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Comparison of Two Theorem Provers: Isabelle/HOL and Coq [article]

Artem Yushkovskiy
2018 arXiv   pre-print
The need for formal definition of the very basis of mathematics arose in the last century.  ...  For this reason, it firstly gives a brief introduction to the bases of formal systems and automated deduction theory, their main problems and challenges, and then provides detailed comparison of most notable  ...  Stavros Tripakis for letting me dive into the exciting world of Logic, for providing feedback on my paper at all stages of the work, for answering all my countless questions and supporting me.  ... 
arXiv:1808.09701v2 fatcat:ffesxt4esffgjottsxmwjykjri

Subject Index

2006 Journal of Applied Logic  
; Supporting the formal veri- fication of mathematical texts, 592 Proof systems A general tableau method for propositional inter- val temporal logics: Theory and implementa- tion, 305 Proofs as programs  ...  TPS: A hybrid automatic-interactive system for developing proofs, 367 Formal digital libraries Innovations in computational type theory using Nuprl, 428 Formalization of mathematics Is ZF a hack  ... 
doi:10.1016/s1570-8683(06)00055-3 fatcat:yeph4llopfdgjoimxiakdzqqxm

Formalizing a Paraconsistent Logic in the Isabelle Proof Assistant [chapter]

Jørgen Villadsen, Anders Schlichtkrull
2017 Lecture Notes in Computer Science  
We show how to use the proof assistant Isabelle to formally prove theorems in the logic as well as meta-theorems about the logic.  ...  In particular, we formalize a meta-theorem that allows us to reduce the infinite number of truth values to a finite number of truth values, for a given formula, and we use this result in a formalization  ...  Also thanks to Hendrik Decker and the anonymous reviewers for many constructive comments.  ... 
doi:10.1007/978-3-662-55947-5_5 fatcat:y2yhllvytrh7vn6rrdzzwph7xy

A view of automated proof checking and proving

Christian Lengauer
1988 Annals of Operations Research  
The main trade-off is between autonomy and flexibility in conducting proofs. Examples of the use of techniques and existing systems are given, but not attempt of an exhaustive overview is made.  ...  The goal is to provide the reader with an idea of what to look for when selecting an approach for his/her application.  ...  , or (3) submitting lemmas, i.e., propositions for proof by the system.  ... 
doi:10.1007/bf02283736 fatcat:gc5ifhb5vje4df3tq33jp3o3v4

From Logicism to Metatheory [chapter]

Patricia Blanchette
2013 The Palgrave Centenary Companion to Principia Mathematica  
. ii The Principia system, with its explicit axiomatic approach to the fundamental principles of logic, was just what was needed in the early years of the 20 th century to make possible the precise formulation  ...  and treatment of meta-logical questions.  ...  for formal systems of logic.  ... 
doi:10.1057/9781137344632_4 fatcat:slpm6riocjfpxl6gsiphi2bmrm

Constructive Formalization of Hybrid Logic with Eventualities [chapter]

Christian Doczkal, Gert Smolka
2011 Lecture Notes in Computer Science  
This paper reports on the formalization of classical hybrid logic with eventualities in the constructive type theory of the proof assistant Coq.  ...  We present our work in three steps: propositional logic, modal logic, and finally hybrid logic.  ...  Proof "⇒" Let M, w ⊨ t and S be some maximal Hintikka system such that H w ∈ S for all w ∈ |M| (Proposition 4.4).  ... 
doi:10.1007/978-3-642-25379-9_3 fatcat:mm7w4itpcfcjvb75vesju6qkzq

Is Mathematical Logic Really Necessary in Teaching Mathematical Proofs?

Michael Aristidou
2019 Athens Journal of Education  
Finally, we argue that even though mathematical logic is central in mathematics, its formal methods are not really necessary in doing and teaching mathematical proofs and the role of those formalities  ...  As it is already observed by mathematicians and educators, there is a discrepancy between the formal techniques of mathematical logic and the informal techniques of mathematics in regards to proof.  ...  for his comments on the book Mathematics B' Lyceum, and Babis Stergiou and Christos Lefkonitziatis for communicating to me some of the curriculum historicals related to the paper.  ... 
doi:10.30958/aje.7-1-5 fatcat:rtox7vhbdjcelc4s7u5jovge5a

Page 22 of Mathematical Reviews Vol. 58, Issue 1 [page]

1979 Mathematical Reviews  
Caicedo, Xavier [Caicedo Ferrer, Xavier] 58 #129 A formal system for the non-theorems of the propositional calculus. Notre Dame J. Formal Logic 19 (1978), no. 1, 147-151.  ...  The undecidability of first-order logic implies the nonexistence of an effective formal system for proving its nontheorems, but it Logic and Foundations 22 would be interesting to extend the author’s system  ... 

