This work investigates a 3-valued semantics, where the third value has the intention ”unimportant” or ”insignificant”. If ”true” and ”unimportant” are the distinguished values, then we can define a tautology in this logics also as follows: a formula is a tautology iff every subformula, which arises by elimination of propositional variables, is a classical tautology. So our system is stable against loosing or missing information and therefore it is called stable.
There are many connections to other non-classical logics. So we can see these truth-tables as counterpart to Bočvar's one. Furthermore they are related to those of RM3, the strongest logic in the family of relevance logics. And finally we can derive them from the interpretation of the multiplicative connectives in the phase semantics.
The classical sequent calculus missing the weakening rule on the right side is sound and complete w.r.t. this semantics, where ”true” and ”unimportant” are distinguished; the calculus missing the weakening rule on the left side is sound and complete w.r.t. this semantics, where ”true” is the only distinguished value.
As in classical propositional logic, where boolean algebra is a counterpart to the two-valued semantics, we define an algebra as a counterpart to this three-valued one. A subclass of the algebra can be interpreted as an algebra of pairs of sets, which gives a very graphic representation of our three-valued connectives.
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- [AB]A.R. Anderson and N.D. Belnap, Entailment Vol.1, Princeton University Press, Princeton, New Jersey, 1975.Google Scholar
- [Gi]J-Y. Girard, Linear Logic, Theoretical Computer Science, vol.50(1987), pp. 1–101.Google Scholar
- [Kl]S.C. Kleene, Introduction of Metamathematics, D. Van Nostrand Company, Princeton, 1952.Google Scholar
- [Ta]G. Takeuti, Proof Theory, North-Holland, Amsterdam, 1975.Google Scholar