Abstract: This chapter discusses the abstract quantification theory. The logical frameworks that arise in the concrete contexts of propositional logic and first order logic possess a certain property, viz. that no formula possesses an infinite descending chain of proper subformulas. The property of well-foundedness is exploited in some completeness proofs, but not in others. As a result, different proofs of the same completeness theorem generalize in the abstract setting to distinct completeness theorems. Many theorems of intuitionistic logic (such as the completeness theorem) do not depend on the elements singled out as special. All theorems about structures also apply to other subsystems of classical logic by simply changing the class of special elements. Kripke's form of the completeness theorem goes through intact for semantically normal regular structures, that is, every consistent set is realizable (and hence every valid element is provable).
Publication Year: 1970
Publication Date: 1970-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 11
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