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# Topic: Proof theory

###### In the News (Wed 19 Jun 19)

 Proof theory Proof theory, a form of metamathematics, studies the ways in which proofs are used in mathematics. As such, proof theory is related to syntax[?] in logic; model theory correspondingly relates to semantics[?]. Proof theory, model theory, axiomatic set theory, and recursion theory are the so-called "four theories" of the foundations of mathematics. www.ebroadcast.com.au /lookup/encyclopedia/pr/Proof_(logic).html   (322 words)

 Mathematical proof In the context of proof theory, where purely formal proofs are considered, such not entirely formal demonstrations are called "social proofs". The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language[?]. Proof by contradiction: where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false. www.ebroadcast.com.au /lookup/encyclopedia/pr/Proof_(math).html   (399 words)

 Proof theory - Wikipedia, the free encyclopedia Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proof theory can also be considered a branch of philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible. Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. en.wikipedia.org /wiki/Proof_theory   (1014 words)

 03: Mathematical logic and foundations proof by contradiction is a proof of the contrapositive). Proof theory is the study of certain kinds of symbol manipulation. To the extent that one "calculates" a proof of a theorem, this is the question of decidability in proof theory. www.math.niu.edu /~rusin/known-math/index/03-XX.html   (2050 words)

 Proof Theory The topic of proof theory is the study of axiom systems, in which mathematical proofs can be formalized, mainly subsystems of set theory and analysis. The main goal of a proof theoretic analysis is the determination of the proof theoretic ordinal of a theory, a measure for its strength. Proof theoretic methods are often the only way to reduce impredicative systems to more constructive principles. www.math.uu.se /~setzer/foerelaesning/bevisteori   (285 words)

 Open Directory - Science: Math: Logic and Foundations: Proof Theory   (Site not responding. Last check: 2007-11-03) The Calculus of Structures - The calculus of structures is a new proof theoretical formalism. Proof Theory as an Alternative to Model Theory - Short article by Dale Miller, arguing that logic programming languages should base their semantics on proof theory, not model theory. Topics in Logic and Proof Theory - Brief introductions to combinatory logic, the incompleteness theorems and independence results, by Andrew D Burbanks. dmoz.org /Science/Math/Logic_and_Foundations/Proof_Theory   (351 words)

 Proof Theory and Programming During the software development process, specification, design and proof search methodologies are essential and have an impact on the activities of verification and validation and then on the quality of the obtained software. A consequence of this approach is that we consider different formal systems from type theory and intuitionistic, classical or linear logics, that allow to represent and to analyze concepts, at a logical level, such as communication, concurrency, sequentiality, control or verification of systems properties. The results will be applied and specialized in domains as the design of process calculi based on proof theory of linear logic, the automated analysis of sentences in natural language, the diagnosis of actions and the synchronization of activities in a network. www.ercim.org /publication/Ercim_News/enw23/galmiche.html   (740 words)

 Syntax and Proof theory   (Site not responding. Last check: 2007-11-03) For example, boolean algebra is of little interest to mathematicians since it is a decidable theory however, it is of considerable interest to engineers who use it to design digital systems. A proof is a sequence of formulas each of which is an axiom, or may be inferred by an inference rule, from formulas appearing earlier in the sequence. A theory is interesting when not all formulas of the language are theorems or the definition of a theorem is not effective. cs.wwc.edu /~aabyan/CII/BOOK/book/node75.html   (765 words)

 in theory One is that the "natural proofs" results show that, assuming strong one-way functions exist (an assumption in the "ballpark" of P \$\neq\$ NP) there are boolean functions that are efficiently computable but have all the efficiently computable properties of random functions. So although the proof is a finite object, it does define an "algorithm" (the one that describes the properties of the function that are used in the proof) and such algorithm cannot be asymptotically efficient. Marge, I agree with you - in theory. in-theory.blogspot.com   (4733 words)

 Amazon.ca: Basic Proof Theory: Books: A. S. Troelstra,H. Schwichtenberg   (Site not responding. Last check: 2007-11-03) Schwichtenberg (Author) "Proof theory may be roughly divided into two parts: structural proof theory and interpretational proof theory..." (more) Examples are given of several areas of application, namely: the metamathematics of pure first-order logic, logic programming theory, category theory, modal logic, linear logic, first-order arithmetic and second-order logic. Proof theory may be roughly divided into two parts: structural proof theory and interpretational proof theory. Read the first page www.amazon.ca /Basic-Proof-Theory-S-Troelstra/dp/0521779111   (547 words)

