Can classical reasoning be recovered through the combination of logics?
; Marcos, J.
Can classical reasoning be recovered through the combination of logics?, Proc International Joint Conf. on Automated Reasoning - IJCAR, Coimbra, Portugal, Vol. 0, pp. 0 - 0, July, 2016.
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We begin by discussing various forms of the so-called collapsing problem, a special case of failure of conservativeness characteristic of certain recipes for the
combination of logics, and that turns out to be very sensitive to the proof-theoretical presentation of the logics involved. Such problem, and its diagnosis, will be seen to bear connections with the question about the existence of a unique connective satisfying certain conditions. Focusing on the fibring of logics as our main combinational mechanism and on sublogics of classical logic as our Guinea pigs, we will show by way of motivational examples that while the fibring of two classical conjunctions obviously produces a full collapse, the fibring of the logic of classical conjunction with the logic of classical disjunction cannot be characterized by a finite-valued matrix, even if non-determinism is allowed at the semantic level. Other examples of logics without a non-deterministic finite-valued characterization include the fibring of the logic of classical implication with itself, and the fibring of the logic of classical implication with the logic of classical disjunction. Reinforcing the persisting challenge concerning the un-
derstanding of the semantics of our main combinational mechanism, while we will show that the fibring of the logic of classical negation with the logic of classical bottom does have a characteristic 3-valued non-deterministic matrix, we will also show that the fibring of the logic of classical disjunction with the logic of classical negation does not give rise to Classical Logic. As we will see, the latter situation is a particular instance of a much more general result: As we will show, Classical Logic cannot be obtained as the disjoint fibring of any two of its sublogics.