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:
“It may comfort you to know that if your child reaches the age of eleven or twelve and you have a good bond or relationship, no matter how dramatic adolescence becomes, you children will probably turn out all right and want some form of connection to you in adulthood.”
—Charlotte Davis Kasl (20th century)
“... instinct is the direct connection with truth.”
—Laurette Taylor (1887–1946)
“What avail all your scholarly accomplishments and learning, compared with wisdom and manhood? To omit his other behavior, see what a work this comparatively unread and unlettered man wrote within six weeks. Where is our professor of belles-lettres, or of logic and rhetoric, who can write so well?”
—Henry David Thoreau (1817–1862)