Cut Elimination inside a Deep Inference System for Classical Predicate Logic

Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivation

Semantics and Ontology:\ud On the Modal Structure of an Epistemic Theory of Meaning

In this paper I shall confront three basic questions.\ud First, the relevance of epistemic structures, as formalized\ud and dealt with by current epistemic logics, for a\ud general Theory of meaning. Here I acknowledge M. Dummett"s\ud idea that a systematic account of what is meaning of\ud an arbitrary language subsystem must especially take into\ud account the inferential components of meaning itself. That\ud is, an analysis of meaning comprehension processes,\ud given in terms of epistemic logics and semantics for epistemic\ud notions.\ud The second and third questions relate to the ontological\ud and epistemological framework for this approach.\ud Concerning the epistemological aspects of an epistemic\ud theory of meaning, the question is: how epistemic logics\ud can eventually account for the informative character of\ud meaning comprehension processes. "Information� seems\ud to be built in the very formal structure of epistemic processes,\ud and should be exhibited in modal and possibleworld\ud semantics for propositional knowledge and belief.\ud However, it is not yet clear what is e.g. a possible world.\ud That is: how it can be defined semantically, other than by\ud accessibility rules which merely define it by considering its\ud set-theoretic relations with other sets-possible worlds.\ud Therefore, it is not clear which is the epistemological status\ud of propositional information contained in the structural\ud aspects of possible world semantics. The problem here\ud seems to be what kind of meaning one attributes to the\ud modal notion of possibility, thus allowing semantical and\ud synctactical selectors for possibilities. This is a typically\ud Dummett-style problem.\ud The third question is linked with this epistemological\ud problem, since it is its ontological counterpart. It concerns\ud the limits of the logical space and of logical semantics for a\ud of meaning. That is, it is concerned with the kind of\ud structure described by inferential processes, thought, in a\ud fregean perspective, as pre-conditions of estentional\ud treatment of meaning itself. The second and third questions\ud relate to some observations in Wittgenstein"s Tractatus.\ud I shall also try to show how their behaviour limits the\ud explicative power of some semantics for epistemic logics\ud (Konolige"s and Levesque"s for knowledge and belief)

Integrating descriptions of knowledge management learning activities into large ontological structures: A case study

Ontologies have been recognized as a fundamental infrastructure for advanced approaches to Knowledge Management (KM) automation, and the conceptual foundations for them have been discussed in some previous reports. Nonetheless, such conceptual structures should be properly integrated into existing ontological bases, for the practical purpose of providing the required support for the development of intelligent applications. Such applications should ideally integrate KM concepts into a framework of commonsense knowledge with clear computational semantics. In this paper, such an integration work is illustrated through a concrete case study, using the large OpenCyc knowledge base. Concretely, the main elements of the Holsapple & Joshi KM ontology and some existing work on e-learning ontologies are explicitly linked to OpenCyc definitions, providing a framework for the development of functionalities that use the built-in reasoning services of OpenCyc in KM ctivities. The integration can be used as the point of departure for the engineering of KM-oriented systems that account for a shared understanding of the discipline and rely on public semantics provided by one of the largest open knowledge bases available

Psychosemantic analyticity

It is widely agreed that the content of a logical concept such as and is constituted by the inferences it enters into. I argue that it is impossible to draw a principled distinction between logical and non-logical concepts, and hence that the content of non-logical concepts can also be constituted by certain of their inferential relations. The traditional problem with such a view has been that, given Quine’s arguments against the analytic-synthetic distinction, there does not seem to be any way to distinguish between those inferences that are content constitutive and those that are not. I propose that such a distinction can be drawn by appealing to a notion of ‘psychosemantic analyticity’. This approach is immune to Quine’s arguments, since ‘psychosemantic analyticity’ is a psychological property, and it is thus an empirical question which inferences have this property

Interactive Destiny

Mitra demonstrates that specific memory erasure causes the observer to be in a different sector of the multiverse, one with a different destiny: events in the future, remote to any possible influence of the observer, having radically different probabilities. The concept only applies to an observer defined by a structure of information, so cannot apply to a human observer as usually defined, as the physical body. However, Everett defines the functional identity of the observer as the contents of the memory, a structure of information. Only such an identity encounters the appearance of collapse. Thus, any observer encountering change of this nature is necessarily of this type, and in principle Mitra's effect would apply. Alteration to the quantum state of the physical environment effective for the observer merely by deletion of a record of observation would seem to require that the universe is primarily an information system, and that physical reality is secondary to the information defining it. This, however, is only the case with respect to the collapse dynamics. The universe is first and foremost a physical reality, as generally understood, defined by the quantum state, with the concomitant linear dynamics. Thus, at any given moment, the effective physical environment of the observer is a Newtonian, relativistic, physical domain, probabilistically defined throughout four-dimensional space-time by the linear dynamics of the quantum state of the environment effective for that observer: here the quantum mechanical frame of reference. With regard to the collapse dynamics, such a domain is of a first, primitive, logical type, while collapse, the change of the quantum mechanical frame of reference, is of a different, second logical type. As Everett makes clear, collapse is a purely subjective phenomenon, and as Tegmark explains, it exists only on the inside view of the quantum mechanical frame of reference. In this regard, and here only, the information process of the collapse dynamics, the establishment of new correlations with the physical environment, is primary, and, in a sense, 'overrules' the linear dynamics of the physical environment

A System of Interaction and Structure II: The Need for Deep Inference

This paper studies properties of the logic BV, which is an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator. BV is presented in the calculus of structures, a proof theoretic formalism that supports deep inference, in which inference rules can be applied anywhere inside logical expressions. The use of deep inference results in a simple logical system for MLL extended with the self-dual non-commutative operator, which has been to date not known to be expressible in sequent calculus. In this paper, deep inference is shown to be crucial for the logic BV, that is, any restriction on the ``depth'' of the inference rules of BV would result in a strictly less expressive logical system

Is Time Travel Too Strange to Be Possible? Determinism and Indeterminism on Closed Timelike Curves

Notoriously, the Einstein equations of general relativity have solutions in which closed timelike curves (CTCs) occur. On these curves time loops back onto itself, which has exotic consequences. However, in order to make time travel stories consistent constraints have to be satisfied, which prevents seemingly ordinary and plausible processes from occurring. This, and several other "unphysical" features, have motivated many authors to exclude solutions with CTCs from consideration, e.g. by conjecturing a chronology protection law. In this contribution we shall investigate the nature of one particular class of exotic consequences of CTCs, namely those involving unexpected cases of indeterminism or determinism. Indeterminism arises even against the backdrop of the usual deterministic physical theories when CTCs do not cross spacelike hypersurfaces outside of a limited CTC-region (such hypersurfaces fail to be Cauchy surfaces). By contrast, a certain kind of determinism appears to arise when an indeterministic theory is applied on a CTC: things cannot be different from what they already were. We shall argue that on further consideration both this indeterminism and determinism on CTCs turn out to possess analogues in other, familiar areas of physics. CTC-indeterminism is close to the epistemological indeterminism we know from statistical physics, while the "fixedness" typical of CTC-determinism is pervasive in physics. CTC-determinism and CTC-indeterminism therefore do not provide incontrovertible grounds for rejecting CTCs as conceptually inadmissible