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Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
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On the Translation of HigherOrder Problems into FirstOrder Logic
 Proceedings of ECAI94
, 1994
"... . In most cases higherorder logic is based on the  calculus in order to avoid the infinite set of socalled comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the calculus, but translate higherorder expressions into firstor ..."
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. In most cases higherorder logic is based on the  calculus in order to avoid the infinite set of socalled comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the calculus, but translate higherorder expressions into firstorder expressions by standard translation techniques, we have to translate the infinite set of comprehension axioms, too. Of course, in general this is not practicable. Therefore such an approach requires some restrictions such as the choice of the necessary axioms by a human user or the restriction to certain problem classes. This paper will show how the infinite class of comprehension axioms can be represented by a finite subclass, so that an automatic translation of finite higherorder problems into finite firstorder problems is possible. This translation is sound and complete with respect to a Henkinstyle general model semantics. 1 Introduction Firstorder logic is a powerful tool for ...
INTENTIONAL PARADOXES AND AN INDUCTIVE THEORY OF PROPOSITIONAL QUANTIFICATION
"... Quantification over propositions is a necessary component of any theory of attitudes capable of providing a semantics of attitude ascriptions and a sophisticated system of reasoning about attitudes. There appear to be two general approaches to propositional quantification. One is developed within a ..."
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Quantification over propositions is a necessary component of any theory of attitudes capable of providing a semantics of attitude ascriptions and a sophisticated system of reasoning about attitudes. There appear to be two general approaches to propositional quantification. One is developed within a first order quantificational language, the other in the language of higher order logic. The first order theory is described in Asher & Kamp (1986), Asher (1988), Asher and Kamp (1989). This paper concentrates on propositional quantification in a higher order framework, the simple theory of types. I propose a method of resolving difficulties noticed by Prior and Thomason with propositional quantification. The method borrows from Kripke's (1975) defintition of truth and results in a partial logic, which I call the simple theory ofpartial types (SPT). SPT offers a tractable, complete logic (with respect to general models) that includes propositional quantification, accomodates a semantics of the attitudes that avoids logical omniscience, and allows for some selfreference. Consider the following examples in which there is apparent quantification over propositions. 1 (1.a) Everything Mary believes is true.
NonStandard Models of Arithmetic: a Philosophical and Historical perspective MSc Thesis (Afstudeerscriptie)
, 2010
"... 1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5 ..."
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1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5