The question as to whether the propositional logic of Heyting, which was a formalization of Brouwer's intuitionistic logic, is finitely many valued or not, was open for a while (the question was asked by Hahn). Kurt Gödel (1932) introduced an infinite decreasing chain of intermediate logics, which are known nowadays as Gödel logics, for showing that the intuitionistic logic is not finitely (man...

We suggest a new basic framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF . The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemente...

The last chapter presented the general concept of a logic. In this chapter we consider a particular logic | rst order predicate calculus, also known as simply rst order logic. First order logic is by far the most widely studied logic. It was rst formulated by Guttlob Frege in 1879. However, the fundamental semantic properties of rst order logic were not really understood until the work of Skole...

It is well known that the conjunction-negation fragment of the classical propositional calculus coincides with the conjunction-negation fragment of the Heyting propositional calculus (Godel, 1933; Kleene, 1952, $81). Since conjunction and negation form a sufficient basis for classical propositional logic, this reduces, in a certain sense, classical propositional logic to a fragment of Heyting p...

Deduktive Datenbanksysteme, die Programmiersprache Godel und die Evolution von Wissensbasen (Abschnitte 2 bis 4).

In an important sequence of papers, Halmos has developed the beautiful and significant theory of algebraic logic. The central mathematical object of this theory is the polyadic algebra, which incorporates in an axiomatic way the concepts of substitution, transformation, and quantification for the lower functional calculus. The beauty of the theory is revealed by the fact that the semantic compl...

One drawback of Hierarchical Task Network (HTN) planning is the difficulty of providing complete domain knowledge, i.e., a complete and correct set of HTN methods for every task. To provide a principled way to overcome this difficulty, we define a simple formalism that extends classical planning to include problem decomposition using methods, and a planning algorithm based on this formalism. In...

In this paper we are interested in using a first order theorem prover to prove theorems that are formulated in some higher order logic. To this end we present translations of higher order logics into first order logic with flat sorts and equality and give a sufficient criterion for the soundness of these translations. In addition translations are introduced that are sound and complete with resp...