The simple substitution property of the intermediate propositional logics on finite slices

The simple substitution property provides a systematic and easy method for proving a theorem by an axiomatic way. The notion of the property was introduced in Hosoi 4] but without a definite name and he showed three examples of the axioms with the property. Later, the property was given it's name as above in Sasaki 7].Our main result here is that the necessary and sufficient condition for a logicL on a finite slice to have the simple substitution property is thatL is finite. Here the necessity part is essentially new, for the sufficiency part has been proved in Hosoi and Sasaki 5]. Also the proof of sufficiency part is improved here.For logics on the -th slice, the condition for them to have the simple substitution property is not yet known.We abbreviate the simple substitution property asSSP.