The term "non-monotonic logic" covers a family of formal frameworks devised to capture and represent defeasible inference , i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called "non-monotonic" because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base (...) itself. This is in contrast to classical (first-order) logic, whose inferences, being deductively valid, can never be "undone" by new information. (shrink)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be found (...) in human reasoning. Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

The article aims to show how the acceptance of non-monotonic logic enables arguments to be held between science and religion in a way that does not exclude either of these two spheres. The starting point of the analyses is the idea of the 13th century Danish philosopher, Boethius of Dacia, who states that it is both acceptable that: a natural scientist negates that the world had a beginning, and a Christian theologian asserts that the world had a beginning, (...) because each of them is basing their statements on the principles of their respective discipline: the first on the principles of nature, and the latter on knowledge supplemented by divine revelation. What is more, analogically: a metaphysician, when limited to his principles, cannot settle the issue, as he takes into account supranatural beings and their powers, but cannot know what God or another powerful supranatural would have decided in a such a case. The paper shows that Boethius’s approach: violates the rule of monotonicity, cannot be finally interpreted in terms of classical logic and assumes certain non-monotonic logic as its inference framework. Other presented examples of arguments between religious beliefs and the statements of natural science are resolved in the same way. Thus, it is shown how non-monotonic thinking allows us to seriously treat both scientific and religious inference as compatible. (shrink)

This paper analyses the logical structure of the balancing of conflicting normative arguments, and asks whether non-monotonic logic is adequate to represent this type of legal or practical reasoning. Norm conflicts are often regarded as a field of application for non-monotonic logics. This paper argues, however, that the balancing of normative arguments consists of an act of judgement, not a logical inference, and that models of deductive as well as of defeasible reasoning do not give an adequate (...) account of its structure. Moreover, it argues that as far as the argumentation consists in logical inferences, deductive logic suffices for reconstructing the argumentation from the internal point of view of someone making normative judgements. (shrink)

A range of formal models of human reasoning have been proposed in a number of fields such as philosophy, logic, artificial intelligence, computer science, psychology, cognitive science, etc.: various logics, probabilistic systems, belief revision systems, neural networks, among others. Now, it seems reasonable to require that formal models of human reasoning be empirically adequate if they are to be viewed as models of the phenomena in question. How are formal models of human reasoning typically put to empirical test? One (...) way to do so is to isolate a number of key principles of the system, and design experiments to gauge the extent to which participants do or do not follow them in reasoning tasks. Another way is to take relevant existing results and check whether a particular formal model predicts these results. The present investigation provides an illustration of the second kind of empirical testing by comparing two formal models for reasoning – namely the non-monotonic logic known as preferential logic; and a particular version of belief revision theories, screened belief revision – against the reasoning phenomenon known as belief bias in the psychology of reasoning literature: human reasoners typically seek to maintain the beliefs they already hold, and conversely to reject contradicting incoming information. The conclusion of our analysis will be that screened belief revision is more empirically adequate with respect to belief bias than preferential logic and non-monotonic logics in general, as what participants seem to be doing is above all a form of belief management on the basis of background knowledge. The upshot is thus that, while it may offer valuable insights into the nature of human reasoning, preferential logic is ultimately inadequate as a formal model of the phenomena in question. (shrink)

A range of formal models of human reasoning have been proposed in a number of fields such as philosophy, logic, artificial intelligence, computer science, psychology, cognitive science, etc.: various logics, probabilistic systems, belief revision systems, neural networks, among others. Now, it seems reasonable to require that formal models of human reasoning be empirically adequate if they are to be viewed as models of the phenomena in question. How are formal models of human reasoning typically put to empirical test? One (...) way to do so is to isolate a number of key principles of the system, and design experiments to gauge the extent to which participants do or do not follow them in reasoning tasks. Another way is to take relevant existing results and check whether a particular formal model predicts these results. The present investigation provides an illustration of the second kind of empirical testing by comparing two formal models for reasoning – namely the non-monotonic logic known as preferential logic; and a particular version of belief revision theories, screened belief revision – against the reasoning phenomenon known as belief bias in the psychology of reasoning literature: human reasoners typically seek to maintain the beliefs they already hold, and conversely to reject contradicting incoming information. The conclusion of our analysis will be that screened belief revision is more empirically adequate with respect to belief bias than preferential logic and non-monotonic logics in general, as what participants seem to be doing is above all a form of belief management on the basis of background knowledge. The upshot is thus that, while it may offer valuable insights into the nature of human reasoning, preferential logic is ultimately inadequate as a formal model of the phenomena in question. (shrink)

This paper reports on an approach to model generalized implicatures using nonmonotonic logics. The approach, called compositional, is based on the idea of compositional semantics, where the implicatures carried by a sentence are constructed from the implicatures carried by its constituents, but it also includes some aspects nonmonotonic logics in order to model the defeasibility of generalized implicatures.

