peculiar introduction to defeasible reasoning.
In this article we observe and analyze Hilbert algebras in a non-classical principled-philosophical-logical framework: We establish the properties of Hilbert algebras within the frames of Bishop's constructive mathematics by considering the carriers of these algebraic structures as sets with apartness relation. In addition to redefining the concept of Hilbert algebras in this environment , we introduce and analyze the concepts of some specific substrates of these algebras, such as co-ideals and co-filters. The concept of co-congruence has been introduced and is associated with co-ideals and quotient algebras, also.
The present paper expounds a preferred models semantics of paraconsistent reasoning. The basic idea of this semantics is that we interpret the language L(V) of a theory T in such a way that the axioms of T are satisfied to a maximal extent. These preferred interpretations are described in terms of a network of partial structures. Upon this semantic analysis of paraconsistent reasoning we develop a corresponding proof theory using adaptive logics.
In order to deal with the possibility of deontic conflicts Lou Goble developed a group of logics (DPM) that are characterized by a restriction of the inheritance principle. While they approximate the deductive power of standard deontic logic, they do so only if the user adds certain statements to the premises. By adaptively strengthening the DPM logics, this chapter presents logics that overcome this shortcoming. Furthermore, these ALs are capable of modeling the dynamic and defeasible aspect of our normative reasoning by their dynamic proof theory. This way they enable us to have a better insight into the relations between obligations and thus to localize deontic conflicts.
In this section sequential combinations of adaptive logics are studied under a generic perspective. We will provide meta-theoretic insights and dynamic proof theories.
In this chapter we generalize the standard format of adaptive logics and thereby introduce an interesting larger class of adaptive logics that can be characterized in a simple and intuitive way. We demonstrate that the new format overcomes the two shortcomings of the standard format. On the one hand, logics with both qualitative and quantitative rationales can be expressed in it. On the other hand, the format is expressive enough to allow for the handling of priorities in various ways. We show that many adaptive logics that have been considered in the literature fall within this larger class -for instance adaptive logics in the standard format, adaptive logics with counting strategies, lexicographic adaptive logics-and that the characterization of this class offers many possibilities to formulate new logics. One of the advantages of the format studied in this chapter is that a lot of meta-theory comes for free for any logic formulated in it. We show that adaptive logics formulated in it are always sound and complete. Furthermore, many of the meta-theoretic properties that are usually associated with the standard format (such as cumulativity, fixed point property, (strong) reassurance, etc.) also hold for rich subclasses of logics formulated in the new format.
The standard format of adaptive logics makes use of two so-called strategies: reliability and minimal abnormality. While these
are fairly well-known and frequently applied, the question of whether and when the two strategies are equi-expressive has
so far remained unaddressed. In this article, we show that for a specific, yet significant class of premise sets, the consequence
set of an adaptive logic that uses the minimal abnormality strategy can be expressed by another adaptive logic that uses the
reliability strategy. The basic idea is that we close the set of abnormalities under conjunction. We show that the consequence
sets obtained by both logics from a premise set Γ is identical if and only if Γ is finite-conditional. The latter property is specified in terms of a well-known characterization of minimal abnormality. In addition, we discuss
other (stronger) properties of premise sets that have been considered in the literature, showing each of them to imply finite-conditionality.
In a rather general setting, we prove a number of basic theorems concerning computational complexity of derivability in adaptive logics. For that setting, the so-called standard format of adaptive logics is suitably adapted, and the corresponding completeness results are established in a very uniform way.
This paper presents an adaptive logic enhancement of conditional logics of normality that allows for defeasible applications of Modus Ponens to conditionals. In addition to the possibilities these logics already offer in terms of reasoning about conditionals, this way they are enriched by the ability to perform default inferencing. The idea is to apply Modus Ponens defeasibly to a conditional and a fact on the condition that it is ‘safe’ to do so concerning the factual and conditional knowledge at hand. It is for instance not safe if the given information describes exceptional circumstances: although birds usually fly, penguins are exceptional to this rule. The two adaptive standard strategies are shown to correspond to different intuitions, a skeptical and a credulous reasoning type, which manifest themselves in the handling of so-called floating conclusions.
