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Implication and analysis in classical frege structures

Robert C. Flagg, John Myhill
1987 Annals of Pure and Applied Logic  
M (5) Implication and analysis in classical Frege structures 37 and M e {x I'"x } (6) are interdeducible.  ...  (11) Implication and analysis in classical Frege structures 39 If now we define I = SKK we get Ix = SgKx = = x. So I is the identity function on everything.  ...  Let us now therefore develop the theory of (Dedekind) real numbers and a bit of elementary analysis using our hierarchy of implications.  ... 
doi:10.1016/0168-0072(87)90040-6 fatcat:iwkfjwsjcfehnmndhkyewotb6y

Page 35 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews  
In this paper, we investigate classical substructural logics consisting of implication and negation. How- ever, our method is general enough to apply to rich systems beyond the fragment.  ...  It is well known that in simply typed lambda calculus, the notion of formulas-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevance logic  ... 

Page 3032 of Mathematical Reviews Vol. , Issue 89F [page]

1989 Mathematical Reviews  
89:0 3046 89f:03046 03E70 03B40 03F60 Flagg, Robert C. (1-OHS); Myhill, John (1-SUNYB) Implication and analysis in classical Frege structures. Ann. Pure Appl. Logic 34 (1987), no. 1, 33-85.  ...  In this paper the authors develop the proof theory and model theory of classical Frege structures, formalize the hierarchy of implications, prove the consistency of the system and outline the resulting  ... 

Deflationary Truth and the Problem of Relativistic Grounds of Emotivism

The essence of the problem is explained, which consists in the impossibility to make a logical conclusion in a situation of mixed contexts, when the antecedent has a value meaning, and the consequent is  ...  The paper describes how recognition of an emotive content by moral statements in the form of attitudes leads to the emergence of the Frege-Geach problem.  ...  The Frege-Geach problem arises when we consider the implication and logical conclusion assuming the non-cognitive thesis that moral statements are not statements, but attitudes.  ... 
doi:10.35940/ijitee.a9222.119119 fatcat:yintntg5jna53busdd2smjlg2y

Page 3539 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
Some approaches in each conception are philosophical and some are mathematical in character, their common assumption being that they reflect a previous intuition—in some cases—already at work in classical  ...  Most of this paper is an intricate and delicate alge- braic analysis of various forms of inverse compositionality.  ... 

Logic as Calculus and Logic as Language: Too Suggestive to be Truthful?

Jan von Plato
2021 Philosophia Scientiæ  
Abstract: The paper focuses on the inferential role of quantifiers in Frege, Peano and Russell.  ...  questions, as embodied in Jean van Heijenoort's well-known dictum about "logic as calculus and logic as language."  ...  C C Even implication elimination and some other rules with independent premisses, say A and B in the introduction of a conjunction A&B are thus displayed, which opens the possibility to analyse the structure  ... 
doi:10.4000/philosophiascientiae.2816 fatcat:uun7ug3ckzfntdjwwvf3lao7ne

Author index volume 34 (1987)

1987 Annals of Pure and Applied Logic  
degrees (1) EPSTEIN, R.L., see COOPER, S.B. (1) FLAGG, R.C. and FRIEDMAN, H., Maximality in modal logic (2) FLAGG, R.C. and MYHILL, J., Implication and analysis in classical Frege structures (1) FRIEDMAN  ...  ., Categoricity in hyperarithmetical degrees (1) BALDWIN, J.T. and HARRINGTON, L., Trivial pursuit: Remarks on the main gap (3) COOPER, S.B. and EPSTEIN, R.L., Complementing below recursively enumerable  ... 
doi:10.1016/0168-0072(87)90071-6 fatcat:sgh6etqckncjhhkjigjimlvemy

Frege on Consistency and Conceptual Analysis

P. A. Blanchette
2007 Philosophia Mathematica  
Analysis in Frege: a Brief Overview Conceptual analysis is crucial to all of Frege's logicist and proto-logicist work.  ...  sees it, Frege and Hilbert have no disagreement over the role of conceptual analysis in logical investigation, and hence their disagreement over consistency-proofs cannot be explained in the way just  ...  structure of the pre-analytic sentences in question.  ... 
doi:10.1093/philmat/nkm028 fatcat:u2rn6myg4jdijoe4bkdbakhojq

Page 619 of Mathematical Reviews Vol. , Issue 92b [page]

1992 Mathematical Reviews  
The axioms for a Frege structure can be proved in this theory.  ...  the truth-table conditions and standard quantifier rules without the grammatical conditions in the corresponding axioms for a Frege structure.  ... 

Page 4221 of Mathematical Reviews Vol. , Issue 89H [page]

1989 Mathematical Reviews  
Frege would only have re- jected model-theoretic characterizations of logical notions, such as logical validity, implication and equivalence.)  ...  Such a subsystem, Friedman’s WKLh, is discussed by the author, along with a list of important theorems of classical analysis and algebra that have been proved in WKL» (mainly by the author and/or D.  ... 

Proof Complexity of Non-classical Logics [chapter]

Olaf Beyersdorff, Oliver Kutz
2012 Lecture Notes in Computer Science  
Due their richer expressivity and numerous applications within computer science, also non-classical logics have been intensively studied from a proof complexity perspective in the last decade, and a number  ...  In these notes we give an introduction to this recent field of proof complexity of non-classical logics. We cover results from proof complexity of modal, intuitionistic, and non-monotonic logics.  ...  Acknowledgements We are very grateful to Rosalie Iemhoff and Emil Jeřábek for numerous helpful comments on an earlier version of these notes.  ... 
doi:10.1007/978-3-642-31485-8_1 fatcat:o2gh5kqzp5fpvb3fataonggkz4

Page 5563 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
It follows from this result that provability in the pure classical implication calculus is co-NP complete. E.  ...  The author proposes that some structures can be described in both “static” terms and in “dynamic” terms using a set of ideal con- structions.  ... 


2021 ARHE  
We will argue that despite the common belief, intensional aspects of mathematical notions can be, and in fact have been successfully described in mathematics.  ...  Our considerations result in defending a) the importance of a rule-based semantical approach and b) the position according to which non-reductive and somewhat circular explanations play an essential role  ...  According to his account, the main difference between an intuitionist and a classical logician is that while in classical logic the implication is understood truth-functionally, the intuitionistic implication  ... 
doi:10.19090/arhe.2020.34.61-83 fatcat:aao5d7jncna53pxipoblya5dfq

Rational hermeneutics and paraconsistency

Helen N. Shulga
2004 Logic and Logical Philosophy  
Frege in his paper from 1918 "The Thought: a Logical Inquiry" [Frege 1967 ], in fact, set an example of such an analysis when he envisages a vicious circle in the definition of truth.  ...  Meanwhile, the recent investigations discovered the presence of paraconsistent structure within the depth of classical logics. As it was show by J.-Y.  ... 
doi:10.12775/llp.1999.016 fatcat:dpju34uesbf4jl2avya3oyrjli

Understanding Gentzen and Frege Systems for QBF

Olaf Beyersdorff, Ján Pich
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
Here we provide a comprehensive analysis of the new extended Frege system from [10], denoted EF + ∀red, which is a natural extension of classical extended Frege EF.  ...  Finally, we show that unconditional lower bounds for EF + ∀red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold  ...  Acknowledgments This research was supported by grant no. 48138 from the John Templeton Foundation and EPSRC grant EP/L024233/1.  ... 
doi:10.1145/2933575.2933597 dblp:conf/lics/BeyersdorffP16 fatcat:drqb2sqeybhzfgbrlwbjaiosni
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