# Semantics of Sequent Calculi with Basic Structural Rules: Fuzziness Versus Non-Multiplicativity

### EasyChair Preprint 4153

15 pages•Date: September 8, 2020### Abstract

The main general\/} result of the paper is that

basic\/} structural rules --- Enlargement, Permutation and Contraction ---

(as well as Sharings) [and Cuts] are derivable in

a \{multiplicative\} propositional two-side sequent calculus

iff there is a class of \{crisp\} (reflexive) [transitive distributive] fuzzy two-side

matrices such that any rule is derivable in the calculus iff

it is true in the class,

the ``\{\}''/``()[]''-optional case being

due to \cite{My-label}/\cite{My-fuzzy}.

Likewise, fyzzyfying the notion of signed matrix \cite{My-label},

we extend the main result obtained therein beyond

multiplicative calculi.

As an application, we prove that

the sequent calculus $\mbb{LK}_\mr{[S/C]}$

resulted from Gentzen's $LK$ \cite{Gen}

by adding the rules inverse to the logical ones

and retaining as structural ones merely basic ones

[and Sharing/Cut] is equivalent

(in the sense of \cite{DEAGLS}) to the bounded version of

Belnap's four-valued logic (cf. \cite{Bel})

[resp., the {\em logic of paradox\/} \cite{Priest}/

Kleene's three-valued logic \cite{Kleene}].

As a consequence of this equivalence,

appropriate generic results of \cite{DEAGLS}

concerning extensions of equivalent calculi

and the advanced auxiliary results on extensions of

the bounded versions of Kleene's three-valued logic

and the logic of paradox proved here

with using the generic algebraic tools elaborated in \cite{LP-ext},

we then prove that extensions of the Sharing/Cut-free version

$\mbb{LK}_\mr{C/S}$ of $LK$ form a three/four-element chain/,

consistent ones having same derivable sequents

that provides a new profound insight into Cut Elimination in $LK$

appearing to be just a consequence of the well-known regularity of

operations of Belnap's four-valued logic.

**Keyphrases**: Calculus, logic, matrix, sequent