[FOM] question about relevance and variable-sharing

[Neil Tennant]

Thanks to Carl Hewitt, Arnon Avron, Ed Mares and Harry Deutsch for their helpful responses. 

I can add to their combined reading list the 1988 book by Stephen Read, in which he proves a variable-sharing result (for his own favored relevance system) that provides information about positive and negative occurrences of atomic sentences (in premises and conclusions of proofs). 


The latter feature can be found also in the relevance result proved in Ch.9 of my 1992 book Autologic (concerning the system IR, or Core Logic). My own result further partitions any non-empty set of premises into non-empty cells in such a way that each cell of the partition is suitably related to the conclusion, by 'same-parity' variable sharing. Within any cell, by contrast, member premises are related by 'opposite-parity' variable-sharing. With proofs of absurdity from a set X, X is the only cell in the partition. With proofs of theorems, i.e. from the empty set of premises, the conclusion must contain at least one variable occuring both positively and negatively within it.


I am currently trying to find a 'best possible' variable-sharing result for relevance logic, that incorporates, as far as is possible, all the various features that have been established for one system or another. Whether Ross Brady's further constraint about shared levels of occurrences (for contraction-free systems) can also be incorporated looks doubtful, since it depends so crucially on the system being contraction-free.


Best,
Neil Tennant