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Refining the arithmetical hierarchy of classical principles [article]

Makoto Fujiwara, Taishi Kurahashi
2022 arXiv   pre-print
We mainly investigate some restricted versions of the law of excluded middle, de Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.  ...  We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic.  ...  The first author was supported by JSPS KAKENHI Grant Numbers JP19J01239 and JP20K14354, and the second author by JP19K14586.  ... 
arXiv:2010.11527v2 fatcat:wzkixuijp5ac7lcpdfc6fyici4

Page 2824 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
and Brouwer’s point of view, and finally quotes Novikov as representing the approach of today’s mathematicians: it is not necessary to assume the logical principles (excluded middle, noncontradiction)  ...  Chapters 6 and 7 provide a discussion of the fragments of arithmetics and the induction schemes based on subrecursive hierarchies.  ... 

Page 5960 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
Cantor’s diagonal argument is, in his view, an application of the law of excluded middle. In many of his proofs, the author uses classical logic, even where this could have been avoided.  ...  The author does not mention DC, and is not very familiar with intuitionistic mathematics. Without explanation, he calls the law of excluded middle “a considerable weakening of the axiom of choice”.  ... 

Mathematics based on incremental learning—Excluded middle and inductive inference

Susumu Hayashi
2006 Theoretical Computer Science  
Mathematical or logical concepts seem to be one of the main research targets of learning theory and its applications.  ...  It suggests that logic and learning theory are related in a still unknown but deep new way. 1 Mathematics based on Learning?  ...  which the law of excluded middle and its equivalents are forbidden.  ... 
doi:10.1016/j.tcs.2005.10.019 fatcat:bjcpur2pxzeihdscoer2fz6ty4

Page 3220 of Mathematical Reviews Vol. 58, Issue 5 [page]

1979 Mathematical Reviews  
Using different terminologies for classical and constructive logi- cal operations, the author treats in the second chapter the law of the excluded middle, double negation, models for the enumeration of  ...  (errata insert) Author’s summary: “We construct a sequential variant of an arithmetical system of R. M.  ... 

The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization

Franz-Benjamin Mocnik
2018 Scientific Reports  
The following investigates the coexistence of local optimization, caused by the principle of connected neigbourhoods, and global optimization, here exemplified at hierarchies.  ...  The local structure of space and global optimization can both be found in transport, brain, and communication networks, which suggests the polynomial volume law, often in combination with hierarchies or  ...  Frank for critical discussions of the Mocnik model and Christian Freksa for his support, as well as Bernhard Höfle, Barnaby Walters, and René Westerholt for critical proofreading.  ... 
doi:10.1038/s41598-018-29131-0 pmid:30054491 pmcid:PMC6063948 fatcat:udtkuwl3bvc5vbp2yiifexpgkm

Notes on the First Chapter of The Continuum: Intension, Extension, and Arithmetism

Julien Bernard
2009 Philosophia Scientiæ  
This basic category is a structure made up of primitive objects and relations. The intuitive knowledge we have of those entities give foundations to the Excluded-Middle Principle.  ...  First, the constructivism of Brouwer calls the Excluded Middle Principle (and so classical logic) into question, whereas Hermann Weyl wants to give consistency to mathematics in maintaining the Excluded  ... 
doi:10.4000/philosophiascientiae.308 fatcat:72qmwl3hxfbbtprupf5qez6qb4

Towards Limit Computable Mathematics [chapter]

Susumu Hayashi, Masahiro Nakata
2002 Lecture Notes in Computer Science  
The notion of Limit-Computable Mathematics (LCM) will be introduced. LCM is a fragment of classical mathematics in which the law of excluded middle is restricted to 1 0 2 -formulas.  ...  We can give an accountable computational interpretation to the proofs of LCM.  ...  The proof is carried out by an application of the law of the excluded middle (LEM) that a zero exists in the progression or not.  ... 
doi:10.1007/3-540-45842-5_9 fatcat:ajugnuaa5fg5bnx7kfxq7kui2e

