Logic: a new beginning
Entailment, contrariety, and contradiction stand in a triangular relation. Given negation of the predicate and given the duality of all and some, two isomorphic triangles arise, together forming an improved notation for the traditional Square of Opposition. Logical operators are treated as (abstract) predicates, definable in terms of satisfaction conditions and shaping the logic in which they take part. Carnapian meaning postulates are discussed. Boolean algebra and corresponding standard set theory are shown to underlie standard propositional and predicate logic. The method of valuation‐space modelling is introduced as a means of providing succinct and complete representations of logical systems in such a way that their properties are open to immediate inspection. A survey is given of Russellian and generalized quantification, of internal negation, and De Morgan's laws.
Keywords: Boolean algebra, contradiction, contrariety, Conversions, De Morgan's Laws, duality, entailment, subcontrariety, internal vs. external negation, meaning postulates, quantifiers, satisfaction conditions, Square of Opposition, valuation‐space modelling
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