In the first part I argue that normic laws are the phenomenological laws of evolutionary systems. If this is true, then intuitive human reasoning should be fit in reasoning from normic laws. In the second part I show that system P is a tool for reasoning with normic laws which satisfies two important evolutionary standards: it is probabilistically reliable, and it has rules of low complexity. In the third part I finally report results of an experimental study which demonstrate that (...) intuitive human reasoning is in well accord with basic argument patterns of system P. (shrink)

The present paper analyses the topic of scientific discovery and the problem of the existence of a logical framework involved in such endeavour. We inquire how several non-monotonic logic frameworks and other formalisms can account for such a task. In the same vein, we analyse some key aspects of the historical and theoretical debate surrounding scientific discovery, in particular, the context of discovery and context of justification context distinction. We present an argument concerning the weakening of the discovery/justification (...) context dichotomy based on the descriptive accent contained in the non-monotonic logic perspective together with its epistemological concerns. (shrink)

The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?

We investigate the application of Courcelle’s theorem and the logspace version of Elberfeld et al. in the context of non-monotonic reasoning. Here we formalize the implication problem for propositional sets of formulas, the extension existence problem for default logic, the expansion existence problem for autoepistemic logic, the circumscriptive inference problem, as well as the abduction problem in monadic second order logic and thereby obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, (...) we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu, resp., XLnu) under standard complexity assumptions. (shrink)

In this paper we present an embedding of abstract argumentation systems into the framework of Barwise and Seligmans logic of information flow. We show that, taking P.M. Dungs characterization of argument systems, a local logic over states of a deliberation may be constructed. In this structure, the key feature of non-monotonicity of commonsense reasoning obtains as the transition from one local logic to another, due to a change in certain background conditions. Each of Dungs extensions of argument (...) systems leads to a corresponding ordering of background conditions. The relations among extensions becomes a relation among partial orderings of background conditions. This introduces a conceptual innovation in Barwise and Seligmans representation of commonsense reasoning. (shrink)

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving (...) several reasoners, not just one as in the standard development. (shrink)

A consequence relation \ is monotonic iff for premise sets \ and conclusion \, if \, \, then \; and non-monotonic if this fails in some instance. More plainly, a consequence relation is monotonic when whatever is entailed by a premise set remains entailed by any of its supersets. From the High Middle Ages through the Early Modern period, consequence in theology is assumed to be monotonic. Concomitantly, to the degree the argument formulated by Anselm at (...) Proslogion 2–4 is taken up by later commentators, it is accepted or rejected in accordance with a monotonic notion of consequence. Examining Anselm’s use of parallelism in the Proslogion, I show Anselm embeds his famous argument in Proslogion 2–4 in a non-monotonic context. The results here presented challenge some deeply ingrained ideas governing the historiography of the long twelfth century, particularly concerning how the theology of the later eleventh through the twelfth century relates to the scholasticism of the thirteenth. (shrink)

In 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 Gödel’s 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. (shrink)

ABSTRACT The major approaches to belief revision and non monotonic reasoning proposed in the literature differ along a number of dimensions, including whether they are “syntax- based” or “semantic-based”, “foundational” or “coherentist”, “consistence-restoring” or “inconsistency-tolerant”. Our contribution towards clarifying the connections between these various approaches is threefold: •We show that the two main approaches to belief revision, the foundations and coherence theories, are mathematically equivalent, thus answering a question left open in [Gar90, Doy92], The distinction between syntax-based approaches to (...) revision and approaches based on (semantic) preferential structures falls along similar lines, and their expressive equivalence is a consequence of this result. •We formally clarify the connection between belief revision and non monotonic reasoning, in a particularly simple way which also throws light on the connection between consistence- restoring and reasoning-from-inconsistency approaches [BDP95]. •As a direct application of the above, we show that Poole's (syntax-based) system of default reasoning and Shoham's preferential semantic for non monotonic reasoning are also expressively equivalent, in that they can represent the same set of non monotonic consequence relations. (shrink)

