
Contents
-
-
-
-
-
-
-
-
-
-
-
-
-
Abstract Abstract
-
9.1 Introduction 9.1 Introduction
-
9.2 From Mathematical Axioms to Mathematical Rules 9.2 From Mathematical Axioms to Mathematical Rules
-
9.3 Derivations in Left and Right Rule Systems 9.3 Derivations in Left and Right Rule Systems
-
9.4 Geometric and Cogeometric Axioms and Rules 9.4 Geometric and Cogeometric Axioms and Rules
-
9.5 Duality of Dependent Types and Degenerate Cases 9.5 Duality of Dependent Types and Degenerate Cases
-
References References
-
-
-
-
-
-
9 THE DUALITY OF CLASSICAL AND CONSTRUCTIVE NOTIONS AND PROOFS
Get access-
Published:October 2005
Cite
Abstract
The method of converting mathematical axioms into rules of sequent calculus reveals a perfect duality between classical and constructive basic notions, such as equality and apartness, and between the respective rules. Derivations with the mathematical rules of a constructive theory are duals of corresponding classical derivations. The class of geometric theories is among those convertible into rules, and the duality defines a new class of ‘co-geometric’ theories. Examples of such theories are projective and affine geometry. The logical rules of classical sequent calculus are invertible. For quantifier-free theories, this has the effect that logical rules can be permuted to apply after the mathematical rules. For mathematical rules involving variable conditions, this separation of logic does not always hold because quantifier rules may fail to permute down. A sufficient condition for the permutability of mathematical rules is determined and applied to give an extension of Herbrand theorem from universal to geometric and co-geometric theories.
Sign in
Personal account
- Sign in with email/username & password
- Get email alerts
- Save searches
- Purchase content
- Activate your purchase/trial code
- Add your ORCID iD
Purchase
Our books are available by subscription or purchase to libraries and institutions.
Purchasing informationMonth: | Total Views: |
---|---|
October 2022 | 1 |
November 2022 | 1 |
March 2023 | 3 |
May 2024 | 3 |
June 2024 | 2 |
July 2024 | 2 |
Get help with access
Institutional access
Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:
IP based access
Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.
Sign in through your institution
Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institution’s website and Oxford Academic.
If your institution is not listed or you cannot sign in to your institution’s website, please contact your librarian or administrator.
Sign in with a library card
Enter your library card number to sign in. If you cannot sign in, please contact your librarian.
Society Members
Society member access to a journal is achieved in one of the following ways:
Sign in through society site
Many societies offer single sign-on between the society website and Oxford Academic. If you see ‘Sign in through society site’ in the sign in pane within a journal:
If you do not have a society account or have forgotten your username or password, please contact your society.
Sign in using a personal account
Some societies use Oxford Academic personal accounts to provide access to their members. See below.
Personal account
A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.
Some societies use Oxford Academic personal accounts to provide access to their members.
Viewing your signed in accounts
Click the account icon in the top right to:
Signed in but can't access content
Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.
Institutional account management
For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.