 Atheistic Manifesto - Proof, Theory, & The Scientific Method It is very unfortunate that a large percentage of our population do not understand the meaning of the words "proof" and "theory". Scientific theories represent conclusions drawn from observation of the natural world and the testing of these conclusions using models which imitate reality. The application of this observation to model systems supports the theory of an expanding Universe where space itself is expanding, but does not support the theory of an explosion in static space. www.chestnutcafe.com /cafe/_manifesto/02_proof.html   (764 words)

 Type Theory (Stanford Encyclopedia of Philosophy) The theory of types was introduced by Russell in order to cope with some contradictions he found in his account of set theory (Russell, 1903). Besides these proof theoretic investigations related to the ramified hierarchy, much work has been devoted in proof theory to analysing the consistency of the axiom of reducibility, or, equivalently, the consistency of impredicative definitions. Type theory can be used as a foundation for mathematics, and indeed, it was presented as such by Russell in his 1908 paper, which appeared the same year as Zermelo's paper, presenting set theory as a foundation for mathematics. plato.stanford.edu /entries/type-theory   (6501 words)

 Peter Suber, "Gödel's Proof" We cannot say that the number theory interpretation is "primary" and the metatheory interpretation "secondary" or nonstandard except in reference to human intentions. Gödel's proof in a nutshell is to create a wff that says in one interpretation, "This wff cannot be proved in S", then to prove that it is undecidable in S, and thereby to prove that it is true. Here, instead of saying that there is no number which is the Gödel number of the proof of G, we are making the separate denials for each natural number: wff-sequence(0) is not the proof of G, wff-sequence(1) is not the proof, wff-sequence(2) is not the proof, and so on. www.earlham.edu /~peters/courses/logsys/g-proof.htm   (4313 words)

 The Epsilon Calculus (Stanford Encyclopedia of Philosophy) Suppose one of the axioms in the proof is the transfinite axiom In addition to the traditional, foundational branch of proof theory, today there is a good deal of interest in structural proof theory, a branch of the subject that focuses on logical deductive calculi and their properties. Girard, J.-Y., 1982, ‘Herbrand's theorem and proof theory’, Proceedings of the Herbrand Symposium, Amsterdam: North-Holland, 29-38 plato.stanford.edu /entries/epsilon-calculus   (6424 words)

 Greg Restall * Proof Theory and Philosophy, in progress. Be a useable textbook in philosophical logic, accessible to someone who’s done only an intro course in logic, covering at least some model theory and proof theory of propositional logic, and maybe predicate logic. Present the duality between model theory and proof theory in a philosophically illuminating fashion. Posted by: Charles Stewart at November 10, 2004 08:11 AM ‘basic proof theory’, on cambridge, presents proofs for normalization and strong normalization, so does ‘structural proof theory’. consequently.org /writing/ptp   (337 words)

 Proof theory Summary Hilbert and Formalism The leading exponent of the formalist philosophy of mathematics was David Hilbert (1862–1943), who pioneered in a development of logic known as proof theory or metamathematics. Proof Theory The background to the development of "proof theory" since 1960 is contained in the entry "Mathematics, Foundations of." Briefly, Hilbert's program (HP), inaugurated in the 1920s, aimed to secure the foundat... Proofs are typically presented as inductively-defined data structures such as plain lists, boxed... www.bookrags.com /Proof_theory   (181 words)

 Proof Theory Forum   (Site not responding. Last check: 2007-11-03) Links: A course on structural proof theory given by Roy Dyckhoff at the University of Dresden. Papers on proof theory by Sara Negri and Jan von Plato. Text of the tutorial (in pdf format) "Five Lectures on Proof Analysis" given by Sara Negri at the Summer School on Proof Theory, Computation and Complexity, Dresden 2003. www.helsinki.fi /~negri/ptforum.html   (214 words)

 Issues in Model Theory and Proof Theory   (Site not responding. Last check: 2007-11-03) The study of formal systems naturally separates into two areas, proof theory which is concerned with language, axioms, inference, and what is provable and model theory which is concerned with the relationship between the structure and the language and whether what is true is provable. Thus, model theory begins with a class of set-theoretic objects called relational structures and constructs a language and a mapping from the language to the structure. We begin with semantics and model theory and follow with syntax and proof theory. cs.wwc.edu /~aabyan/CII/BOOK/book/node73.html   (460 words)