In order to deal with the possibility of deontic conflicts Lou Goble developed a group of logics (DPM) that are characterized by a restriction of the inheritance principle. While they approximate the deductive power of standard deontic logic, they do so only if the user adds certain statements to the premises. By adaptively strengthening the DPM logics, this paper presents logics that overcome this shortcoming. Furthermore, they are capable of modeling the dynamic and defeasible aspect of our normative reasoning by their dynamic proof theory. This way they enable us to have a better insight in the relations between obligations and thus to localize deontic conflicts.
A broad range of defeasible reasoning forms has been explicated by prioritized adaptive logics. However, the relative lack
in meta-theory of many of these logics stands in sharp contrast to the frequency of their application. This article presents
the first comparative study of a large group of prioritized adaptive logics. Three formats of such logics are discussed: superpositions
of adaptive logics, hierarchic adaptive logics from F. Van De Putte (2011, Log. J. IGPL, doi:10.1093/jigpal/jzr025) and lexicographic adaptive logics from F. Van De Putte and C. Straßer (2012, Log. Anal., forthcoming). We restrict the scope to logics that use the strategy Minimal Abnormality. It is shown that the semantic characterizations
of these systems are equivalent and that they are all sound with respect to either of these characterizations. Furthermore,
sufficient conditions for the completeness and equivalence of the consequence relations of the three formats are established.
Some attractive properties, including Fixed Point and the Deduction Theorem, are shown to hold whenever these conditions are
obeyed.
For 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.
In 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 P11\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.
The lexicographic closure of any given finite set D of normal defaults is defined. A conditional assertion a b is in this lexicographic closure if, given the defaults D and the fact a, one would conclude b. The lexicographic closure is essentially a rational extension of D, and of its rational closure, defined in a previous paper. It provides a logic of normal defaults that is different from the one proposed by R. Reiter and that is rich enough not to require the consideration of non-normal defaults. A large number of examples are provided to show that the lexicographic closure corresponds to the basic intuitions behind Reiter's logic of defaults. 1 Plan of this paper Section 2 is a general introduction, describing the goal of this paper, in relation with Reiter's Default Logic and the program proposed in [12] by Lehmann and Magidor. Section 3 first discusses at length some general principles of the logic of defaults, with many examples, and, then, puts this paper in perspe...
this paper, we adopt the former perspective. In order to distinguish operations on syntactic descriptions -- on belief bases -- from operations on belief sets, belief base changes are called base revision and base contraction.
Lehmann and Magidor have proposed a powerful approach to model defeasible conditional knowledge: Rational Closure. Their account significantly strengthens previous proposals based on the so-called KLM-properties. In this chapter a dynamic proof theory for Rational Closure is presented.
This paper presents the new adaptive logic APT. APT has the peculiar property that it enables one to interpret a (possibly inconsistent) theory Γ ‘as pragmatically as possible’. The aim is to capture the idea of a partial structure (in the sense of da Costa and associates) that adequately models a (possibly inconsistent) set of beliefs Γ. What this comes to is that APT localizes the ‘consistent core’ of Γ, and that it delivers all sentences that are compatible with this core. For the core itself, APT is just as rich as classical logic. APT is defined from a modal adaptive logic APV that is based itself on two other adaptive logics. I present the semantics of all three systems, as well as their dynamic proof theory. The dynamic proof theory for APV is unusual (even within the adaptive logic programme) in that it incorporates two different kinds of dynamics.
What philosophers call defeasible reasoning is roughly the same as nonmonotonic reasoning in AI. Some brief remarks are made about the nature of reasoning and the relationship between work in epistemology, AI, and cognitive psychology. This is followed by a general description of human rational architecture. This description has the consequence that defeasible reasoning has a more complicated structure than has generally been recognized in AI. We define a proposition to be warranted if it would be believed by an ideal reasoner. A general theory of warrant, based on defeasible reasons, is developed. This theory is then used as a guide in the construction of a theory of defeasible reasoning, and a computer program implementing that theory. The theory constructed deals with only a subset of defeasible reasoning, but it is an important subset.
