experchange > comp.theory

peteolcott (12-20-18, 07:48 PM)
So as I have been (somewhat ineptly) arguing all along is that when we
encode the generic notion of provability as a BASE CLASS within a knowledge
ontology, then the symbolic logic notion of provability diverges from
this BASE semantic meaning in exactly the same sort of way that the
concept of a set totally containing itself diverges from a key axiom
of the BASE CLASS semantic specification of Total_Containment:

This Rudolf Carnap Meaning Postulate defines one of the semantic properties
of the concept of {Totally_Contains} for every element (including physical
and conceptual) of the set of all things which logically entails that no
conceptual thing (such as a set) can conceptually contain itself.

∀x ∈ Thing
~Totally_Contains(x,x) ⊢ (x ∉ x)

In this exact same way the symbolic logic concept of provability must be
corrected so that all of the concepts of provability can directly plug
into this same knowledge ontology inheritance hierarchy.

There exists only a single coherent BASE meaning of the generic concept
of provability in the universe and the symbolic logic notion of provability
diverges from this BASE notion in the exact same way that a set containing
itself diverges from the axiom of Totally_Containment.

Whom-so-ever thought up the concept of a set totally containing itself
simply did not bother to think this ALL-THE-WAY-THROUGH or they would
have seen the logical incoherence derived. Likewise with the symbolic logic
concept of Provability.

So we can begin on the basis of the symbolic logic concept of provability
and augment this to create a better concept of provability named:
{semantically sound provability} that simply forbids extraneous premises
that are not any element of the inference chain from Γ to C in: Γ ⊢ C.

Ultimately the symbolic logic concept of provability will be discarded
as an earlier misconception in the same way that a set containing itself
can be discarded as a misconception.
peteolcott (12-22-18, 07:00 AM)
On 12/20/2018 12:24 PM, Ben Bacarisse wrote:
> peteolcott <Here@Home> writes:
>> ∀x ∈ Thing
>> ~Totally_Contains(x,x) ⊢ (x ∉ x)


In the above case "⊢" means that the RHS inherits its BASE semantics
from the LHS, as a kind of semantic logical entailment specified syntactically.

If nothing can totally contain itself then
nothing can totally physically contain itself
nothing can totally conceptually contain itself then
a set cannot totally contain itself.

> Which of these symbols have your own meanings? In particular is ⊢ the
> one you think will
>> Ultimately the symbolic logic concept of provability will be discarded

> or is it the ⊢ from Mendelson (and others) that's actually useful?


All of the above symbols have their conventional meanings except "⊢".
I am adapting it to unify across formal language and formalized natural
language, and thus forbids anything to be inferred on the basis of False
besides False itself.

This symbol: "⊢" is used to syntactically formalize semantics.
Ben Bacarisse (12-22-18, 03:08 PM)
peteolcott <Here@Home> writes:

> On 12/20/2018 12:24 PM, Ben Bacarisse wrote:
> In the above case "⊢" means that the RHS inherits its BASE semantics
> from the LHS, as a kind of semantic logical entailment specified
> syntactically.


Why should anyone care about your recently made up and badly explained
notation?

> This symbol: "⊢" is used to syntactically formalize semantics.


Mathematicians use ⊨ for that.
peteolcott (12-22-18, 05:24 PM)
On 12/22/2018 7:08 AM, Ben Bacarisse wrote:
> peteolcott <Here@Home> writes:
> Why should anyone care about your recently made up and badly explained
> notation?


It is eventually going to provide the software basis of a fully operational
human mind.

>> This symbol: "⊢" is used to syntactically formalize semantics.

> Mathematicians use ⊨ for that.


Extraneous complexity that Rudolf Carnap proves could be eliminated
by his 1952 Meaning Postulates.
peteolcott (12-22-18, 07:12 PM)
On 12/22/2018 8:08 AM, Antnio Marques wrote:
> peteolcott <Here@Home> wrote:
> Whatever your point is, making an improper analogy is not a way to prove
> it. Tho it can be useful; in this case, you illustrate how your view of the
> principle of explosion has nothing to do with what the principle of
> explosion is [to anyone else].


I was just trying to show the gist of how the principle of explosion can
derive obviously incorrect conclusions. The analogy may be less than perfectly
precise yet it does show the gist of the problem.


The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)",

If literally anything follows from falsehood, then this man's innocence
would logically follow from falsehood.

> You keep thinking that you can avoid paradoxes by redefining language so
> that it can't express those paradoxes. That's like making circles go away
> by forbidding the word, saying it's all polygons with many sides.