TRUTH, LOGICAL VALIDITY AND DETERMINATENESS: A COMMENTARY ON FIELD'S SAVING TRUTH FROM PARADOX

P. D. WELCH
2011 The Review of Symbolic Logic  
We try to explicate his notion of determinate truth by clarifying the path dependent hierarchies of his determinateness operator.  ...  We consider notions of truth and logical validity defined in various recent constructions of Hartry Field.  ...  when we are in a position to assert the law of excluded middle for the defining relations.)  ... 
doi:10.1017/s1755020311000049 fatcat:ojbuxauksreidhmjrjsk3mbto4

INTERRELATION BETWEEN WEAK FRAGMENTS OF DOUBLE NEGATION SHIFT AND RELATED PRINCIPLES

MAKOTO FUJIWARA, ULRICH KOHLENBACH
2018 Journal of Symbolic Logic (JSL)  
AbstractWe investigate two weak fragments of the double negation shift schema, which are motivated, respectively, from Spector's consistency proof of ACA0 and from the negative translation of RCA0, as  ...  well as double negated variants of logical principles.  ...  The second author has been partially supported by the German Science Foundation (DFG Project KO 1737/5-2).  ... 
doi:10.1017/jsl.2017.63 fatcat:p42tmohzvvgmrnxbbboubfcjhq

Page 1770 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews  
It starts with Kol- mogorov’s 1925 paper, “On the law of excluded middle”, and goes beyond 1969 only to describe work of A. A. Markov and N. A.  ...  The authors concentrate on two main areas: hierarchies of primitive recursive functions and hier- archies of provably total functions (in Peano arithmetic).  ... 

Constructive Ackermann's interpretation [article]

Hanul Jeon
2020 arXiv   pre-print
We also examine bi-interpretability between subtheories of finitary 𝖢𝖹𝖥 and Heyting arithmetic based on the modification of Fleischmann's hierarchy of formulas, and the set of hereditarily finite sets  ...  The main goal of this paper is to formulate a constructive analogue of Ackermann's observation about finite set theory and arithmetic.  ...  Acknowledgement The author would like to thank the author's advisor Otto van Koert for improving the grammatical and style of this paper.  ... 
arXiv:2010.04270v2 fatcat:fjpsakez6rgo5fv2glrvdheb5m

Kronecker in Contemporary Mathematics. General Arithmetic as a Foundational Programme

Yvon Gauthier
2013 Reports on Mathematical Logic  
, following Kronecker; for Hilbert, the objective was to preserve the laws of finite logic with the excluded middle principle in the transfinite (set-theoretic) realm of ideal elements (ideale Elemente  ...  The idea of the epsilon-calculus for the -symbol was to enable the extension of the simple laws of Aristotelian logic, excluded middle and universal instantiation with existential import, to the transfinite  ... 
dblp:journals/rml/Gauthier13 fatcat:vt6whunfanf55k4mb44x6g6r5u

Naïve validity

Julien Murzi, Lorenzo Rossi
2017 Synthese  
The aim of this paper is to respond to Field's objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi's principles: grounded validity.  ...  Beall and Murzi (J Philos 110(3):143-165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules (  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s11229-017-1541-6 pmid:35068595 pmcid:PMC8755702 fatcat:mxeit5fwizhehkxwz5m3k66ste

A Revenge-Immune Solution to the Semantic Paradoxes

Hartry Field
2003 Journal of Philosophical Logic  
The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True( A ) with A within  ...  We can in fact define a hierarchy of "defectiveness" predicates within the language; contrary to claims that any solution to the paradoxes just breeds further paradoxes ("revenge problems") involving defectiveness  ...  , then induction needs to be put as a rule Some non-laws: Certain laws that are valid for the classical '⊃' (and hence valid for '→' in the context of excluded middle) fail for '→' when excluded middle  ... 
doi:10.1023/a:1023027808400 fatcat:zjdt4u7nfza2dapa54vwnzkydi
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