Abstract
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 ω-th slice, the condition for them to have the simple substitution property is not yet known.
We abbreviate the simple substitution property asSSP.
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Sasaki, K. The simple substitution property of the intermediate propositional logics on finite slices. Stud Logica 52, 41–62 (1993). https://doi.org/10.1007/BF01053063
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DOI: https://doi.org/10.1007/BF01053063