Simplest possible notion of a formal system

[RCSaunders]

I redefine all this stuff from first principles.
Formal and natural languages are simply stipulated relations between finite strings.
Concepts are the relations of these stipulated relations.

Certainly no one should object to you or anyone else defining the terms they use in any way they like. It is going to very difficult to discuss things with others unless you make them privy to your own private definitions though, isn't it?

What, "first principles," do you base your definitions of concepts, etc. on? I'm not asking for a treatise. If you can't explain briefly, perhaps you can provide a reference, or just skip the question. I'd like to know but will demand anything.

"... finite strings," of what? Words? Words are only symbols for concepts, else you are stringing together meaningless symbols, aren't you?

[PeteOlcott]

I redefine all this stuff from first principles.
Formal and natural languages are simply stipulated relations between finite strings.
Concepts are the relations of these stipulated relations.

Certainly no one should object to you or anyone else defining the terms they use in any way they like. It is going to very difficult to discuss things with others unless you make them privy to your own private definitions though, isn't it?

What, "first principles," do you base your definitions of concepts, etc. on? I'm not asking for a treatise. If you can't explain briefly, perhaps you can provide a reference, or just skip the question. I'd like to know but will demand anything.

"... finite strings," of what? Words? Words are only symbols for concepts, else you are stringing together meaningless symbols, aren't you?

Finite strings has the same meaning that it has always had in mathematical formalist school:
sequences of characters.

https://en.wikipedia.org/wiki/Formalism ... thematics)
In the philosophy of mathematics, formalism is the view that holds that statements of
mathematics and logic can be considered to be statements about the consequences of
the manipulation of strings using established manipulation rules.

When we say that "cats are animals" we mean that the <is a type of> relation applies
between the finite strings of "cats" and "animals".

Every meaning of every expression of any formal or natural language is defined in this
same way.

[Skepdick]

When we say that "cats are animals" we mean that the <is a type of> relation applies
between the finite strings of "cats" and "animals".

When you say "cats are animals" you are using "are" to mean "type of". That is instantiation.

But when you say "the kids are hungry", or "lead atoms are heavier than oxygen atoms" then te verb "are" doesn't mean type relation. It means something else.

You can't formalise those colloquial verbs in a way they have uniform, formal meaning. The very notion of equality, as expressed in Mathematical notation as the symbol = is ambiguous.

More importantly, you can't side-step the fact that a cat is a type of animal, but it is ALSO a type of pet, as well as a type of carnivore.

And so you have to account for many-sorted logic and the diamond problem.

You are trying to formalise English. A language which is stubbornly inconsistent in the application of its own grammar rules. You have already set yourself up for failure.

[PeteOlcott]

When we say that "cats are animals" we mean that the <is a type of> relation applies
between the finite strings of "cats" and "animals".

When you say "cats are animals" you are using "are" to mean "type of". That is instantiation.

But when you say "the kids are hungry", or "lead atoms are heavier than oxygen atoms" then te verb "are" doesn't mean instantiation. It means something else.

You can't formalise those colloquial verbs in a way they have uniform, formal meaning. The very notion of equality, as expressed in Mathematical notation as the symbol = is ambiguous.

More importantly, you can't side-step the fact that a cat is a type of animal, but it is ALSO a type of pet.

And so you have no answers to the many-sorted logic, or the diamond problem.

The colloquial verbs are translated into formal predicates.

A knowledge ontology inheritance hierarchy can be defined without either of the two issues that you posed.
Many-sorted when organized properly has no issues:
inheritance hierarchy
dog ⊆ mammal ⊆ animal ⊆ organism

set of properties of animal
animal.ingestion_type.carinovore

Multiple inheritance is disallowed.

[Skepdick]

Multiple inheritance is disallowed.

Disallowed by whom?

The more limits you impose on a formal system, the less powerful and less useful it becomes.
I am not in the habit of giving up expressive power for the sake of formality.

If you are asking me to give up multiple inheritance - what are you offering in return?

[PeteOlcott]

Multiple inheritance is disallowed.

Disallowed by whom?