I provide the formal means to recognize and reject otherwise undetected
logical incoherence. I am wrapping up my work on Pathological self-reference
(Olcott 2004) that encompasses:
(1) Liar Paradox
(2) 1931 Incompleteness Theorem
(3) 1936 Tarski Undefinability Theorem
(4) 1936 Halting Problem

My refutation of the Halting Problem proofs will bridge my credibility gap
in that I actually wrote an actual (fully executable) halt decider that
actually decides halting for all of the halting problem proof counter-examples.


It took me 14 years since 2004 carefully studying nothing more than
pages 318-319 of the above textbook to derive this correct refutation.

Because of the simplicity of the language of Turing machine descriptions
and higher level abstractions based on this language it may not take
very much more work beyond the correct refutation of the halting problem
proofs to actually solve the halting problem by creating a fully executable
universal halt decider.

>> is that nothing is semantically entailed by False besides False
>> thus symbolic logic errs when it says otherwise.

> All of this time and you haven't yet noticed that the principle of
> explosion is based not on 'false' but 'true AND false'?
> Nobody said that 'Peter has been to Mars' entails everything. EFQ says just that


> It's 'Peter
> has been to Mars AND Peter hasn't been to Mars' that does - ECQ says that.


> in other words,
> 'if we believe something and its opposite, than anything goes'. Note the
> *if*. We're not saying we believe it. We're saying that if we did, we might
> as well believe anything.




Both (1) and (2) and I am focusing on (1) for now.

(1) The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)",

(2) ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"),
Ben Bacarisse (12-22-18, 08:25 PM)
peteolcott <Here@Home> writes:

> On 12/22/2018 7:08 AM, Ben Bacarisse wrote:
> It is eventually going to provide the software basis of a fully operational
> human mind.


No it isn't. There, two totally unsupported points of view for other
reader to choose between. This is the new post-truth mathematics.
peteolcott (12-29-18, 07:39 AM)
On 12/22/2018 11:37 AM, Peter Percival wrote:
> peteolcott wrote:
> So establishing a falsehood would establish the man's innocence? But how is the falsehood to be established? I.e., you have
> falsehood -> the man is innocent
> and you need falsehood to conclude that the man is innocent by modus ponens. You seem to think that
> falsehood -> the man is innocent
> on its own implies (or is even equivalent to)
> the man is innocent
> it does (it isn't).
> You actually wrote it, did you?

Yes. I have been working on it a little bit each day most all week.
I have several other very high priority things that take up most of
my time.
peteolcott (12-29-18, 06:02 PM)
On 12/29/2018 1:07 AM, Dan Christensen wrote:
> See my use of the principle of explosion in my proof of the existence of the so-called empty function at on lines 56 and 71.
> Dan
> Download my DC Proof 2.0 freeware at
> Visit my Math Blog at


I don't have time for this much detail.
Doe you system reject POE, and it so what is its basis?

I simply require that every premise be included in an inference
step from the LHS to the RHS consequence, thus making it impossible
to conclude anything besides False from False or a Contradiction.

Proof of the Error of the Principle of Explosion
p = “I am going to the store”
q = “I am going to the store to buy groceries”
~p ⊢ q
The fact that I am not going to the store at all proves that I am going to the store to buy groceries.

Copyright 2018 Pete Olcott
All rights reserved
peteolcott (12-30-18, 07:39 AM)
On 12/29/2018 2:19 PM, Dan Christensen wrote:
> On Saturday, December 29, 2018 at 11:02:22 AM UTC-5, peteolcott wrote:
> A poor example. If you are not going to the store, you cannot use POE to infer anything of whether you are buying groceries or not. Your groceries could be delivered to your house. Or maybe you are simply not buying any groceries.


YOU CANNOT IGNORE THIS PART !!!
If p and q have the semantics that I specified

then POE is proved to be semantically incoherent because it PROVES
that the fact [that I am not going to the store] PROVES
[I am going to the store to buy groceries].

I am defining the inherent mathematics of knowledge ontologies for
both formal and formalized natural language semantics and I just
PROVED that POE is necessarily incorrect.
peteolcott (12-30-18, 05:51 PM)
On 12/29/2018 11:53 PM, exflaso.quodlibet wrote:
> On Sunday, December 30, 2018 at 12:39:16 AM UTC-5, peteolcott wrote:
> It doesn't matter what their semantics are. The principle of explosion is valid regardless of how you interpret p and q.
> No, it's not. You haven't not even bothered to formally define semantics, let alone semantic incoherence.
> The fact remains that if you have two sentences p and q, such that p is a contradiction, then there are two things we can guarantee to be true:


So that fact that [that I am not going to the store]
really does prove that [I am going to the store to buy groceries] ???

