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)
“Much is made of the accelerating brutality of young people’s crimes, but rarely does our concern for dangerous children translate into concern for children in danger. We fail to make the connection between the use of force on children themselves, and violent antisocial behavior, or the connection between watching father batter mother and the child deducing a link between violence and masculinity.”
—Letty Cottin Pogrebin (20th century)
“The much vaunted male logic isn’t logical, because they display prejudices—against half the human race—that are considered prejudices according to any dictionary definition.”
—Eva Figes (b. 1932)