The more limits you impose on your formal system, the less powerful it becomes.
I am not in the habit of giving up expressive power for the sake of formality.

If you are asking me to give up multiple inheritance - what are you offering in return?

The natural preexisting order of the body of conceptual knowledge has no multiple inheritance.

[Skepdick]

The natural preexisting order of the body of conceptual knowledge has no multiple inheritance.

Literally any ancestral hierarchy has multiple inheritance! Your genes come from mommy AND daddy.

And you are ignoring all the counter-examples I offered you where that's clearly not true.

A dog is a carnivore.
A dog is a pet.
A dog is a mammal.
A dog is an animal.
A dog is a rabies carrier.
A dog is a delicacy.

So I ask again: If you insist that I should give up the ability to formally express multiple inheritance - what are you offering in return?

[PeteOlcott]

The natural preexisting order of the body of conceptual knowledge has no multiple inheritance.

Literally any ancestral hierarchy has multiple inheritance! Your genes come from mommy AND daddy.

And you are ignoring all the counter-examples I offered you where that's clearly not true.

A dog is a carnivore.
A dog is a pet.
A dog is a mammal.
A dog is an animal.
A dog is a rabies carrier.
A dog is a delicacy.

So I ask again: If you insist that I should give up the ability to formally express multiple inheritance - what are you offering in return?

I am saying that this is simply not the way that it actually works.
The fundamental nature of truth itself IS NOT negotiable.
The collection of concepts decide for themselves how they fit together.

[Skepdick]

The fundamental nature of truth itself IS NOT negotiable.

I am showing you that it is.

If truth is merely linguistic, and language is constructed, then so is truth.

The collection of concepts decide for themselves how they fit together.

Concepts, symbols and words have no decision-agency in this regard.

[PeteOlcott]

The fundamental nature of truth itself IS NOT negotiable.

I am showing you that it is.

If truth is merely linguistic, and language is constructed, then so is truth.

Truth is not merely linguistic. When we change the word labels between dogs and cats the underlying semantics remains unchanged.
A rose by any other name would never become a pile of dog shit.

[PeteOlcott]

The collection of concepts decide for themselves how they fit together.

Concepts, symbols and words have no decision-agency in this regard.

Concepts only fit together coherently according to the stipulated relations
between them that define their semantics.

If we label a {dog} as a "boxcar" and an {animal} as an "office building" A {dog} is still
an {animal} . We are not free to say that {dogs} {ride on rails} or that {animals} {have windows}.

[Eodnhoj7]

Yes, it is...but is it limited to mathematics alone?

Mathematics is never limited to mathematics when it's applied to real-world problems.

I know.

[PeteOlcott]

Yes, it is...but is it limited to mathematics alone?

Mathematics is never limited to mathematics when it's applied to real-world problems.

I know.

The formal system that I propose can even formalize natural language semantics within its stipulated relations between finite strings.

[Eodnhoj7]

Mathematics is never limited to mathematics when it's applied to real-world problems.

I know.

The formal system that I propose can even formalize natural language semantics within its stipulated relations between finite strings.

Maybe, maybe not. What function symbols are you using?

I cannot see a formalized system for language outside of some advanced geometric/hieroglyphs strictly because any formal system needs a universal axiomatic base. You would need geometric symbols for it....it would look like something you would see in a movie about aliens.

This is strictly because space is the most universal axiomatic state there is. It strictly "just is". A dot will always be a dot, same with a line or triangle, etc. ...no matter where you go.

However this formal system would need an underlying base metaphysics to support it considering language is a perception of how reality works...it is a philosophy of being.

Do you have examples?

[Eodnhoj7]

The fundamental nature of truth itself IS NOT negotiable.

I am showing you that it is.

If truth is merely linguistic, and language is constructed, then so is truth.

Truth is not merely linguistic. When we change the word labels between dogs and cats the underlying semantics remains unchanged.
A rose by any other name would never become a pile of dog shit.

True, but we are left with sub symbols.

A label is a symbol.
The object is a symbol.
A dog as a name and a dog as empirical are a set of intertwined symbols.

One abstract (the name) one physical (the entity).