Journal of Symbolic Logic

SN and {CR} for free-style {${\bf LK}\sp {tq}$}: linear decorations and simulation of normalization

Jean-Baptiste Joinet, Harold Schellinx, and Lorenzo Tortora de Falco

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Abstract

The present report is a, somewhat lengthy, addendum to "A new deconstructive logic: linear logic," where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.

The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.

We give a detailed description of the simulation of tq-reductions by means of reductions of the interpre- tation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.

We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e., by only one specific combination, and structural rules.

Article information

Source
J. Symbolic Logic, Volume 67, Issue 1 (2002), 162-196.

Dates
First available in Project Euclid: 18 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1190150036

Digital Object Identifier
doi:10.2178/jsl/1190150036

Mathematical Reviews number (MathSciNet)
MR1889543

Zentralblatt MATH identifier
1015.03053

Subjects
Primary: 03F05: Cut-elimination and normal-form theorems
Secondary: 03B10: Classical first-order logic 03B35: Mechanization of proofs and logical operations [See also 68T15] 03F07: Structure of proofs 03F52: Linear logic and other substructural logics [See also 03B47]

Citation

Joinet, Jean-Baptiste; Schellinx, Harold; Tortora de Falco, Lorenzo. SN and {CR} for free-style {${\bf LK}\sp {tq}$}: linear decorations and simulation of normalization. J. Symbolic Logic 67 (2002), no. 1, 162--196. doi:10.2178/jsl/1190150036. https://projecteuclid.org/euclid.jsl/1190150036


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