The article presents a unifying adaptive logic framework for abstract argumentation. It consists of a core system for abstract argumentation and various adaptive logics based on it. These logics represent in an accurate sense all standard extensions defined within Dung’s abstract argumentation system with respect to sceptical and credulous acceptance. The models of our logics correspond exactly to specific extensions of given argument systems. Additionally, the dynamics of adaptive proofs mirror the argumentative reasoning of a rational agent. In particular, the presented logics allow for external dynamics, i.e. they are able to deal with the arrival of new arguments and are therefore apt to model open-ended argumentations by providing provisional conclusions.
This paper proposes two adaptive approaches to inconsistent priori- tized belief bases. Both approaches rely on a selection mechanism, that is not applied to the premises as they stand, but to the consequence sets of the belief levels. One is based on classical compatibility, the other on the modal logic T of Feys. For both approaches the two main strategies of inconsistency adaptive logics are formulated: the reliability strategy and the minimal abnormality strategy. All four systems are compared and found useful.
In this paper, I illustrate the main characteristics of abductive reasoning processes by means of an example from the history of the sciences. The example is taken from the history of chemistry and concerns a very small episode from Lavoisier's struggle with the 'air' obtained from mercury oxide. Eventually, this struggle would lead to the discovery of oxygen. I also show that Lavoisier's reasoning process can be explicated by means of a particular formal logic, namely the adaptive logic LA r . An important property of LA r is that it not only nicely integrates deductive and abductive steps, but that it moreover has a decent proof theory. This proof theory is dynamic, but warrants that the conclusions derived at a given stage are justified in view of the insight in the premises at that stage. Another advantage of the presented logic is that, as compared to other existing systems for abductive reasoning, it is very close to natural reasoning.
In this paper we characterize the six (basic) signed systems from [18] in terms of adaptive logics. We prove the characterization correct and show that it has a number of advantages. 1 Aim of This Paper In [18], signed propositional systems for paraconsistent reasoning were intro-duced and six central consequence relations were defined and studied. Three of the consequence relations are called signed, the three others are called un-signed. Some further consequence relations were defined in [18] and extended and studied in later works, for example [19]. In this paper we characterize the six central consequence relations in terms of adaptive logics. Doing so has a number of advantages. First, it avoids several complications (the preparation of the premises in negation normal form, their translation to a signed language, reasoning in terms of extensions) and hence makes the consequence relations more transparent. Next, it provides them with dynamic proofs, with a characteristic semantics, and with decision methods at the propositional level. Third, it makes several extensions obvious—we shall present the extension with a detachable implication and the extension to the predicative level. The extensions are absolutely straightforward whereas they are tiresome (if at all possible) on the signed approach. Even at the predicative level, there are partial decision methods and criteria (see [14] and [15] for tableau methods and [11] for procedural proofs). Fourth, it gives the consequence re-lations a place in a unified framework, which facilitates the comparison with other inconsistency-handling logics. Fifth, it provides them with easy proofs of many metatheoretic properties.
Inconsistency-adaptive logics isolate the inconsistencies that are deriv-able from a premise set, and restrict the rules of Classical Logic only where inconsistencies are involved. From many inconsistent premise sets, disjunctions of contradictions are derivable no disjunct of which is itself derivable. Given such a disjunction, it is often justified to introduce new premises that state, with a certain degree of confidence, that some of the disjuncts are false. This is an important first step on the road to consistency: it narrows down suspicion in inconsistent premise sets and hence locates the real problems among the possible ones. In this paper I present two approaches for handling such new premises in the context of the original premises. The first approach may apparently be combined with all paraconsistent logics. The second approach does not have the same generality, but is decidedly more elegant.
In this paper, adaptive logics are studied from the viewpoint of universal logic (in the sense of the study of common structures
of logics). The common structure of a large set of adaptive logics is described. It is shown that this structure determines
the proof theory as well as the semantics of the adaptive logics, and moreover that most properties of the logics can be proved
by relying solely on the structure, viz. without invoking any specific properties of the logics themselves.
