User blog:Emlightened/Extensions to the SK Calculus

Terms are identified up to equivalence ≡. A model is consistent iff K ≡ SK is false.


 * Sabc ≡ ac(bc)
 * Kab ≡ a
 * If ac ≡ bc, c is a free variable and c does not occur in a or b, a ≡ b

The extensions additionally include some of the following terms. It is not guaranteed that all extensions or combinations of extensions are consistent.


 * Qab ≡ K if a≡b and Qab ≡ SK otherwise
 * Eab ≡ K if there is some c such that ac ≡ b, and Eab ≡ SK otherwise
 * Cab ≡ Kc for some c such that ac ≡ b (if it exists), and Cab ≡ SKK otherwise
 * Mabc ≡ K if there is some d such that bd ≡ c and ad ≡ K, and Mabc ≡ SK otherwise
 * Nabc ≡ Kd for some d such that bd ≡ c and ad ≡ K, and Nabc ≡ SKK otherwise

First off, K and SK represent the truth values true and false, respectively. Given this, it is somewhat clear what each of these does:


 * Q is a term which tests for equivalence, so Qab represents a≡b
 * E is a term that represents an existential quantifier, so Eab represents ∃c(ac≡b)
 * C represents some choice function corresponding to E. Multiple may satisfy the definition.
 * M is a term that represents the image function, and corresponds to conditional existential quantification: Mabc represents ∃d(ad and bd ≡ c)
 * N represents some choice function corresponding to N. Multiple may satisfy the definition.

It is clear that some of these can be used to represent others. In particular:


 * Qa ≡ E(Ka)
 * Eab ≡ (Q(Cab)SKK)(SK)K
 * E ≡ M(KK)
 * C ≡ N(KK)
 * Mabc ≡ (Q(Nabc)SKK)(SK)K

This has been provided mainly for interest purposes. SKQ is not contradictory, despite the similarity to the supposedly contradictory SKIΩ calculus. I suspect SKM (and hence SKQN) is contradictory, but have no proof. If it isn't, it could serve as a foundation to some of mathematics, which would be pretty cool.