It is generally recognized that the possibdity of detecting contradictions and identifying their sources is an important feature of an intelligent system. Systems that are able to detect contradictions, identify their causes, or readjust their knowledge bases to remove the contradiction, called Belief Revision Systems. Truth Maintenance Systems, or Reason Maintenance Systems. have been studied by several researchers in Artificial bttelligence ( AI). In this paper, we present a logic suitable for supporting belief revision systems, discuss the properties that a belief revision system based on this logic will exhibit, and present a particular intplementation of our model of a belief revision system. The system we present differs from most of the systems developed so far in three respects: First, it is baseti on a logic that was developed to support belief revision systems. Second, it uses the rules of inference of the logic to automatically compute the dependencies among propositions rather than having to force the user to do titis, as in many existing systems. Third, it was the first belief revision system whose implementation relies on the manipulation of sets of assumptions, not justifications.

Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their

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It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

A key cognitive faculty that enables humans to communicate with each other is their ability to incrementally construct and use models describing the mental states of others. Every such model describing some other cognitive agent will realistically contain only a finite number of sentences in some language of thought, hence, assuming sufficiently powerful inference rules, some of its consequences will remain implicit. To make them explicit, the person holding the model could employ a kind of reasoning that can be paraphrased as "what would I believe if I were the other person believing everything I believe that person believes", a strategy that can be viewed as a simulation of the other person's reasoning using the model of that person in conjunction with the reasoning abilities of the simulator. If we want to equip an artificial cognitive agent with such a simulative reasoning ability we have to cope with problems such as simulation at various levels of nesting, meta-reasoning to make ...

We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proof-checking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.

A key cognitive faculty that enables humans to communicate with each other is their ability to incrementally construct and use models describing the mental states of others, in particular, models of their beliefs. Not only do humans have beliefs about the beliefs of others, they can also reason with these beliefs even if they do not hold them themselves. If we want to build an artificial or computational cognitive agent that is similarly capable, we need a formalism that is fully adequate to represent the beliefs of other agents, and that also specifies how to reason with them. Standard formalizations of knowledge or belief, in particular the various epistemic and doxastic logics, seem to be not very well suited to serve as the formal device upon which to build an actual computational agent. They neglect either representation problems, or the reasoning aspect, or the defeasibility that is inherent in reasoning about somebody else's beliefs, or they use idealizations which are problema...

Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.

The primary focus of this thesis is the semantic gap between a fine-grained structural operational semantics and a set of rely/guarantee-style development rules. The semantic gap is bridged by considering the development rules to be a part of the same logical framework as the operational semantics, and a set of soundness proofs show that the development rules, though making development easier for a developer, do not add any extra power to the logical framework as a whole. The soundness proofs given are constructed to take advantage of the structural nature of the language and its semantics; this allows for the addition of new development rules in a modular fashion. The particular language semantics allows for very fine-grained concurrency. The language itself includes a construct for nested parallel execution of statements, and the semantics is written so that statements can interfere with each other between individual variable reads. The language also includes an atomic block construct for which the semantics is an embodiment of a form of software transactional memory. The inclusion of the atomic construct helps illustrate the inherent expressive weakness present in the rely/guarantee rules with respect to termination properties. As such, two development rules are proposed for the atomic construct, one of which has serious restrictions in its application, and another for which the termination property does not hold.

We present a formalization of propositional modal logic in the framework of Labelled Deductive Systems (LDS) in which modal theory is presented as a "configuration" of several "local actual worlds". We define a natural deduction style proof system for a propositional modal labelled deductive system (MLDS). We describe a model--theoretical semantics (based on first--order logic) and we show that the natural deduction proof system is sound and complete with respect to this semantics. We also show that the semantics given here is equivalent to Kripke semantics for a normal modal logic whenever the initial configuration is a single point. Finally we discuss how this logic can be extended to the predicate case, wesketch some natural deduction rules for quantifiers and we discuss how such rules solve certain problems associated with the nesting of quantifiers within the scope of modal operators. 1 Contents 1 Basic definitions concerning MLDS 5 2 A natural deduction system for MLDS 11 3 A ...