# Re: Re: intuitional logic and comp

```Hi Bruno Marchal

Perhaps one can say that intuitionist logic is a personalized modal logic,
while classic logic is impersonal necessary logic ?```
```

[Roger Clough], [rclo...@verizon.net]
11/28/2012
"Forever is a long time, especially near the end." -Woody Allen

----- Receiving the following content -----
From: Bruno Marchal
Time: 2012-11-27, 14:08:13
Subject: Re: intuitional logic and comp

On 27 Nov 2012, at 11:58, Roger Clough wrote:

Hi Bruno Marchal

Then since the brain is earth, shouldn't we use intuitionist logic ?

To get money, or to build bridges, or to put a man on the moon, we can argue
that this is what we do.
To search aliens on other planet, or to explore the realm of elementary
particles, or to search a coherent view on the mind and body, we have to bet on
non constructive object, like the "others", the "unknown" etc.

Once I have the time I can give an example of a simple non intuitionist proof
in math, so you can appreciate better.

Bruno

[Roger Clough], [rclo...@verizon.net]
11/27/2012
"Forever is a long time, especially near the end." -Woody Allen

----- Receiving the following content -----
From: Bruno Marchal
Time: 2012-11-26, 12:49:38
Subject: Re: intuitional logic and comp

On 26 Nov 2012, at 12:34, Roger Clough wrote:

Hi Bruno,

Since intuitional logic seems to be a form of "pure logic", inapplicable to
the outside world,

Why do you say that? On the contrary, most people believe that intuitionist
logic is the logic most suited for the application in the real world. I tend to
think that woman and engineers are intuitionist by nature. The believe in what
they can construct, where a classical logician extends its beliefs into what is
impossible to not exist, even if we can't construct it. Intuitionism is Earth
logic, classical logic is Heaven logic.

can or does comp implement intuitional logic ?

Yes, in many ways, but what is nice is that we get freely an intuitionistic
subject  associated to the machine, by applying the theaetus' definition of the
knower (true justified belief) with "justified belief" interpreted by the
machine (sound) provability ability. It is the []p : Bp & p definition that I
mentioned earlier. But this can'be made clear without doing a bit more of
logic. This works because Bp -> p, although true, is unprovable by the machine,
so that from both her 3p and 1p self views, they behave differently.  Bp and Bp
& p leads to two veru different view on the Arithmetical truth that he machine
can be aware of.

Bruno

[Roger Clough], [rclo...@verizon.net]
11/26/2012
"Forever is a long time, especially near the end." -Woody Allen

----- Receiving the following content -----
From: Platonist Guitar Cowboy
Time: 2012-11-25, 20:00:12
Subject: Re: arithmetic truth

On Sun, Nov 25, 2012 at 5:10 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:

Hi Cowboy,

On 24 Nov 2012, at 19:52, Platonist Guitar Cowboy wrote:

Hi Everybody,

At several points the discussions of the list led us to hypothesis of
arithmetic truth. Bruno mentioned once that the basis for this hypothesis was
quite strong, requiring studies in logic to grasp.

You might quote the passage. Comp (roughly "I am machine", with the 3-I, the
body) is quite strong, compared to "strong AI" (a machine can be conscious).
Although the comp I use is the weaker of all comp; as it does not fix the
substitution level. But logically it is still stronger than strong AI.

But arithmetical truth itself is easy to grasp. Even tribes having no names for
the natural numbers get it very easily, and basically anyone capable of given
sense (true or false or indeterminate, it does not matter) to sentences like
"I will have only a finite number of anniversary birthdays", already betrays
his belief in arithmetical truth (the intuitive concept). So I would say it is
assumed and know by almost everybody, more or less explicitly depending on
education.

I still have difficulty with intuition as "ability to understand something no
reasoning" in this loose linguistic sense and how mathematicians frame that.
When Kleene makes this precise in "The Foundations of Intuitionistic
Mathematics"... this is a bit too much for cowboys with guitars, but for some
reason I am intrigued.

But as a non-logician, I have some trouble wrapping my brain around G?el and
Tarski's Papers concerning this.

Well, this is quite different. It concerns what machine and theories can said
about truth. This is far more involved and requires some amount of study of
mathematical logic. I will come back on this, probably in the FOAR list (and
not soon enough, as we have to dig a bit on the math needed for this before).

What I do see is that Tarski generalizes the notion and its difficulties to all
formal languages: truth isn't arithmetically definable without higher order
language. Post attacking the problem with Turing degrees also resonates with
this in that no formula can define truth for arbitrarily large n.

My question as non-logician therefore is: don't these results weaken the basis
for such a hypothesis or at least make it completely inaccessible for us?

No, it is totally accessible to us, but by intuition only. You can be sure that
music is very similar. We are all sensible to it, but to explain this is beyond
the formal method; neither a brain nor a computer might ever been able to do
that.

That is so strange and amazing. Especially that weird parallel to music. And
"might" is a very large word there to me because don't composers or
mathematicians of, I'll say vaguely, "similar approaches to their craft"
already agree on certain facts about objects and their properties already? I
know, I do with certain musicians/composers, without total certainty though.

Now, comp needs only the sigma_1 truth, which is machine definable, to proceed.
I use the non-definability of truth only to see the relation with God, and for
the arithmetical interpretation of Plotinus, as the numbers themselves will
have to infer more than Sigma_1 truth (actually much more).

But it is clear that consciousness is also not definable, yet we have all
access to it, very easily. It is the same for arithmetical truth. The notion is
easy, the precise content is infinitely complex, non computable, unsolvable,
not expressible in arithmetic, etc.

Only "philosophers" can doubt about the notion of arithmetical truth. In math,
both classical and intuitionist, arithmetical truth is considered as the easy
sharable part (even if interpreted differently). COMP is strong because "yes
doctor" involves a risky bet, and the Church thesis requires a less risky bet
but is still logically strong, but the Arithmetical realism is very weak: it is
assumed by every scientists and lay men, and disputed only by philosophers (and

I have never heard about something like a student abandoning school and
thinking his teacher is mad when he heard him saying that there is no bigger
prime number. It *is* a bit extraordinary, when you think twice, but we are
used to this.

But isn't this like informally stating that Euclid proof "there will always be
larger prime". So it's more like a proof than intuition? Like you have to know
what prime is, natural numbers are infinite etc., if natural numbers are
infinite than there will always be one more? So this you can fomalize and state
loosely in language, but what the student dreamed at the last concert he
enjoyed is not. It is not clear to me why the prime statement is intuition.

Cowboy regards :)

Best,

Bruno

http://iridia.ulb.ac.be/~marchal/

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