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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: The Foundation of Linguistic truth is stipulated relations
 between finite strings
Date: Sat, 14 Sep 2024 09:01:31 -0500
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On 9/14/2024 3:26 AM, Mikko wrote:
> On 2024-09-13 14:38:02 +0000, olcott said:
> 
>> On 9/13/2024 6:52 AM, Mikko wrote:
>>> On 2024-09-04 03:41:58 +0000, olcott said:
>>>
>>>> The Foundation of Linguistic truth is stipulated relations
>>>> between finite strings.
>>>>
>>>> The only way that we know that "cats" <are> "animals"
>>>> (in English) is the this is stipulated to be true.
>>>>
>>>> *This is related to*
>>>> Truth-conditional semantics is an approach to semantics of
>>>> natural language that sees meaning (or at least the meaning
>>>> of assertions) as being the same as, or reducible to, their
>>>> truth conditions. This approach to semantics is principally
>>>> associated with Donald Davidson, and attempts to carry out
>>>> for the semantics of natural language what Tarski's semantic
>>>> theory of truth achieves for the semantics of logic.
>>>> https://en.wikipedia.org/wiki/Truth-conditional_semantics
>>>>
>>>> *Yet equally applies to formal languages*
>>>
>>> No, it does not. Formal languages are designed for many different
>>> purposes. Whether they have any semantics and the nature of the
>>> semantics of those that have is determined by the purpose of the
>>> language.
>>
>> Formal languages are essentially nothing more than
>> relations between finite strings.
> 
> Basically a formal language is just a set of strings, usually defined
> so that it is easy to determine about each string whether it belongs
> to that subset. Relations of strings to other strings or anything else
> are defined when useful for the purpose of the language.
> 

Yes.

>> Thus, given T, an elementary theorem is an elementary
>> statement which is true.
> 
> That requires more than just a language. Being an elementary theorem means
> that a subset of the language is defined as a set of the elementary 
> theorems

a subset of the finite strings are stipulated to be elementary theorems.

> or postulates, usually so that it easy to determine whether a string is a
> member of that set, often simply as a list of all elementary theorems.
> 

Yes.

>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>
>> Some of these relations between finite strings are
>> elementary theorems thus are stipulated to be true.
> 
> No, that conficts with the meanings of those words. Certain realtions
> between strings are designated as inference rules, usually defined so
> that it is easy to determine whether a given string can be inferred
> from given (usually one or two) other strings. Elementary theorems
> are strings, not relations between strings.
> 

One elementary theorem of English is the {Cats} <are> {Animals}.
The only way that way know that the set named "cats" is a subset
of the set named "animals" is that it is stipulated to be true is
that it is stipulated. The set of properties that belong to the
named set of "cats" and the set of "animals" is also stipulated
to be true. "cats" <have> "lungs".

>> Thus True(L,x) merely means there is a sequence of truth
>> preserving operations from x in L to elementary theorems
>> of L.
> 
> Usually that prperty of a string is not called True. Instead, a non- 
> empty sequence of strings where each member is an elementary theorem or 
> can be
> inferred from strings nearer the beginning of the sequence by the inference
> rules is called a proof. The set of theorems is the set that contains every
> string that is he last members of a proof and no other string.
> 

The elementary theorems (ET) are stipulated to have the semantic 
property of Boolean true. Other expressions x are only true when
x can be derived by applying a sequence of truth preserving operations 
to (ET) (typically back-chained inference).

> Postulates, theoresm, inference rules and theorems are not parts of a
> language but together with language constritue a large system that is
> called a theory. 

That is typically the way it is done yet becomes difficult to understand
when applied to natural language. We never think of English as dividable
into separate theories.

We construe English as also containing all of the semantics of English.
We never have systems of English whether the same expression is the
truth in one system and a lie in another system.

> In order to discuss meanings and truth a still larger
> system is needed where the strings of a theory are related to something
> else (for example real world objects or strings of another language).
> 

Not really. When we have a separate model theory then crucial
details get overlooked.

When we look at a language (including all of its semantics as)
relations between finite strings then we can see all of the
details with none overlooked.

 From Tarski's perspective this would mean that a language
is its own metal-language.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer