A Theory of Structured Propositions

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus

A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte

Classical Opacity

Philosophy and Phenomenological Research, EarlyView

Fixed point combinators as fixed points of higher-order fixed point generators

Corrado B\"ohm once observed that if $Y$ is any fixed point combinator (fpc), then $Y(\lambda yx.x(yx))$ is again fpc. He thus discovered the first "fpc generating scheme" -- a generic way to build new fpcs from old. Continuing this idea, define an $\textit{fpc generator}$ to be any sequence of terms $G_1,\dots,G_n$ such that $$Y \in FPC \Rightarrow Y G_1 \cdots G_n \in FPC$$ In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several robust classes of such generators, by examining their elementary properties like injectivity and (weak) constancy. We provide sufficient conditions for existence of fixed points of a given generator $(G_1,\cdots,G_n)$: an fpc $Y$ such that $Y = Y G_1 \cdots G_n$. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This statement generalizes Statman's conjecture on non-existence of "double fpcs": fixed points of the generator $(G) = (\lambda yx.x(yx))$ discovered by B\"ohm. Finally, we define and make a few observations about the monoid of (weak) fpc generators. This enables us to formulate new a conjecture about their structure

Discriminating Lambda-Terms Using Clocked Boehm Trees

As observed by Intrigila, there are hardly techniques available in the lambda-calculus to prove that two lambda-terms are not beta-convertible. Techniques employing the usual Boehm Trees are inadequate when we deal with terms having the same Boehm Tree (BT). This is the case in particular for fixed point combinators, as they all have the same BT. Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model P-omega of lambda-calculus, and hence valid with respect to BT-equality but nevertheless the terms are beta-inconvertible. To prove such beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey information of the tempo in which the data in the BT are produced. Boehm Trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for lambda-terms. The corresponding equality is strictly intermediate between beta-convertibility and Boehm Tree equality, the equality in the model P-omega. An analogous approach pertains to Levy-Longo and Berarducci Trees. Our refined Boehm Trees find in particular an application in beta-discriminating fixed point combinators (fpc's). It turns out that Scott's equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a variety of production schemes of which the simplest was found by Boehm, stating that new fpc's are obtained by postfixing the term SI, also known as Smullyan's Owl. We prove that all these newly generated fpc's are indeed new, by considering their clocked BT's. Even so, not all pairs of new fpc's can be discriminated this way. For that purpose we increase the discrimination power by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257

Exploring the ontological links between Human Ecodynamics and field archaeology through the integration of archaeological reports into DataARC’s landscape ontology

This research focuses on the processes of knowledge creation within a framework of Big Data. Concretely, my project focus on grey archaeological data and their underlying ontologies, and how to interlink such type of data with other data in a Computational Ontology. This research was carried out within the dataARC Project, which tries to represent Human Ecodynamics for the North Atlantic context using data generated over 25 years by NABO (North Atlantic Biocultural Organisation). The project has been developing a cyberinfrastructure whose main tool is a computational ontology. We try to include in this ontology diverse conceptual models (from archaeological and historical, to ecological or geological data) from different grey sources. In my case, the main issue is the creation of knowledge using multiple datasets that comes from grey literature sources. Said another way, my thesis explores how to create a dataset by extracting data and their underlying ontologies from grey-archaeological reports with the final aim of interlinking them with other datasets. For so doing, these reports are transformed into a dataset, which in turn is mapped to the interdisciplinary ontology. This thesis, therefore, presents a prototype dataset and the process of conceptual mappings. In so doing, I explore how to decompose archaeological reports that are in Open Access to make their data FAIRer and (inter)linked. Human Ecodynamics are of special importance for NABO and, consequently, for the DataARC project. Our computational ontology, therefore, had to be developed for representing Human Ecodynamics in a rigorous and efficient way, yet capable of engaging a broad audience. This, along the use multiple, interconnected, datasets in a concrete relational manner, forces us to develop an ontology capable of representing very abstract themes while representing small details that affect these ecodynamics. This led me to try to develop a theoretical framework which could allow me to contextualise DataARC's ontology and my own method. The theoretical framework blends Latourian Actor-Network-Theory and some basic ontological principles (in the theoretical-philosophical sense) drawn from Human Ecodynamics. This might arguably open new avenues for developing powerful ontologies capable of representing complex knowledge. Problems such as overlappings or identifying the right number of hierarchical levels will be discussed, as well as some procedures that might help in rethinking computational ontologies. Digital Archaeology's ethical challenges is a final interesting point touched by this thesis. Here it is argued that Digital Archaeology, specifically if we convey inter/transdisciplinary knowledge such as Human Ecodynamics or Local and Traditional Knowledge, carries quite an important ethical responsibility. This leads me to offer new possible pathways by openly engaging theoretical critical schools of thought such as Social Ecology -which, in turn, might shed light on problems related to Climate Change and similar current issues