> On 24 Oct 2018, at 23:02, Tomas Pales <[email protected]> wrote:
>
> Well, you must assume the principle of identity at the beginning,
The logical point is that we can assume something more general the projection
(x y) —> x, which is what the combinator K does.
At the meta intuitive level, I assume such more thing, but when making the
point technical, the devil is in the detail.
In all reversible algebra, identity has to be assume, but allowing (non
reversible) projection make the transitive axiom sufficient; we get the
reflexivity and the symmetry as a gift.
> otherwise all your other assumptions will be their own negations.
At the meta-level, yes. In the formal theory, it happens you can derive it from
projection and transitivity.
That is already true with the axioms of the SK-combinators:
Kxy = x
Sxyz = xz(yz)
But without elimination (like K eliminate y), we need to assume the identity
combinators.
We can add, in the formal theory, the axiom S ≠ K, to avoid having only one
bird, the identity bird. Indeed, we have trivially
III = I
IIII = II(II)
So that {I} is a trivial combinatory algebra. It plays the role of a
contradiction in an equality theory.
Nevertheless a L_obian combinator will have to believe in first order logical
specification, and have the ability to make induction on all combinators.
An ontology without induction can be Turing universal, and allow “creature”
believing in induction, and to extract physics from self-reference (as we have
to for soling the mechanist mind-latter problem) we interview machines
believing in the induction axioms (the self-referentially correct one obeys to
G and G*, from which the logic of the observable is derived. It makes
(classical) computationalism testable.
Bruno
>
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