Highest Abstraction for Logic and Math?

[Eodnhoj7]

This is a simple question.

My stance, as stated elsewhere, is that the language for it has not been fully invented yet. Considering the nature of truth is subject to context, the primary symbols would be:

"( )" for "context"
"{ }" for "context of contexts"
"[ ]" for "transitional contexf"
"/" "modality of context"
"-->" for "transition of one context to another"
"•" as the "fundamental variable"


A simple statement such as "The cat eats cat food therefore we bought cat food" would be expressed as:

{(C)[E-->](F/C)}-->{(W)[B-->](C/F)}

Or "The sky is blue"
(S)-->(B)


Or for math

1+2=3
{+1-->(+1-->+1)}-->+3

4÷2=2
(+4/+2) --> +2




All inference and implication shows a probabilistic nature; therefore would be expressed as modalities as all modalities are fractions and fractals:

{({(In)(Im)}/A) [S-->] (N/P)} [E-->] (M)[A-->]{(Fn)(Fl)}

"The cat eating the food implies the cat is hungry"
{(C/E)(F)}/(C/H)



The logic is primitive yet seems to represent the basic underlying form of all propositions. I cannot seem to break it down to any deeper basics. Thoughts for or against? Discuss.

[Skepdick]

The highest abstraction requires no grammar, because it's conceptual not linguistic.

https://en.wikipedia.org/wiki/Monad_(fu ... ogramming)

Once you understand what a Monad IS you could express it (define it?) in any notation you want.

So your possible candidates for "the highest abstraction" are limitless. Because you could be talking about ontology; in which case your candidates are, The Universe, Reality, Existence etc.

Or you could be talking about your values; in which case your candidates may be God/Consciousness, The Self, Truth etc.

[Eodnhoj7]

The highest abstraction requires no grammar, because it's conceptual not linguistic.

https://en.wikipedia.org/wiki/Monad_(fu ... ogramming)

Once you understand what a Monad IS you could express it (define it?) in any notation you want.

So your possible candidates for "the highest abstraction" are limitless. Because you could be talking about ontology; in which case your candidates are, The Universe, Reality, Existence etc.

Or you could be talking about your values; in which case your candidates may be God/Consciousness, The Self, Truth etc.

The highest most universal abstraction, with highest meaning an underlying centerpoint from which all things stem, is a contextual loop. It can be subject to language but not limited to it. Any higher language would have to underlie all possible languages, in which case we are left with a loop between the languages and we ironically go back to a language emphasizing context again.

In trying to escape and language we use a series of symbols to emphasize it.

Context cannot seem to be escaped from without creating an ultimate context. If all being is composed of a loop, then the highest abstraction is the monad as a symbol ⊙ with all grammar being a variation of it.

[Arising_uk]

The logic is primitive yet seems to represent the basic underlying form of all propositions. I cannot seem to break it down to any deeper basics. Thoughts for or against? Discuss.

It's not even a primitive logic, it's just a bunch of random symbols. You'll need syntax rules for your operators and then maybe some semantics before you can even use your 'therefore' and 'implies' and create a logic.

[Eodnhoj7]

The logic is primitive yet seems to represent the basic underlying form of all propositions. I cannot seem to break it down to any deeper basics. Thoughts for or against? Discuss.

It's not even a primitive logic, it's just a bunch of random symbols. You'll need syntax rules for your operators and then maybe some semantics before you can even use your 'therefore' and 'implies' and create a logic.

Each variable is a context with verbs , "[x -->]", representing a transitional context to another context, and "/" representing modality. Each context is observed as a set of contexts { }, thus meta recursion is observed.


Syntax rules would require a regress outside the system leading to a variation of Godel's incompleteness theorem. The rules would have to be self referencing, and a context within context observes this, the framework would have to be descriptive by nature. As self referencing it would be subject to double positives and double negative simultaneously. Its truth value lies in is descriptive properties.

For example:

(-True) --> (-True)
(True) --> (True)
((True)True)--->(True & -True)
((-True)-True) --> (True & -True)

[Skepdick]

It's not even a primitive logic, it's just a bunch of random symbols. You'll need syntax rules for your operators and then maybe some semantics before you can even use your 'therefore' and 'implies' and create a logic.

The above is almost true, so long as you manage to stay on the right side of the rule-following paradox. But as a shift of perspective consider: Locus solum: from the rules of logic to the logic of rules

We've been swirling around the drain between syntax/sementics for a while.

My view is that if I had to choose between the traditional meanings of these terms, syntax is meaningful, semantics is not. Which always lands us in the swamp of "but semantics literally means 'meaning'". Except when the word "semantics" has different meanings. Formal grammars are ONLY that - grammars. Platonism. Semantics come from interpretation.

I have been harping on about the distinction between operational (engineering) and denotational (mathematical) semantics. Which is practically - the distinction between "symbols have interpretations" (operational semantics) and "symbols have meaning" (denotational semantics). The latter is not even a "semantic" - it's prescriptive, not descriptive.

Somebody actually formalised "operational semantics" and now the game starts again. There will be a semantic/viewpoint (higher level of abstraction) from which to interpret "operational semantics"

https://ncatlab.org/nlab/show/Geometry+of+Interaction

What has been called Geometry of Interaction (Girard 89) is a kind of semantics for linear logic/linear type theory that is however different in method from the usual categorical semantics in monoidal categories. Instead of interpreting a proof of a linear entailment A⊢B as a morphism between objects A and B in a monoidal category as in categorical semantics, the Geometry of Interaction interprets it as an endomorphism on the object A⊸B. This has been named operational semantics to contrast with the traditional denotational semantics.

I found my people.

Geometry of interaction VI: a blueprint for transcendental syntax

[Skepdick]

The bounded version of the "Geometry of Interaction" is called "The Geometry of Synthesis"

the Geometry of Synthesis has been used to compile higher-order programming languages directly into static circuits

In plain English: That is direct reduction from a programming languages to hardware circuit!

http://www.veritygos.org/

We are getting closer and closer to cyborg life