This paper concerns a (prospective) goal directed proof procedure for the propositional fragment of the inconsistency-adaptive logic ACLuN1. At the propositional level, the procedure forms an algorithm for final derivability. If extended to the predicative level, it provides a criterion for final derivability. This is essential in view of the absence of a positive test. The procedure may be generalized to all flat adaptive logics.
A uniform approach to constructing and understanding nonmonotonic logics is proposed. This framework subsumes existing nonmonotonic formalisms, and yet is remarkably simple, adding almost no extra baggage to traditional logic. In the framework, classical notions such as satisfiability and entailment are redefined, and what happens to some standard results from classical logic, such as the deduction theroem, are examined. The author also briefly discusses how existing nonmonotonic logics are special cases of the proposed general formulation, and how this analysis serves to clarify them and relate them to one another.
In this paper, we present an adaptive logic for deontic conflicts, called P2.1r, that is based on Goble’s logic SDLaPe — a bimodal extension of Goble’s logic P that invalidates aggregation for all prima facie obligations. The logic P2.1r has several advantages with respect to SDLaPe. For consistent sets of obligations it yields the same results as Standard Deontic Logic and for inconsistent sets of obligations, it validates aggregation “as much as possible”. It thus leads to a richer consequence set than SDLaPe. The logic P2.1r avoids Goble’s criticisms against other non-adjunctive systems of deontic logic. Moreover, it can handle all the ‘toy examples’ from the literature as well as more complex ones.
When a conflict of duties arises, a resolution is often sought by determining an ordering of priority or importance. This paper ex- amines how such a conflict resolution works, compares mechanisms that have been proposed in the literature, and gives preference to one de- veloped by Brewka and Nebel. I distinguish between two cases - that some conflicts may remain unresolved, and that a priority ordering can be determined that resolves all - and provide semantics and axiomatic systems for accordingly defined dyadic deontic operators.
Since our ethical and behavioral norms have a conditional form, it is of great importance that deontic logics give an account of deontic commitments such as “A commits you to do/bring about B”. It is commonly agreed that monadic approaches are suboptimal for this task due to several shortcomings, for instance their falling short of giving a satisfactory account of “Strengthening the Antecedent” or their difficulties in dealing with contrary-to-duty paradoxes. While dyadic logics are more promising in these respects, they have been criticized for not being able to model “detachment”: A and the commitment under A to do B implies the actual obligation to do B. “We seem to feel that detachment should be possible after all. But we cannot have things both ways, can we? This is the dilemma on commitment and detachment.” (Lennart Åqvis. Deontic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, p. 199, Kluwer, Dordrecht, 2002).In this paper I answer Åqvist's question with “Yes, we can”. I propose a general method to turn dyadic deontic logics in adaptive logics allowing for a defeasible factual detachment while paying special attention to specificity and contrary-to-duty cases. I show that a lot of controversy about detachment can be resolved by analysing different notions of unconditional obligations. The logical modeling of detachment is paradigmatically realized on basis of one of Lou Goble's conflict tolerant CDPM logics.
The paper contains a survey of (mainly unpublished) adaptive logics of inductive generalization. These defeasible logics are precise formula- tions of certain methods. Some attention is also paid to ways of handling background knowledge, introducing mere conjectures, and the research guiding capabilities of the logics.
This article discusses the proof theory, semantics and meta-theory of a class of adaptive logics, called hierarchic adaptive logics. Their specific characteristics are illustrated throughout the article with the use of one exemplary logic HKx, an explicans for reasoning with prioritized belief bases. A generic proof theory for these systems is defined, together
with a less complex proof theory for a subclass of them. Soundness and a restricted form of completeness are established with
respect to a non-redundant semantics. It is shown that all hierarchic adaptive logics are reflexive, have the strong reassurance
property and that a subclass of them is a fixed point for a broad class of premise sets. Finally, they are compared to a different
yet related class of adaptive logics.
Introducing techniques deriving from dynamic proofs in proofs for propositional classical logic is shown to lead to a proof format that enables one to push search paths into the proofs themselves. The resulting goal directed proof format is shown to provide a decision method for A1 , . . . An B and a positive test for # A.