Adams' famous thesis that the probabilities of conditionals are conditional probabilities is incompatible with standard probability theory. Indeed it is incompatible with any system of monotonic conditional probability satisfying the usual multiplication rule for conditional probabilities. This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength. It shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, but that a standard (...) complementation operation cannot be defined within them, on pain of collapsing probability into bivalence. (shrink)

Expected Utility Theory (EUT) has anomalies when interpreted descriptively and tested empirically. Experiments show that the way in which options are formulated is, in most cases, relevant for decision-making. This kind of anomaly is directly related, however, not with a proper axiom of EUT but rather with the logical principle of extensionality and its decision theoretic version: the principle of invariance. This paper focuses on the phenomenon of framing effects (FE) and the associated failures of invariance. FE arise when different (...) descriptions of the same object lead to different preferences, beliefs, or choices. A formal reconstruction of the Gain–Loss (G-L) frame is proposed, expanding the domain of standard decision theory to incorporate not only prospects but also descriptions of prospects. The system is extended by adding non-monotonic logical resources so as to account for the ceteris paribus character of FE. Finally, the same method is applied to sketching a formal representation for an anomalous case that involves subjective probabilities. The logical method is a general one that can be applied to most cases of FE. The philosophical consequences of the strategy pursued are also discussed. (shrink)

Some interesting cases of non-monotonic reasoning have already been identified in medieval theological texts. Jacob Archambault proved in 2015 that the argumentation presented by St Anselm of Canterbury in his Proslogion has non-monotonic “embeddings”. My own contribution from 2011 indicated that we can argue that a non-monotonic logic underlies some discussions provided by St Thomas Aquinas in his Summa theologiae, and showed that Boethius of Dacia used non-monotonic reasoning in his De aeternitate mundi. In this (...) article, I would like to briefly present these examples and verify whether we can speak about similar cases in medieval Biblical exegesis. My aim is to outline particular problems connected with the identification of non- monotonicity which are specific to theology, as well as assumptions that should be adopted to successfully discuss this issue. (shrink)

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov's positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov's theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its name (...) suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov's other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov's theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works. (shrink)

ABSTRACT The major approaches to belief revision and non monotonic reasoning proposed in the literature differ along a number of dimensions, including whether they are ?syntax- based? or ?semantic-based?, ?foundational? or ?coherentist?, ?consistence-restoring? or ?inconsistency-tolerant?. Our contribution towards clarifying the connections between these various approaches is threefold: ?We show that the two main approaches to belief revision, the foundations and coherence theories, are mathematically equivalent, thus answering a question left open in [Gar90, Doy92], The distinction between syntax-based approaches to (...) revision and approaches based on (semantic) preferential structures falls along similar lines, and their expressive equivalence is a consequence of this result. ?We formally clarify the connection between belief revision and non monotonic reasoning, in a particularly simple way which also throws light on the connection between consistence- restoring and reasoning-from-inconsistency approaches [BDP95]. ?As a direct application of the above, we show that Poole's (syntax-based) system of default reasoning and Shoham's preferential semantic for non monotonic reasoning are also expressively equivalent, in that they can represent the same set of non monotonic consequence relations. (shrink)

In this paper, it is argued that Ferguson’s (2003, Argumentation 17, 335–346) recent proposal to reconcile monotonic logic with defeasibility has three counterintuitive consequences. First, the conclusions that can be derived from his new rule of inference are vacuous, a point that as already made against default logics when there are conflicting defaults. Second, his proposal requires a procedural “hack” to the break the symmetry between the disjuncts of the tautological conclusions to which his proposal leads. Third, Ferguson’s (...) proposal amounts to arguing that all everyday inferences are sound by definition. It is concluded that the informal logic response to defeasibility, that an account of the context in which inferences are sound or unsound is required, still stands. It is also observed that another possible response is given by Bayesian probability theory (Oaksford and Chater, in press, Bayesian Rationality: The Probabilistic Approach to Human Reasoning, Oxford University Press, Oxford, UK; Hahn and Oaksford, in press, Synthese). (shrink)

We present a method for relevance sensitive non-monotonic inference from belief sequences which incorporates insights pertaining to prioritized inference and relevance sensitive, inconsistency tolerant belief revision. Our model uses a finite, logically open sequence of propositional formulas as a representation for beliefs and defines a notion of inference from maxiconsistent subsets of formulas guided by two orderings: a temporal sequencing and an ordering based on relevance relations between the putative conclusion and formulas in the sequence. The relevance relations are (...) ternary (using context as a parameter) as opposed to standard binary axiomatizations. The inference operation thus defined easily handles iterated revision by maintaining a revision history, blocks the derivation of inconsistent answers from a possibly inconsistent sequence and maintains the distinction between explicit and implicit beliefs. In doing so, it provides a finitely presented formalism and a plausible model of reasoning for automated agents. (shrink)

