General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

which we can always do, since we have an axiom for this formula, then we apply the rule subst,
where A is this result, B is C, and X is the variable P to obtain
(C ⇒ Q ⇒ R) ⇒ (C ⇒ Q) ⇒ C ⇒ R (2.2)
Next we apply the rule subst to this where B is C ⇒ C and X is the variable Q this time to
obtain
(C ⇒ (C ⇒ C) ⇒ R) ⇒ (C ⇒ C ⇒ C) ⇒ C ⇒ R (2.3)
And again, we apply the rule subst this time, B is C and X is the variable R yielding the first
formula in our proof on the slide. To conserve space, we have combined these three steps into one
in the slide. The next steps are done in exactly the same way.
2.6.2 A digression on Names and Logics
The name MP comes from the Latin name “modus ponens” (the “mode of putting” [new facts]),
this is one of the classical syllogisms discovered by the ancient Greeks. The name Subst is just
short for substitution, since the rule allows to instantiate variables in formulae with arbitrary
other formulae.
Digression: To understand the reason for the names of K and S we have to understand much
more logic. Here is what happens in a nutshell: There is a very tight connection between types
of functional languages and propositional logic (google Curry/Howard Isomorphism). The K and
S axioms are the types of the K and S combinators, which are functions that can make all other
functions. In SML, we have already seen the K in Example 97
val K = fn x => (fn y => x) : ‘a -> ‘b -> ‘a
Note that the type ‘a -> ‘b -> ‘a looks like (is isomorphic under the Curry/Howard isomorphism)
to our axiom P ⇒ Q ⇒ P . Note furthermore that K a function that takes an argument n
and returns a constant function (the function that returns n on all arguments). Now the German
name for “constant function” is “Konstante Function”, so you have letter K in the name. For the
S aiom (which I do not know the naming of) you have
val S = fn x => (fn y => (fn z => x z (y z))) : (‘a -> ‘b -> ‘c) - (‘a -> ‘c) -> ‘a -> ‘c
Now, you can convince yourself that SKKx = x = Ix (i.e. the function S applied to two copies
of K is the identity combinator I). Note that
val I = x => x : ‘a -> ‘a
where the type of the identity looks like the theorem C ⇒ C we proved. Moreover, under the
Curry/Howard Isomorphism, proofs correspond to functions (axioms to combinators), and SKK
is the function that corresponds to the proof we looked at in class.
We will now generalize what we have seen in the example so that we can talk about calculi and
proofs in other situations and see what was specific to the example.
2.6.3 Logical Systems and Calculi
Calculi: general
A calculus is a systems of inference rules: A1 · · ·
CR and Ax
A
A1: assumptions, C: conclusion (axioms have no assumptions)
A Proof of A from hypotheses in H (H ⊢ A) is a tree, such that its
nodes contain inference rules
leaves contain formulae from H
root contains A
94
An