Many-Valueness and Modality
DOI:
https://doi.org/10.12775/RF.2019.020Keywords
logic, extensionality, two-valuedness, logical valueAbstract
Modal propositional logic can be obtained either by extending non-modal propositional logic (this is the case of Lewis’ systems) or by using many-valued logic as the basic system. The second route was taken by Jan Łukasiewicz, who proved that modalities cannot be defined within two-valued logic. The principle of extensionality was a tacit Łukasiewicz’s assumption. If we compare Lewis’ modal systems with that of Łukasiewicz, we see that both solutions share most logical principles. Perhaps the most important difference concerns the formula ◊(A ^ ¬A), in words (A and ¬A) is possible. Łukasiewicz argued that this formula has the value 1/2, if A and ¬A have this value as well. I argue that Łukasiewicz’s argument is not correct.
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