A logic of diagnosis proceeds in terms of a set of data and one or more (prioritized) sets of expectancies. In this paper we generalize the logics of diagnosis from [26] and present some alternatives. The former operate on the premises and expectancies themselves, the latter on their consequences.
Handling a possibly inconsistent prioritized belief base can be done in terms of consistent subsets. Humans do not compute consistent subsets, they just start reasoning and when confronted with inconsistencies in the course of their reasoning, they may adjust their interpretation of the information. In logics this behaviour corresponds to the mechanisms of dynamic proof theories. The aim of this paper is to transform known consequence relations for inconsistent prioritized belief bases in terms of consistent subsets, into dynamic proof theories that are a more faithful representation of human reasoning processes.
It was shown in [6] that the at Rescher{Manor consequence relations| the Free, Strong, Argued, C-Based, and Weak consequence relation|are all characterized by special applications of inconsistency-adaptive logics dened from the paraconsistent logic CLuN. As as result, these consequence relations are provided with a dynamic proof theory. In the present paper we show that the detour via an inconsistency-adaptive logic is not necessary. We present a direct dynamic proof theory, formulated in the language of Classical Logic, and prove its adequacy. 1 Aim of this Paper Rescher{Manor consequence relations constitute an approach to handling inconsistency. The underlying idea is that inconsistent sets of sentences are divided into maximal consistent subsets|henceforth MCS|and that what `follows ' from the inconsistent set is dened in terms of the classical consequences of the MCS or of a selection of them|Classical Logic will henceforth be abbreviated as CL. Such consequence relati...
Inconsistency-adaptive logics Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa
- Diderik Batens
Diderik Batens. Inconsistency-adaptive logics. In Ewa Orłowska, editor, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pages 445–472. Physica Verlag (Springer), Heidelberg, New
York, 1999.
Adaptive C n logics The Many Sides of Logic Forthcoming
- Diderik Batens
Diderik Batens. Adaptive C n logics. In Walter Carnielli, Marcello E.
Coniglio, and Itala M. Loffredo D'Ottaviano, editors, The Many Sides
of Logic, pages 27–45. College Publications, 2009. Forthcoming.
Mastering the Dynamics of Reasoning, with Special Attention to Handling Inconsistency
- Diderik Batens
Diderik Batens. Adaptive Logics and Dynamic Proofs. Mastering the
Dynamics of Reasoning, with Special Attention to Handling Inconsistency. In Progress.
A formal explication of the search for explanations: The adaptive logics approach to abductive reasoning
- Hans Lycke
Hans Lycke. A formal explication of the search for explanations: The
adaptive logics approach to abductive reasoning. In progress.
Hierarchic adaptive logics doi: 10.1093, 2011. Forthcoming. Published online at http://jigpal.oxfordjournals.org/content
- Frederik Van De Putte
Frederik Van De Putte. Hierarchic adaptive logics. Logic Journal
of IGPL 2011; doi: 10.1093, 2011. Forthcoming. Published online at http://jigpal.oxfordjournals.org/content/
early/2011/04/13/jigpal.jzr025.full.pdf.
Inconsistency-adaptive logics In Ewa Or? lowska, editor, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa
- Diderik
Diderik Batens. Inconsistency-adaptive logics. In Ewa Or? lowska, editor, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pages 445–472. Physica Verlag (Springer), Heidelberg, New York, 1999.
In Ewa Orłowska, editor, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa
- Diderik Batens
Diderik Batens. Inconsistency-adaptive logics. In Ewa Orłowska, editor, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pages 445-472. Physica Verlag (Springer), Heidelberg, New
York, 1999.
An adaptive logic for pragmatic truth
- A Walter
- Marcelo E Carnielli
- Itala M Coniglio
- Loffredo D'ottaviano
Joke Meheus. An adaptive logic for pragmatic truth. In Walter A.
Carnielli, Marcelo E. Coniglio, and Itala M. Loffredo D'Ottaviano,
editors, Paraconsistency. The Logical Way to the Inconsistent, pages
167-185. Marcel Dekker, New York, 2002.