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. 

Semantic Consistency Proofs for Systems of Illative Combinatory Logic

TLDR
The strongest of these systems essentially incorporates full extensional classical higher-order logic extended with dependent function types, dependent sums, subtypes and W-types, which allows to interpret a great deal of mathematics in this system.

Semantic Consistency Proofs for Systems of Illative Combinatory Logic

Illative systems of combinatory logic or lambda-calculus consist of type-free combinatory logic or lambda-calculus extended with additional constants intended to represent logical notions. In fact,

BUNDER’S PARADOX

TLDR
It is shown that a paradox that arises in illative logic is thought to be inconsistent, but this isn’t so, by providing a nonempty class of models that establishes its consistency.

Partiality and Recursion in Higher-Order Logic

TLDR
This paper presents an illative system of classical higher-order logic with subtyping and basic inductive types, and shows conservativity of $\ensuremath{{\cal I}}_s$ over classical first- order logic.

A Shallow Embedding of Pure Type Systems into First-Order Logic

TLDR
A shallow embedding of logical proof-irrelevant Pure Type Systems into minimal first-order logic is defined, which forms a basis of the translations used in the recently developed CoqHammer automation tool for dependent type theory.

Semantyczne dowody niesprzeczności systemów Illatywnej

Illatywne systemy logiki kombinatorycznej bądź rachunku lambda rozszerzają beztypowy rachunek kombinatorów bądź rachunek lambda o dodatkowe stałe mające na celu reprezentację pojęć logicznych. W

References

SHOWING 1-10 OF 12 REFERENCES

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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

Higher-order illative combinatory logic

  • 2013

A Combinatory Logic