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Syntax versus Semantics [article]

Reinhard Kahle, Wilfried Keller
2015 arXiv   pre-print
formal livery at all. 15 It is worth mentioning that dialogical logic has a rather interesting status concerning the syntax and semantics distinction, which doesn't seem to be properly explored yet, see Kahle  ... 
arXiv:1507.04678v1 fatcat:rt3vtq7brner5eprsr5g43kij4

Die Gödelschen Unvollständigkeitssätze [chapter]

Reinhard Kahle
2017 Facetten der Mathematik  
doi:10.1007/978-3-662-55656-6_16 fatcat:z32f3klxqzaufj23xktrigcfoa

Die Gödelschen Unvollständigkeitssätze

Reinhard Kahle
2007 Mathematische Semesterberichte  
doi:10.1007/s00591-006-0012-9 fatcat:y7muau2xgnaq7iiapa4cen65wy

Hilbert's Paradox

Volker Peckhaus, Reinhard Kahle
2002 Historia Mathematica  
doi:10.1006/hmat.2002.2345 fatcat:fsah733w4ngdtjl3enjtdhj3fi

Structured belief bases

Reinhard Kahle
2004 Logic and Logical Philosophy  
© 2002 by Nicolaus Copernicus University Reinhard Kahle  ... 
doi:10.12775/llp.2002.003 fatcat:5q5edjjcxjfdjmzhx3jwvrs6i4

Introduction: Proof-theoretic Semantics

Reinhard Kahle, Peter Schroeder-Heister
2006 Synthese  
A Na2CO 3 precipitation method to prepare powder of Na-13-Alumina is presented. Comparisons have been made with powder of the same composition obtained by two other methods: the traditional one and a slurry-solution method. After phase characterisation by X-ray, the powders were calcined at different temperatures. The resulting products were characterised by TGA, DSC and XRD analysis. Finally, density and ionic conductivity of sintered pellets were measured.
doi:10.1007/s11229-004-6292-5 fatcat:fcuy7nn7anhrncb6m3pdromqba

Dedekinds Sätze und Peanos Axiomata

Reinhard Kahle
2021 Philosophia Scientiæ  
1905, 250] 31 Hier sollten wir bemerken, daß es neben der logizistischen und der (später so charakterisierten) f (...) 32 Siehe [Kahle  ...  Ausführungen in seinem Brief an Keferstein deutlich wird.21 Dabei hatte er insbesondere schon die Konstruktion von heute sogenannten Nichtstandardmodellen der (erststufigen) Peano-Arithmetik vor Augen [siehe Kahle  ... 
doi:10.4000/philosophiascientiae.2846 fatcat:5hakbfavrjgl3l3nfcbctzzrvi

k-Provability in $$\hbox {PA}$$

Paulo Guilherme Santos, Reinhard Kahle
2021 Logica Universalis  
AbstractWe study the decidability of k-provability in $$\hbox {PA}$$ PA —the relation 'being provable in $$\hbox {PA}$$ PA with at most k steps'—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of $$\hbox {PA}$$ PA is still an open problem, but it is known that the proof-skeleton problem is
more » ... decidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.
doi:10.1007/s11787-021-00278-1 fatcat:poqulusw7zfgbaf3y7lhygfz6a

Towards the Structure of Mathematical Proof

Reinhard Kahle
2014 International Conference on Intelligent Computer Mathematics  
This proof should be understandable for everybody with basic mathematical knowledge. However, the editor draws attention to the fact, that a simple test of correctness of this proof is out of reach for the current computer provers [Wie06, p. 3]: "Ideally, a computer should be able to take this text as input and check it for its correctness. We clearly are not yet there." Thus, he advances with the representation of proofs of this theorem given, or implemented, in the theorem provers presented
more » ... the book. The outcome is puzzling. Of course, all the theorem provers provide us with "correct proofs", but none of them would even come close to a human readable text, comparable with the proof given in the book of Hardy and Wright. This is addressed by Scott when he writes in the preface of the book, [Sco06, p. viiif]: "We can also Research supported by the Portuguese Science Foundation, FCT, through the projects Hilbert's Legacy in the Philosophy of Mathematics, PTDC/FIL-FCI/ 109991/2009 and The Notion of Mathematical Proof, PTDC/MHC-FIL/5363/2012. 1 The same example was chosen by Lamport in his note [Lam95] which, along with its "update" [Lam12], provides a good motivation for our work. 2 Apparently, there took no major advance place in the last 8 years with respect to the question under consideration here, although, of course, many of the provers improved a lot.
dblp:conf/mkm/Kahle14 fatcat:ulqf4nhpavf7hav6awlz7jdqje

