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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge
Date: Sat, 5 Apr 2025 15:58:17 -0500
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On 4/5/2025 2:20 AM, Mikko wrote:
> On 2025-04-03 19:33:41 +0000, olcott said:
> 
>> On 4/3/2025 2:09 AM, Mikko wrote:
>>> On 2025-04-03 02:51:32 +0000, olcott said:
>>>
>>>> On 4/2/2025 8:56 PM, Richard Damon wrote:
>>>>> On 4/2/25 9:30 PM, olcott wrote:
>>>>>> On 4/2/2025 5:05 PM, Richard Damon wrote:
>>>>>>> On 4/2/25 11:59 AM, olcott wrote:
>>>>>>>> On 4/2/2025 4:20 AM, Mikko wrote:
>>>>>>>>> On 2025-04-01 17:51:29 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> All we have to do is make a C program that does this
>>>>>>>>>> with pairs of finite strings then it becomes self-evidently
>>>>>>>>>> correct needing no proof.
>>>>>>>>>
>>>>>>>>> There already are programs that check proofs. But you can make 
>>>>>>>>> your own
>>>>>>>>> if you think the logic used by the existing ones is not correct.
>>>>>>>>>
>>>>>>>>> If the your logic system is sufficiently weak there may also be 
>>>>>>>>> a way to
>>>>>>>>> make a C program that can construct the proof or determine that 
>>>>>>>>> there is
>>>>>>>>> none.
>>>>>>>>>
>>>>>>>>
>>>>>>>> When we define a system that cannot possibly be inconsistent
>>>>>>>> then a proof of consistency not needed.
>>>>>>>
>>>>>>> But you can't do that unless you limit the system to only have a 
>>>>>>> finite number of statements expressible in it, and thus it can't 
>>>>>>> handle most real problems
>>>>>>>
>>>>>>>>
>>>>>>>> A system entirely comprised of Basic Facts and Semantic logical 
>>>>>>>> entailment cannot possibly be inconsistent.
>>>>>>>>
>>>>>>>
>>>>>>> Sure it can.
>>>>>>>
>>>>>>> The problem is you need to be very careful about what you allow 
>>>>>>> as your "Basic Facts", and if you allow the system to create the 
>>>>>>> concept of the Natural Numbers, you can't verify that you don't 
>>>>>>> actually have a contradition in it.
>>>>>>>
>>>>>>
>>>>>> It never has been that natural numbers have
>>>>>> ever actually had any inconsistency themselves
>>>>>> they are essentially nothing more than an ordered
>>>>>> set of finite strings of digits.
>>>>>
>>>>> No, but any logic system that can support them
>>>>
>>>> Can be defined in screwy that has undecidability
>>>> or not defined in this screwy way.
>>>
>>> And you can't define it otherwise.
>>>
>>
>> Yes it free to keeps its screwy definition just like
>> set theory until a superior alternative comes along,
>> then it may be renamed naive formal systems.
>>
>> A consistent set of stipulated axioms combined with
>> semantic logical entailment as the only inference step
>> makes undecidability impossible.
> 
> If semantic logical entaillment is allowed as an inference rule
> the system is not formal. In order to be formal the system must
> define "proof" as any string that satiisfies the syntactic rules
> that the system specifies for proofs.
> 

This "baffled" Richard
https://en.wikipedia.org/wiki/Montague_grammar
https://plato.stanford.edu/entries/montague-semantics/
Semantics as rich as natural language fully formalized
syntactically.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer