The following foundation of correct reasoning simultaneously gets rid of
Gödel Incompleteness Tarski Undefinability the principle of explosion
and every other divergence from correct reasoning that is allowed by
logic systems since the syllogism.

Just like with syllogisms conclusions a semantically necessary
consequence of their premises

Semantic Necessity operator: ⊨□

(a) Some expressions of language L are stipulated to have the property
of Boolean true.
(b) Some expressions of language L are a semantically necessary
consequence of others.
P is a subset of expressions of language L
T is a subset of (a)

Provable(P,X) means P ⊨□ X
True(T,X) means X ∈ (a) or T ⊨□ X
False(T,X) means T ⊨□ ~X

The above system only applies to the analytic side of the analytic
synthetic distinction which includes all of math and logic yet excludes
expressions of language that can only be verified as true with input
from the sense organs.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 4/1/23 5:14 PM, olcott wrote:
> The following foundation of correct reasoning simultaneously gets rid of
> Gödel Incompleteness Tarski Undefinability the principle of explosion
> and every other divergence from correct reasoning that is allowed by
> logic systems since the syllogism.
>
> Just like with syllogisms conclusions a semantically necessary
> consequence of their premises
>
> Semantic Necessity operator: ⊨□
>
> (a) Some expressions of language L are stipulated to have the property
> of Boolean true.
> (b) Some expressions of language L are a semantically necessary
> consequence of others.
> P is a subset of expressions of language L
> T is a subset of (a)
>
> Provable(P,X)   means P ⊨□ X

Which differs from classical logic, so you have lost all ability to
refer to ANYTHING from classical logic until you reestablih it, since
classical logic requries that Provable means a FINITE chain of logic
from P to X, not just "any" chain.

Also, Provable tends to be based not a arbitrary subset of expressions
of languge (which doesn't even require them to be known true) but is
provable in a "Theory" where we start from a subset of the truth-makers
(a). of the Theory.

> True(T,X)       means X ∈ (a) or T ⊨□ X
> False(T,X)      means T ⊨□ ~X
>
> The above system only applies to the analytic side of the analytic
> synthetic distinction which includes all of math and logic yet excludes
> expressions of language that can only be verified as true with input
> from the sense organs.
>

But since you disagree on key points to the existing system, you need to
start at the bottom of the logic tree, you don't even have 1+1 = 2
anymore until you actually prove it.

On 4/1/2023 4:14 PM, olcott wrote:
> The following foundation of correct reasoning simultaneously gets rid of
> Gödel Incompleteness Tarski Undefinability the principle of explosion
> and every other divergence from correct reasoning that is allowed by
> logic systems since the syllogism.
>
> Just like with syllogisms conclusions a semantically necessary
> consequence of their premises
>
> Semantic Necessity operator: ⊨□
>
> (a) Some expressions of language L are stipulated to have the property
> of Boolean true.
> (b) Some expressions of language L are a semantically necessary
> consequence of others.
> P is a subset of expressions of language L
> T is a subset of (a)
>
> Provable(P,X)   means P ⊨□ X
> True(T,X)       means X ∈ (a) or T ⊨□ X
> False(T,X)      means T ⊨□ ~X
>
> The above system only applies to the analytic side of the analytic
> synthetic distinction which includes all of math and logic yet excludes
> expressions of language that can only be verified as true with input
> from the sense organs.
>

When this is called the foundation of correct reasoning it is called
that because this is the actual way that analytic truth really works.

When any existing formal system violates the way that analytic truth
really works then it is incorrect in the absolute sense because all
formal systems and mathematics are instances of analytic truth.

When other formal systems are requited to inherit True(x), Provable(x)
and False(x) from this foundation then only their divergence is
abolished.

Semantic relations that are specified syntactically are still allowed
and long as they are semantically coherent.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 4/1/23 5:58 PM, olcott wrote:
> On 4/1/2023 4:14 PM, olcott wrote:
>> The following foundation of correct reasoning simultaneously gets rid of
>> Gödel Incompleteness Tarski Undefinability the principle of explosion
>> and every other divergence from correct reasoning that is allowed by
>> logic systems since the syllogism.
>>
>> Just like with syllogisms conclusions a semantically necessary
>> consequence of their premises
>>
>> Semantic Necessity operator: ⊨□
>>
>> (a) Some expressions of language L are stipulated to have the property
>> of Boolean true.
>> (b) Some expressions of language L are a semantically necessary
>> consequence of others.
>> P is a subset of expressions of language L
>> T is a subset of (a)
>>
>> Provable(P,X)   means P ⊨□ X
>> True(T,X)       means X ∈ (a) or T ⊨□ X
>> False(T,X)      means T ⊨□ ~X
>>
>> The above system only applies to the analytic side of the analytic
>> synthetic distinction which includes all of math and logic yet excludes
>> expressions of language that can only be verified as true with input
>> from the sense organs.
>>
>
> When this is called the foundation of correct reasoning it is called
> that because this is the actual way that analytic truth really works.
>

And, since you are changing the foundations of the existing logic
system, you need to throw them ALL out and start fresh.

> When any existing formal system violates the way that analytic truth
> really works then it is incorrect in the absolute sense because all
> formal systems and mathematics are instances of analytic truth.

And since you have made a change at the fundamental level, you need to
throw out ALL previous work in logic and start over. PERIOD.

>
> When other formal systems are requited to inherit True(x), Provable(x)
> and False(x) from this foundation then only their divergence is
> abolished.

And since you are changing the base of ALL formal systems, you need to
verify EVERY proposition proven in them is still valid under your new
rules. Depending on your meaning of "Semantic" it may be easy to move
through many of them, or it may be very slow.

YOU need to do this before you can processed.

Show us that in your system 1 + 1 is still 2, remember, you need to go
back to the full set of first principles in each system it is based on,
not just assume you can use the existing logical results.

This means you are first going to need to start at basic logic and
deteremine what can actually be shown to still work.

>
> Semantic relations that are specified syntactically are still allowed
> and long as they are semantically coherent.
>

You like throwing out the words, what do you actually MEAN.

I think you don't understand the actual problem, because you don't
actually underestand what you are doing.

Is "Semantic", based on "The Meaning of the Word", (and how can a
syntactically defined relationship be coherent for all meaning of the
words?)

Or, is "Semantic" what is classically defined, which basically negates
all your meaning as it basically means that which can be shown as
necessity by the syntactic rules.