In most logical systems, inferences cannot be invalidated simply by the addition of new premises. If an inference can be drawn from a set of premises S, then it can also be drawn from any larger set incorporrating S. The truth of the original premises guarantees the truth of the inferred conclusion, and the addition of extra premises cannot undermine it. This property is known as monotonicity. Nonmonotonic inference lacks this property. The conclusions drawn are provisional, and new information may (...) lead to the withdrawal of a previous conclusion, even though none of the original premises is retracted. (shrink)

In this paper we present a framework for the dynamic and automatic generation of novel knowledge obtained through a process of commonsense reasoning based on typicality-based concept combination. We exploit a recently introduced extension of a Description Logic of typicality able to combine prototypical descriptions of concepts in order to generate new prototypical concepts and deal with problem like the PET FISH (Osherson and Smith, 1981; Lieto & Pozzato, 2019). Intuitively, in the context of our application of this (...) class='Hi'>logic, the overall pipeline of our system works as follows: given a goal expressed as a set of properties, if the knowledge base does not contain a concept able to fulfill all these properties, then our system looks for two concepts to recombine in order to extend the original knowledge based satisfy the goal. (shrink)

In this paper we will show that Hempel's covering law model can't deal very well with explanations that are based on incomplete knowledge. In particular the symmetry thesis, which is an important aspect of the covering law model, turns out to be problematic for these explanations. We will discuss an example of an electric circuit, which clearly indicates that the symmetry of explanation and prediction does not always hold. It will be argued that an alternative logic for causal explanation (...) is needed. And we will investigate to what extent non-monotonic epistemic logic can provide such an alternative logical framework. Finally we will show that our non-monotonic logical analysis of explanation is not only suitable for simple cases such as the electric circuit, but that it also sheds new light on more controversial causal explanations such as Milton Friedman's explanation of the business cycle. (shrink)

I will try to do three things in this paper. First, I want to situate certain problems in natural language semantics with respect to larger trends in logicism, including: (i) Attempts by positivist philosophers earlier in this century to provide a logical basis for the physical sciences; (ii) Attempts by linguists and logicians to develop a “natural language ontology” (and, presumably, a logical language that is related to this ontology by formally explicit rules) that would serve as a framework for (...) natural language semantics. (shrink)

ABSTRACT This paper defines a new modal logic based theory for non-monotonic reasoning. This logic expresses notions about hypotheses and known information. These notions are defined in the framework of the modal system τ. A translation of default logic in terms of hypothesis theory is given with which it is possible to fully characterize default logic by giving a necessary and sufficient criterion for the existence and the non-existence of extensions. Moreover several problems relating to (...) non-monotonic reasoning are discussed (and solutions are presented), such as case analysis, multiple extensions and contraposition. (shrink)

In this paper we address the modelling of reasoning processes in natural agents. We focus on a very basic kind of non-monotonic inference for which we identify a simple and plausible underlying process, and we develop a family of logical models that allow to match this process. Partial worlds models, as we call them, are a variant of Kraus, Lehmann and Magidor’s cumulative models. We show that the inference relations they induce form a strict subclass of cumulative relations and (...) tackle the issue of providing sound and complete sets of rules to characterise them. Taking inspiration from the work of Gabbay and Schlechta, we analyse the question in terms of definable sets of partial worlds and conclude that completeness is probably unreachable using a standard propositional language. This brings us to enrich our language with an additional connective, which makes it possible to distinguish between two kinds of disjunction in the partial worlds context. Within this renewed framework, we provide two representation theorems: one for inference relations induced by precisification-free smooth models, and the other for inference relations induced by precisification-free ranked models. Finally we give an aperçu of how partial worlds models lend themselves to the modelling of learning. (shrink)

Any inferential system in which the addition of new premises can lead to the retraction of previous conclusions is a non-monotonic logic. Classical conditional probability provides the oldest and most widely respected example of non-monotonic inference. This paper presents a semantic theory for a unified approach to qualitative and quantitative non-monotonic logic. The qualitative logic is unlike most other non- monotonic logics developed for AI systems. It is closely related to classical (i.e., Bayesian) (...) probability theory. The semantic theory for qualitative non-monotonic entailments extends in a straightforward way to a semantic theory for quantitative partial entailment relations, and these relations turn out to be the classical probability functions. (shrink)