## A general characterization of the variable-sharing property by means of logical matrices.(English)Zbl 1254.03041

The authors consider: the standard (weak) variable sharing property; the strong variable sharing property: if $$A \to B$$ is provable, some variable occurs as an antecedent part (ap) (i.e. negatively) or as a consequent part (cp) (i.e. positively) in both $$A$$ and $$B$$; the no loose pieces property (known also as the 2-property): if $$A$$ is provable and contains no conjunction as an ap or disjunction as a cp, then every variable occurs once as an ap and once as a cp. All subsystems of the relevance logic R have these properties, however there are, as the authors show, other systems that have one or more of these properties. They define matrices, which, if they verify the logic, determine which of these properties hold.

### MSC:

 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)

### Keywords:

variable sharing; logical matrices; relevant logics
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### References:

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