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This essay is an exploration of the ontological landscape of reality. Its aim is to construct an ontological theory which will do justice to reality, and more precisely to those portions or levels of reality which are captured in our ordinary, common-sense or ‘folk ’ conceptual scheme. 2

Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λ-calculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced

We discuss the development of metamathematics in the Hilbert school, and Hilbert's proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.

Abstract. The Linked Data paradigm introduces the possibility to share machine-readable data across numerous Web resources, thus enabling applications that are traditionally only possible in corporate intranets to be realized on a Web scale. Due to the creation of an increasing number of publicly available Linked Open Data resources, the Web of Data has become a major application area for semantic technologies. This work introduces a recently published data set LON of non-lexical entities (NLEs) that can be used for numerous tasks of quantitative modeling on the Semantic Web. The size of the published data increases the magnitude of the public Linked Data significantly, yet we show how it can be seamlessly integrated into current application architectures for the Web of Data. 1

The Principle of Semantic Compositionality is the principle that the meaning of an expression is a function of, and only of, the meanings of its parts together with the method by which those parts are combined. 1 As stated, The Principle is vague or underspecified at a number of points such as "what counts as a part", "what is a meaning", "what kind of function is allowed " and the

This thesis provides a reconstruction of the B-method and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (first-order) logic is also considered: both its development and important points relevant to formal methods. Automated reasoning, particularly its theoretical limits as well as unification and resolution, is discussed. The main part of this thesis is a systematic reconstruction of the B-method, starting from its version of untyped predicate calculus and typed set theory, continuing with the Generalized Substitution Language (GSL) and finishing with the Abstract Machine Notation (AMN). Specification, refinement and implementation of a simple example using the B-method is presented. Both validation and verification of specifications, refinements and implementations using the B-method is discussed. The thesis concludes with a report of the current state of the effort (by the author) to implement the tool support of the B-method, as the Ebba Toolset. The main design decisions are discussed. The use of Unicode as the primary input encoding of AMN and GSL in Ebba is described.

. There are conflicting intuitions concerning the status of a boundary separating two adjacent entities (or two parts of the same entity). The boundary cannot belong to both things, for adjacency excludes overlap; and it cannot belong to neither, for nothing lies between two adjacent things. Yet how can the dilemma be avoided without assigning the boundary to one or the other thing at random? Some philosophers regard this as a reductio of the very notion of a boundary, which should accordingly be treated a mere faon de parler. In this paper I resist this temptation and examine some ways of taking the puzzle at face value within a realist perspective---treating boundaries as ontologically on a par with (albeit parasitic upon) voluminous parts. 1. Introduction The world of everyday experience is mostly a world of physical things separated in various ways from their environment: things with surfaces, skins, crusts, boundaries of some sort. These may not always be sharply defined, or so...

Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topic-neutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this

In this paper we put forward a new hypothesis on compositionality of meaning, namely that compositionality is bidirectional optimization.