> • syntactically, we know that {p} ⊢ q, that is, there is a proof <B1,...,Bk>, such that Bk = q, and every Bi in <B1,...,Bk> is at least one of (i) an axiom, (ii) p, or (iii) the result of applying a rule of inference to some subset of {B1,...,B(i-1)}
> • semantically, we know that {p} ⊨ q, that is, there are no models in which V(p)=1 and V(q)=0, which is trivially obvious, because there are no models in which V(p)=1


I could find no source that uses that notational convention, neither Mendelson nor SEP uses V(p)=1 in their model theory.

if [I am going to the store to buy groceries] this PROVES [I am going to the store] and
if [I am not going to the store] this PROVES [I am not going to the store to buy groceries]

Thus ~p PROVES q is wrong.
[..]
Alan Smaill (12-30-18, 08:20 PM)
[snip sci.lang]

peteolcott <Here@Home> writes:

> On 12/29/2018 11:53 PM, exflaso.quodlibet wrote:
> So that fact that [that I am not going to the store]
> really does prove that [I am going to the store to buy groceries] ???


Are you telling us that you think that Ye Old
propositional logic proves:

not Going_to_store |-
Going_to_store & Buying_Groceries

??

That's what it looks like.

p here is a fact, you tell us; so not a contradiction.
peteolcott (12-31-18, 07:28 PM)
On 12/30/2018 12:24 PM, Peter Percival wrote:
> peteolcott wrote:
> Here is a proof of POE.  Where are the errors?
>   P
> -----
> P v Q, ~P
> ---------
>     Q


Proof of the Error of the Principle of Explosion
p = “I am going to the store”
q = “I am going to the store to buy groceries”
~p ⊢ q
The fact that I am not going to the store at all proves that I am going to the store to buy groceries.

To say that the fact that {I am not going to the store at all} <proves>
{that I am gong to the store to buy groceries} is schizophrenic.

The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)"

That EFQ did not even know his own namesake is pretty telling.
peteolcott (12-31-18, 08:04 PM)
On 12/31/2018 11:05 AM, exflaso.quodlibet wrote:
> On Monday, December 31, 2018 at 11:43:36 AM UTC-5, peteolcott wrote:
> No, you haven't.
> No wonder you think it's false -- you don't actually understand how it works!
> Typical.
> EFQ


You made the mistake that it only applied to contradictions,
proving that you don't even know your own namesake.
The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)"



If I made a mistake than the specific error could be pointed out.
Since not error has been pointed out it is reasonable for me to continue
assuming that there is no mistake.

Even a computer program with fay less intelligence than


Could spout off baseless assertions that I am wrong:

void main()
{
while true)
{
char string[1024];
scanf( "%s" , string );
printf("You are wrong!\n");
}
}
peteolcott (01-01-19, 07:39 PM)
On 1/1/2019 8:08 AM, Dan Christensen wrote:
> On Monday, December 31, 2018 at 6:20:13 PM UTC-5, peteolcott wrote:
> If you use the generally accepted principles of (1) direct proof (=> intro), (2) proof by contradiction (~ intro), (3) joining statements (& intro) and (4) double negation (~~ elim), then POE is inevitable. Which of these principles do you wish to ban or restrict?
> IIRC intuitionists ban (2) and (4). (3) is restricted in so-called relevance logic. Either seems rather arbitrary to me. What exactly is your "solution?"


Anything that accepts as proof that {I WENT TO THE STORE} on the basis
of the premise that {I DID NOT GO TO THE STORE} is necessarily incorrect.

I really don't see how that isn't totally obvious to everyone over three years old.
peteolcott (01-01-19, 08:18 PM)
On 1/1/2019 11:43 AM, Peter Percival wrote:
> peteolcott wrote:
>> Anything that accepts as proof that {I WENT TO THE STORE} on the basis
>> of the premise that {I DID NOT GO TO THE STORE} is necessarily incorrect.

> What has that got to do with POE?


False does not carry along with it sufficient semantics to derive anything besides False itself.

We cannot make any syllogism of expressions of language connected together by rules-of-inference
with False as a premise. Unless a premises is connected by rule-of-inference to the consequence
it is extraneous and must be discarded. Formally this is call the non sequitur error.

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