Higher-Order Illative Combinatory Logic

@article{Czajka2013HigherOrderIC,
title={Higher-Order Illative Combinatory Logic},
author={Lukasz Czajka},
journal={The Journal of Symbolic Logic},
year={2013},
volume={78},
pages={837 - 872}
}
• Lukasz Czajka
• Published 16 February 2012
• Mathematics
• The Journal of Symbolic Logic
Abstract We show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

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References

SHOWING 1-10 OF 12 REFERENCES

Systems of Illative Combinatory Logic Complete for First-Order Propositional and Predicate Calculus

• Philosophy
J. Symb. Log.
• 1993
The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus and fulfills the program of Church, Church, Curry and Curry to base logic on a consistent system of A-terms or combinators.

A Semantic Approach to Illative Combinatory Logic

This work provides a semantic interpretation for a formal framework in which both logic and computation may be expressed in a unified manner and gives a consistency proof for first-order illative combinatory algebras.

Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus

• Mathematics
Arch. Math. Log.
• 1998
This paper proves completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations, which fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$-terms or combinators.

Completeness of the Propositions-as-Types Interpretation of Intuitionistic Logic into Illative Combinatory Logic

• Mathematics, Philosophy
J. Symb. Log.
• 1998
It is proved that also the two indirect translations are complete and one of the systems of illative combinatory logic is also complete for predicate calculus with higher type functions.

Equivalences between Pure Type Systems and Systems of Illative Combinatory Logic

• Mathematics
Notre Dame J. Formal Log.
• 2005
It is shown that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold.

Pure type systems with more liberal rules

• Mathematics
Journal of Symbolic Logic
• 2001
This paper considers a simplification of the start and weakening rules of PTSs which allows contexts to be sets of statements, and a generalisation of the conversion rule which produces Set-modified PTSs or SPTSs, which are closer to standard logical systems.

The Logic of Church and Curry

• J. Seldin
• Computer Science
Logic from Russell to Church
• 2009

• 2013