Towards a Proof-Theoretic Semantics of Equalities [chapter]

Reinhard Kahle
2015 Advances in Proof-Theoretic Semantics  
We have a fresh look on Frege's mode of presentation, taking into account proofs of equalities as a key concept. Revisiting the classical example of Morning star and Evening star the account leads to a proposal for a proof-theoretic semantics of equalities.
doi:10.1007/978-3-319-22686-6_9 fatcat:zt54fdleyna5hldc3sjhki7bou

Alkoholkonsum unter Berliner Schülern

Heike Heß, Matthias Richter, Reinhard Dortschy, Heidrun Kahl, Konrad Tietze, Hans Hüther, G. Röseler
1998 Bundesgesundheitsblatt  
Zusammenfassung Der Alkoholkonsum Berliner Schª der siebten bis zehnten Klassen, untersucht an Haupt-und Gymnasialschª hat sich in den vergangenen zehn Jahren ver~indert: Der Anteil Abstinenter ist insgesamt gestiegen, die H~iufigkeit und Menge des getrunkenen Alkohols sind leicht zurª gen. Doch ist der Gesamtverbrauch an Alkohol unter den von uns untersuchten Schª der siebten bis zehnten Klassen um 32 % gestiegen. Dies ist vor allem ron den Vieltrinkern verursacht; es gibt einen Anteil von
more » ... 6 % der Schª die t~iglich trinken und damit in Gefahr sind, eine Alkoholikerkarriere zu beginnen. Die wichtigsten Korrelate des Trinkens sind der eigene Spaf~ am Trinken und ein Fre¨ der am Trinken Gefallen findet. Schª in Ostberlin sind st~irker gef~ihrdet als solche in Westberlin, M~idchen weniger als Jungen. Die Trinkmenge steigt mit dem Altersgang rasch an. Summary Alcohol consumption among Berlin school children -Results of the Berlin study >~Health of children~< (GIK II) 1994/95
doi:10.1007/bf03044413 fatcat:vn6toirym5c7xgk42r2j3deywu

Proof-theoretic notions for software maintenance

Reinhard Kahle
2001 Electronical Notes in Theoretical Computer Science  
Kahle The first condition is needed to avoid pathological cases. In fact, (here) we are not interested in necessity for unprovable formulae.  ... 
doi:10.1016/s1571-0661(05)80048-9 fatcat:xue7emzwobejvavmyxvkzwsnoa

Universes in explicit mathematics

Gerhard Jäger, Reinhard Kahle, Thomas Studer
2001 Annals of Pure and Applied Logic  
In Kahle [18] , universes are studied for Frege structures, i.e. truth theories corresponding to explicit mathematics.  ...  Transfinitely iterated fixed point theories are introduced and analyzed in Jäger, Kahle, Setzer and Strahm [14] . Theorem 12 1.  ... 
doi:10.1016/s0168-0072(00)00057-9 fatcat:7alxn3zpknapnp4ok4hnmzu3bu

A Theory of Explicit Mathematics Equivalent to ID 1 [chapter]

Reinhard Kahle, Thomas Studer
2000 Lecture Notes in Computer Science  
We show that the addition of name induction to the theory EETJ + (L EM -I N ) of explicit elementary types with join yields a theory proof-theoretically equivalent to ID1.
doi:10.1007/3-540-44622-2_24 fatcat:qkhnepvdqjek5oxtcnlv76dqli

Two function algebras defining functions inNCkboolean circuits

Guillaume Bonfante, Reinhard Kahle, Jean-Yves Marion, Isabel Oitavem
2016 Information and Computation  
His ideas have been reworked by Guillaume Bonfante, Reinhard Kahle, Jean-Yves Marion and Isabel Oitavem in [BKMO08] where mutual in place recursion (mip) is introduced.  ... 
doi:10.1016/j.ic.2015.12.009 fatcat:apylzxh4nrcsxe63ccsllkhtna
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