Classical Brouwer-Heyting-Kolmogorov Interpretation

The Brouwer-Heyting-Kolmogorov interpretation explains the meaning of logical operations as operators that construct proofs from proofs of the operands. The BHK interpretation is usually understood as giving intuitionistic interpretation for the logical operators, but, as pointed out by Troelstra and van Dalen [12], it is possible to understand the BHK interpretation classically. We elaborate this idea and develop a classical theory of proofs as abstract mathematical entities where the truth of a proposition becomes equivalent to the existence of proofs of the proposition.

[1]  Hiroshi Nakano The Non-deterministic Catch and Throw Mechanism and Its Subject Reduction Property , 1994, Logic, Language and Computation.

[2]  M. Sørensen,et al.  The λΔ-calculus , 1994 .

[3]  Timothy G. Griffin,et al.  A formulae-as-type notion of control , 1989, POPL '90.

[4]  Stefano Berardi,et al.  A Symmetric Lambda Calculus for Classical Program Extraction , 1994, Inf. Comput..

[5]  Martín Abadi,et al.  Theoretical Aspects of Computer Software , 1991, Lecture Notes in Computer Science.

[6]  C.-H. Luke Ong,et al.  A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[7]  A. Troelstra,et al.  Constructivism in Mathematics: An Introduction , 1988 .

[8]  Michel Parigot,et al.  Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction , 1992, LPAR.

[9]  Makoto Tatsuta,et al.  A New Formulation of the Catch/Throw Mechanism , 1997 .

[10]  Hiroshi Nakano A constructive formalization of the catch and throw mechanism , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[11]  Makoto Tatsuta,et al.  Intuitionistic and Classical Natural Deduction Systems with the Catch and the Throw Rules , 1995 .

[12]  Chetan R. Murthy An evaluation semantics for classical proofs , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[13]  C.-H. Luke Ong,et al.  A Curry-Howard foundation for functional computation with control , 1997, POPL '97.