## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 57, Number 1 (2016), 105-125.

### Rule-Irredundancy and the Sequent Calculus for Core Logic

#### Abstract

We explore the consequences, for logical system-building, of taking seriously (i) the aim of having *irredundant* rules of inference, and (ii) a preference for proofs of *stronger* results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT.

REFLEXIVITY survives in the minimally necessary form $\phi :\phi $. Proofs have to get started.

CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is redundant, and should not be a rule *of* the system.

CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT *or* THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic.

Given any intuitionistic Gentzen proof of $\Delta :\phi $, one can determine from it a Core proof of some subsequent of $\Delta :\phi $. Given any classical Gentzen proof of $\Delta :\phi $, one can determine from it a classical Core proof of some subsequent of $\Delta :\phi $. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is $\phi :\phi $.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 57, Number 1 (2016), 105-125.

**Dates**

Received: 16 July 2013

Accepted: 4 October 2013

First available in Project Euclid: 24 November 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1448323485

**Digital Object Identifier**

doi:10.1215/00294527-3346463

**Mathematical Reviews number (MathSciNet)**

MR3447728

**Zentralblatt MATH identifier**

06550123

**Subjects**

Primary: 03F03: Proof theory, general 03F05: Cut-elimination and normal-form theorems

**Keywords**

Structural rules logical rules Cut Thinning Reflexivity Cut-elimination Thinning-elimination logical strength

#### Citation

Tennant, Neil. Rule-Irredundancy and the Sequent Calculus for Core Logic. Notre Dame J. Formal Logic 57 (2016), no. 1, 105--125. doi:10.1215/00294527-3346463. https://projecteuclid.org/euclid.ndjfl/1448323485