Some Remarks on the Collapse Issues of Merging Different Logics ― with Classical, Intuitionistic, Finitist and Linear Logics

Abstract

We first analyze MacFarlane's Natural Deduction style presentation of Williamson's proof of collapsing of the classical negation and intuitionistic negation in a merged language; we show how the collapsing proof process is blocked by making the hidden (classical and intuitionistic) contexts explicit. Then, we consider a safe (conservative) merge between the classical rule system and a finitist (implication) rule, system and claim that the approach of safely merging the rules does not necessarily help the mutual understanding of the difference between different logical standpoints in general. MacFarlane (2020), in Sec.6, suggests the mutual embedding approach as an alternative approach for mutual understanding of the difference between the classical logicians and the intuitionistic logicians. At the Appendix Section, we argue on this. Then, we show that the situation is different in the linear logic setting; the embedding approach provides a new positive insight. Importance of semantic consideration is claimed in this paper.