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-It is interesting to see that my understanding about implication (see first paper) comes near the logic called “intuitionist”, in which took part no one less that Goedel himself.

At the contrary, the so-called “classical” logic - the one used to make people ill and let the “high” spheres abuse of them - benefited of Hilbert as highest personnage.

Believing to know (I have not yet read these papers) that Hilbert had several of his works distructed by the philosopher Miles Mathis, and moreover that I accord a infinitely higher thrust in Mathis, than in Hilbert, this state of affairs makes me feel rather well.

Disproof of the called simple “proof(s)” of the contraposition:

First “simple proof” :

Mainstream definition:

A → B ⇔ notA ∨ B

with the commutation of the “Or”,

⇔ B ∨ notA

Now, I take notB Or notA; so with the mainstream definition about this last line: That yields directly to the second result here above.

But this is ridiculous. Why?

-Two reasons. Firstly, when I am considering logic, I am not allowed to use within it, any symbol meaning “logically equivalent”, because if we qualify this equivalence as being logical, I need a larger frame;

as such, this frame cannot anymore be part of this logic.

So how do we resolve the problem, in order to mean that different members will ever have the same value in the same cases ?

-I suggest a simple equality symbol. This one let even the reading be easier.

Also, the so-called “proof” has a dependancy on the mainstream definition for the implication, which I consider as being false. Let us see! And try the definition I found for this implication.

Indeed, I call this one rather as being a balance sheet. So let's try my balance sheet:

Because of my balance sheet for the implication:

[A → B] = [A ∨ notB]

-by commutation of the Or:

= [notB ∨ A]

-So we try to see what give the contraposition to us, in applying the balance sheet done in first line:

[notB → notA] = [notB ∨ A]

Morality :

We see that the contraposition is equal to the original establishement. But the part which pretend to constitute a “proof”, used the mainstream definition, which is false.

Addendum - What is a (mathematical) demonstration :

(In the mathematical meaning), “demonstration” is a contradiction in terms. The prefix “de-” has obviously a retro sense, when the reamining “-monstration” evocate that we show, what constitute in re-doing the process, in a prograde way, in order to convince.

Moreover, in maths, a demonstration call the most generally the result of an annex, an external result. But that is not what is required, in order to convince or to provide a proof.

When we require a proof, we ask for the act of showing, knowing: the process, in itself.

-So, I give the example, as we convince this way (depending on Albert Schweitzer): A. Courvoisier implication :

A imply B
false valid false
false invalid true
true valid false
true valid true

-and let's try its contraposition (in respecting the correspondance with variables values) :

notB imply notA
true valid true
false invalid true
true valid false
false valid false

-As we see, in the colon of the middle, which shall be read last, I obtain the same values in validity, in the same places (and this, in using my balance sheet for the implication).

This is that, a proof, for me; this is the fact of showing.


© Alexandre Courvoisier, 1028 Préverenges.

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