scholarly journals Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

Synthese ◽  
2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
General Introduction ◽  
Main Method ◽  
Philosophical Importance ◽  
Subformula Property

AbstractThis paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformula property.

scholarly journals Normalisation and subformula property for a system of classical logic with Tarski’s rule

2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Classical Logic ◽  
Similar Form ◽  
General Introduction ◽  
Intuitionist Logic ◽  
Introduction Rule ◽  
Definition Of ◽  
Subformula Property

AbstractThis paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic.

scholarly journals A Prolog-based Proof Tool for Type-Theory TA-lambda and Implicational Intuitionistic-Logic

10.29007/8ttk ◽  
2018 ◽  
Author(s):  
L. Yohanes Stefanus ◽  
Ario Santoso
Keyword(s):  
Normal Form ◽  
Computer Science ◽  
Intuitionistic Logic ◽  
Type Theory ◽  
General Type ◽  
Search Algorithm ◽  
Principal Type ◽  
Web Based ◽  
Type Algorithm ◽  
Important Variant

Studies on type theory have brought numerous important contributions to computer science. In this paper we present a GUI-based proof tool that provides assistance in constructing deductions in type theory and validating implicational intuitionistic logic formulas. As such, this proof tool is a testbed for learning type theory and implicational intuitionistic logic. This proof tool focuses on an important variant of type theory named TA<sub>λ</sub>, especially on its two core algorithms: the principal-type algorithm and the type inhabitant search algorithm. The former algorithm finds a most general type assignable to a given λ-term, while the latter finds inhabitants (closed λ-terms in β-normal form) to which a given type can be assigned. By the Curry-Howard correspondence, the latter algorithm provides provability for implicational formulas in intuitionistic logic. We elaborate on how to implement those two algorithms declaratively in Prolog and the overall GUI-based program architecture. In our implementation, we make some modification to improve the performance of the principal-type algorithm. We have also built a web-based version of the proof tool called λ-Guru.

Proofs, Snakes and Ladders

Dialogue ◽  
1974 ◽  
Vol 13 (4) ◽  
pp. 723-731 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Tree Structure ◽  
Deduction System ◽  
Syntactical Structure ◽  
Form Theorem ◽  
Structural Identity ◽  
Natural Deduction System ◽  
Normal Form Theorem

Anyone who has worked at proving theorems of intuitionistic logic in a natural deduction system must have been struck by the way in which many logical theorems “prove themselves.” That is, proofs of many formulas can be read off from the syntactical structure of the formulas themselves. This observation suggests that perhaps a strong structural identity may underly this relation between formulas and their proofs. A formula can be considered as a tree structure composed of its subformulas (Frege 1879) and by the normal form theorem (Gentzen 1934) every formula has a normalized proof consisting of its subformulas. Might we not identify an intuitionistic theorem with (one of) its proof(s) in normal form?

ANALYTIC CUT AND INTERPOLATION FOR BI-INTUITIONISTIC LOGIC

2016 ◽  
Vol 10 (2) ◽  
pp. 259-283 ◽  
Author(s):  
TOMASZ KOWALSKI ◽  
HIROAKIRA ONO
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Property ◽  
Separation Property ◽  
Variable Separation ◽  
Craig Interpolation ◽  
Craig Interpolation Property ◽  
Sequent Systems ◽  
Subformula Property

AbstractWe prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Halldén completeness follows.

Intuitionistic logic freed of all metarules

2007 ◽  
Vol 72 (4) ◽  
pp. 1204-1218 ◽  
Author(s):  
Giovanna Corsi ◽  
Gabriele Tassi
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Proof Search ◽  
Subformula Property

AbstractIn this paper we present two calculi for intuitionistic logic. The first one. IG, is characterized by the fact that every proof-search terminates and termination is reached without jeopardizing the subformula property. As to the second one, SIC, proof-search terminates, the subformula property is preserved and moreover proof-search is performed without any recourse to metarules, in particular there is no need to back-track. As a consequence, proof-search in the calculus SIC is accomplished by a single tree as in classical logic.

A MODAL TRANSLATION FOR DUAL-INTUITIONISTIC LOGIC

2016 ◽  
Vol 9 (2) ◽  
pp. 251-265 ◽  
Author(s):  
YAROSLAV SHRAMKO
Keyword(s):  
Intuitionistic Logic ◽  
Entailment Relations ◽  
Philosophical Importance

AbstractWe construct four binary consequence systems axiomatizing entailment relations between formulas of classical, intuitionistic, dual-intuitionistic and modal (S4) logics, respectively. It is shown that the intuitionistic consequence system is embeddable in the modal (S4) one by the usual modal translation prefixing □ to every subformula of the translated formula. An analogous modal translation of dual-intuitionistic formulas then consists of prefixing ◊ to every subformula of the translated formula. The philosophical importance of this result is briefly discussed.

Bronchogenic carcinoma. General introduction

JAMA ◽  
1966 ◽  
Vol 195 (8) ◽  
pp. 653-654 ◽  
Author(s):  
P. Rubin
Keyword(s):  
Bronchogenic Carcinoma ◽  
General Introduction

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Sign in / Sign up

Export Citation Format

Share Document