Connection To Intuitionistic Logic
The combinators K and S correspond to two well-known axioms of sentential logic:
AK: A (B A),
AS: (A (B C)) ((A B) (A C)).
Function application corresponds to the rule modus ponens:
MP: from A and A B, infer B.
The axioms AK and AS, and the rule MP are complete for the implicational fragment of intuitionistic logic. In order for combinatory logic to have as a model:
- The implicational fragment of classical logic, would require the combinatory analog to the law of excluded middle, e.g., Peirce's law;
- Complete classical logic, would require the combinatory analog to the sentential axiom F A.
Read more about this topic: SKI Combinator Calculus
Famous quotes containing the words connection to, connection and/or logic:
“One must always maintain one’s connection to the past and yet ceaselessly pull away from it. To remain in touch with the past requires a love of memory. To remain in touch with the past requires a constant imaginative effort.”
—Gaston Bachelard (1884–1962)
“... instinct is the direct connection with truth.”
—Laurette Taylor (1887–1946)
“The logic of worldly success rests on a fallacy: the strange error that our perfection depends on the thoughts and opinions and applause of other men! A weird life it is, indeed, to be living always in somebody else’s imagination, as if that were the only place in which one could at last become real!”
—Thomas Merton (1915–1968)