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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
A Lost Proof
 IN TPHOLS: WORK IN PROGRESS PAPERS
, 2001
"... We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which t ..."
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We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which the proof is carried through to the order of the logic in which the problem is formulated in the first place can result in unmanageable long proofs, although there are short proofs in a logic of higher order. Our motivation in this paper is of practical nature and its aim is to sketch the implications of this example to current technology in automated theorem proving, to point to related questions about the foundational character of type theory (without explicit comprehension axioms) for mathematics, and to work out some challenging aspects with regard to the automation of this proof – which, as we belief, nicely illustrates the discrepancy between the creativity and intuition required in mathematics and the limitations of state of the art theorem provers.
Predicative Fragments of Frege Arithmetic
, 2003
"... Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply al ..."
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Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply all of secondorder Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying secondorder logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. 1
Frege's judgement stroke
 Australasian Journal of Philosophy
, 2000
"... This paper brings to light a new puzzle for Frege interpretation, and offers a solution to that puzzle. The puzzle concerns Frege's judgementstroke ('r), and consists in a tension between three of Frege's claims. First, Frege vehemently maintains that psychological considerations sho ..."
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This paper brings to light a new puzzle for Frege interpretation, and offers a solution to that puzzle. The puzzle concerns Frege's judgementstroke ('r), and consists in a tension between three of Frege's claims. First, Frege vehemently maintains that psychological considerations should have no place in logic. Second, Frege regards the judgementstrokeand the associated dissociation of assertoric force from content, of the act of judgement from the subject matter about which judgement is madeas a crucial part of his logic. Third, Frege holds that judging is an inner mental process, and that the distinction marked by the judgementstroke, between entertaining a thought and judging that it is true, is a psychological distinction. I argue that what initially looks like confusion here on Frege's part appears quite reasonable when we remind ourselves of the differences between Frege's conception of logic and our own. I. The Tension This paper is concerned with an apparent tension in Frege's work. On the one hand, Frege vigorously opposes any incursion on the part of psychology into the realm of logic. His attitude is summed up in the statement: 'it is the business of the logician to conduct an
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.
Certified by..........................................................
, 2008
"... Dealing with data in open, distributed environments is an increasingly important problem today. The processing of heterogeneous data in formats such as RDF is still being researched. Using rules and rule engines is one technique that is being used. In doing so, the problem of handling heterogeneous ..."
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Dealing with data in open, distributed environments is an increasingly important problem today. The processing of heterogeneous data in formats such as RDF is still being researched. Using rules and rule engines is one technique that is being used. In doing so, the problem of handling heterogeneous rules from multiple sources becomes important. Over the course of this thesis, I wrote several kinds of reasoners including backward, forward, and hybrid reasoners for RDF rule languages. These were used for a variety of problems and data in a wide range of settings for solving real world problems. During my investigations, I learned several interesting problems of RDF. First, simply making the term space big and well namespaced and the language low enough expressivity did not make computation necessarily easier. Next, checking proofs in an RDF environment proved to be hard because the basic features of RDF
Learning from Paradox
, 1995
"... The danger of paradoxes teaches us to check for consistency in (natural language) semantics. Paradoxes typically involve an element of selfreference (the settheoretic paradoxes) or an element of selfreference combined with reference to truth (the semantic paradoxes). Selfreference need not be vi ..."
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The danger of paradoxes teaches us to check for consistency in (natural language) semantics. Paradoxes typically involve an element of selfreference (the settheoretic paradoxes) or an element of selfreference combined with reference to truth (the semantic paradoxes). Selfreference need not be vicious and talking about truth need not be glib, but linguists who allow selfreference or a truth predicate in their representation languages should be aware of the dangers involved. 1 Russell's Paradox The logical and semantical paradoxes that were discovered at the beginning of this century arose in a context where formal languages were employed in a very loose sense. If Gottlob Frege is hailed in logic textbooks as the inventor of first order predicate logic, then it should be noted that in his Begriffsschrift (see [7]) proposal no distinction is made between language and metalanguage, and indeed, no formal definition of the language is given. Here is a rational reconstruction of Frege'...