Novel Didactic Proof Assistant for First-Order Logic Natural Deduction [chapter]

Jorge Pais, Álvaro Tasistro
2014 Lecture Notes in Computer Science  
We present a proof assistant designed to help learning formal proof, particularly in the system of Natural Deduction for First-Order Logic.  ...  It possesses a graphical interface presenting proofs as trees and handles multiple simultaneous derivations that can be dragged and dropped into one another.  ...  First, we shall take First-Order Logic as the formal language for expressing propositions.  ... 
doi:10.1007/978-3-319-07482-5_42 fatcat:tloqap55g5b23dm44ysd475bli

Uniform provability realization of intuitionistic logic, modality and λ-terms

Sergei N. Artemov
1999 Electronical Notes in Theoretical Computer Science  
This question turned out to be a part of the more general problem of the intended realization for G odel's modal logic of provability S4 which w as open since 1933.  ...  The problem of nding an adequate formalization of the provability s e m a ntics and establishing the completeness of the intuitionistic logic Int was rst raised by G odel in 1933.  ...  Heyting suggested the axiom system Int for intuitionistic logic ( 8] ) 3 .  ... 
doi:10.1016/s1571-0661(04)00100-8 fatcat:ms423mkdgjh2lgj5oume2ouv4a

N.G. de Bruijn's contribution to the formalization of mathematics

Herman Geuvers, Rob Nederpelt
2013 Indagationes mathematicae  
In [McCarthy 1962] , he makes another clear case for using a computer for checking proofs, especially for the verification of programs: It is part of the definition of formal system that proofs be machine  ...  In [McCarthy 1960] , he explicitly states that Lisp was (among other things) developed for • Writing a program to check proofs in a class of formal logical systems. • Writing programs to realize various  ... 
doi:10.1016/j.indag.2013.09.003 fatcat:tgs7v5i6t5gxjl3s5c472obqje

Formalizing a Seligman-Style Tableau System for Hybrid Logic [chapter]

Asta Halkjær From, Patrick Blackburn, Jørgen Villadsen
2020 Lecture Notes in Computer Science  
Hybrid logic is modal logic enriched with names for worlds. We formalize soundness and completeness proofs for a Seligman-style tableau system for hybrid logic in the proof assistant Isabelle/HOL.  ...  Moreover, the completeness proof we formalize is synthetic which suggests we can extend this work to prove a wider range of results about hybrid logic.  ...  The present work appears to be the first proof system for hybrid logic with a formalized completeness proof.  ... 
doi:10.1007/978-3-030-51074-9_27 fatcat:klxbbkjmwzhh7kfxm365meu4ai

Isabelle/HOL as a Meta-Language for Teaching Logic

Asta Halkjær From, Jørgen Villadsen, Patrick Blackburn
2020 Electronic Proceedings in Theoretical Computer Science  
Proof assistants are important tools for teaching logic. We support this claim by discussing three formalizations in Isabelle/HOL used in a recent course on automated reasoning.  ...  The first is a formalization of System W (a system of classical propositional logic with only two primitive symbols), the second is the Natural Deduction Assistant (NaDeA), and the third is a one-sided  ...  Acknowledgements We thank Alexander Birch Jensen and Anders Schlichtkrull for discussions.  ... 
doi:10.4204/eptcs.328.2 fatcat:2o7vflmdlfgelehpeeequl6v5i

A formal logical framework for Cadiag-2

Agata Ciabattoni, Thomas Vetterlein, Klaus-Peter Adlassnig
2009 Studies in Health Technology and Informatics  
Cadiag-2-where "Cadiag" stands for "computer-assisted diagnosis"- is an expert system based on fuzzy logic assisting in the differential diagnosis in internal medicine.  ...  The aim is to have a formal logical calculus at hand which corresponds to the mode of operation of Cadiag-2 and which is among others needed to perform consistency checking of Cadiag-2's medical knowledge  ...  Ciabattoni et al. / A Formal Logical Framework for Cadiag-2 650 atomic propositions by means of the connectives , , and ~.  ... 
pmid:19745391 fatcat:p4baxtkrubfi7mpholdjh3x5fa

Formalizing the Curry-Howard Correspondence [article]

Juan Ferrer Meleiro, Hugo Luiz Mariano
2019 arXiv   pre-print
We restate the tradicional correspondence between typed λ-calculi and propositional logics inside this formalism, and indicate possible directions in which it could foster new and more structured generalizations  ...  In the current work, we introduce the formalism of p-institutions that could unify previous aproaches.  ...  From a logical point of view, there are multiple formalisms for such systems, but we'll use the one in which proofs look like trees $ A prop A $ A true $ A prop $ B prop A $ B prop A, B $ A true A $ B  ... 
arXiv:1912.10961v1 fatcat:fgimqmoudbdexjfkqzsesf4q7q
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