We define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200004990

Proof Theory

An untyped higher order logic with Y combinator

Generalisation is currently a major theorem-proving problem. This paper proposes a new method of generalisation, involving the use of explanation-based generalisation within a new domain, which may succeed when other methods fail. The method has been implemented for simple arithmetical examples.

/pdf/a-new-application-for-explanation-based-generalisation-5b4r3525kn.pdf

A New Application for Explanation-Based Generalisation within Automated Deduction

It is shown that LJ with a generalized Dummett rule, LJ with a generalized Peirce rule and LJ with a specialized Peirce rule have the cut-elimination property. It is also shown that the third system has a weak subformula property and Craig’s interpolation property. The systems presented are versions of single-succedent sequent systems for classical logic. The cut-elimination results for the singlesuccedent systems can also be extended to modal logics.

/pdf/cut-free-single-succedent-systems-revisited-avdfwcq7r9.pdf

Cut-free single-succedent systems revisited

Formalizations of certain intermediate logics Part I

In this article we argue in favour of Natural Deduction Systems as a basis for formal proof explanations. We illustrate our choice presenting a Natural Deduction for ALCQI and use it to help explain UML reasoning.

/pdf/providing-a-proof-theoretical-basis-for-explanation-a-case-4m3rr4bxo0.pdf

Proof Theory

Citations

An untyped higher order logic with Y combinator

A New Application for Explanation-Based Generalisation within Automated Deduction

Cut-free single-succedent systems revisited

Formalizations of certain intermediate logics Part I

Providing a Proof-Theoretical Basis for Explanation: A Case Study on UML and ALCQI Reasoning

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