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Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... 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 λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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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 λcalculus 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 λcalculus (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
The meaning of ‘most’: semantics, numerosity and psychology. Mind and Language 24:554
, 2009
"... Abstract: The meaning of ‘most ’ can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: ‘most ’ is understo ..."
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Abstract: The meaning of ‘most ’ can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: ‘most ’ is understood in terms of onetoone correspondence. Adults evaluated ‘Most of the dots are yellow’, as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a ‘onetoone with remainder ’ strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that ‘most ’ is understood in terms of cardinality comparison, even when counting is impossible. How is the word ‘most ’ related to human capacities for detecting and comparing numerosities? One might think the answer is obvious, and explicit in standard semantic theories: ‘most ’ is understood in terms of a capacity to compare cardinal numbers; e.g. ‘Most of the dots are yellow ’ means that the number of yellow dots
Leveraging NonLexical Knowledge for the Linked Open Data Web
"... Abstract. The Linked Data paradigm introduces the possibility to share machinereadable 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 av ..."
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Cited by 8 (0 self)
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Abstract. The Linked Data paradigm introduces the possibility to share machinereadable 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 nonlexical 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
On Substances, Accidents and Universals. In defence of a constituent Ontology. Philosophical Papers 26
, 1997
"... 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, commonsense or ‘folk ’ conceptual scheme. 2 ..."
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Cited by 7 (4 self)
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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, commonsense or ‘folk ’ conceptual scheme. 2
The Principle of Semantic Compositionality
 Topoi
, 1994
"... 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 " ..."
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Cited by 6 (1 self)
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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
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic 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 ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic 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.
What does it mean to say that logic is formal?
, 2000
"... 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 ..."
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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 “topicneutral”) (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
The formal method known as B and a sketch for its implementation
, 2002
"... This thesis provides a reconstruction of the Bmethod 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 (firstorder) logic is also conside ..."
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This thesis provides a reconstruction of the Bmethod 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 (firstorder) 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 Bmethod, 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 Bmethod is presented. Both validation and verification of specifications, refinements and implementations using the Bmethod 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 Bmethod, 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.
Boundaries, Continuity, and Contact
 Time, Space and Movement: Meaning and Knowledge in the Sensible World (Proceedings of the 5th International Workshop
, 1997
"... . 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 ..."
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. 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 perspectivetreating 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...