 Martin-Löf Type Theory: Semantics and Proof Theory   (Site not responding. Last check: 2007-11-03) The formal system of Martin-Löf Type Theory (henceforth ``Type Theory'') was developed together with its semantics and proof theory. Proof theory refers to the part of logic in which proofs are considered as mathematical objects and as such become the subject of metamathematical study. The development of Type Theory was influenced by the proof-theoretic tradition and early papers often discuss typical proof-theoretic issues such as proof normalization and proof-theoretic strength (ordinal analysis). www.cs.chalmers.se /~coquand/Sem.html   (212 words)

 Amazon.com: Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science): Books: A. S. Troelstra,H. ...   (Site not responding. Last check: 2007-11-03) Proof theory may be roughly divided into two parts: structural proof theory and interpretational proof theory. Considerations about lengths of proofs are undeniably important when the proofs in question are infinitely long; yet students of the subject should be allowed to see that the considerations that apply here are just generalizations of the same considerations as they apply to finitely long proofs. People doing research in proof theory might also welcome the fact that the authors discuss quite a wide variety of logical systems, thus giving the reader a chance to weigh up the merits and disadvantages of each. amazon.com /Theory-Cambridge-Theoretical-Computer-Science/dp/0521779111   (1640 words)

 Proof Theory Logic and Foundations Math Science - Basic material on proof theory and the home page of the only mailing list devoted to proof theory, with hundreds of experts. Its thematic is focused on developing the theory and the applications of Linear Logic. - The calculus of structures is a new proof theoretical formalism. www.iaswww.com /ODP/Science/Math/Logic_and_Foundations/Proof_Theory   (235 words)

 Abstracts of Proof Theory Papers In this paper we show that there are two notions of goal-directed proof in classical logic, both of which are suitably weaker than that for intuitionistic logic. Uniform proofs have been presented as a basis for logic programming, and it is known that by restricting the class of formulae it is possible to guarantee that uniform proofs are complete with respect to provability in intuitionistic logic. It is known that by restricting the class of formulae it is possible to guarantee that a certain class of proofs, known as uniform proofs, are complete with respect to provability in intuitionistic logic. goanna.cs.rmit.edu.au /~jah/publications/proofs-abstracts.html   (1467 words)

 Equivalence of Logics: the categorical proof theory perspective (Invited talk) The categorical proof theory approach to logic has been around since at least the early sixties (cf. In this ``propaganda'' talk I will describe the basic ideas of categorical proof theory, some of its successes and some of its possibilities as far as applications back in logic are concerned. Thus I hope to explain the identity criteria that we arrive at, when confronted with the problem of deciding when two logical systems should be taken as ``the same'', using the perspective of categorical proof theory. www.parc.xerox.com /research/publications/details.php?id=5402   (258 words)

 Structural Proof Theory - Cambridge University Press   (Site not responding. Last check: 2007-11-03) Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. Back to natural deduction; Conclusion: diversity and unity in structural proof theory; Appendix A. Simple type theory and categorical grammar; Appendix B. Proof theory and constructive type theory; Appendix C. A proof editor for sequent calculus. www.cambridge.org /catalogue/catalogue.asp?isbn=0521793076   (213 words)

 Workshop on Proof Theory and Computation The focus of the workshop is on proof theory and computation: we aim at a small, informal event where people have plenty of time to exchange ideas. By applying insights from proof theory and the philosophy of language, the theory of harmony in logic and type theory hopes to provide a more natural approach than either denotational or operational approaches. This workshop is organized by the AI Institute at TU Dresden, and sponsored by IQN (Rational mobile agents and systems of agents) and Graduiertenkolleg 334 (Specification of discrete processes and systems of processes by operational models and logics). www.ki.inf.tu-dresden.de /~guglielm/WPT   (1217 words)

 BRICS Theme: Proofs and Complexity The main activity will be the workshop Proof Theory and Complexity held August 3-7, 1998. The course by Ulrich Kohlenbach on Proof Interpretations. During the last years the connections between proof theory and theoretical computer science have become more and more intensive in both directions: proof theoretic techniques are central tools in e.g. www.brics.dk /Activities/98/PAC   (761 words)

 Just What is it that Makes Martin-Lof's Type Theory so Different, so Appealing? | Lambda the Ultimate Constructive proofs are important for several reasons: they are usually clearer than pulling a rabbit out of a hat, they state a "cause" for truth, they form programs, and they are required in some instances (as for example Montague semantics). A notion of proof equivalence is important because it tells you, if someone claims to have found a "new" proof, whether it is genuinely new or just a different perspective on an old proof. Since you seem knowledgeable about ML type theory, maybe you can clarify a puzzle for me. Whenever I read about W-types, I see mysterious comments to the effect that there are problems with their induction principles due to too much "junk" being around. lambda-the-ultimate.org /node/view/1078   (2920 words)

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