Re: How a True(X) predicate can be defined for the set of analytic knowledge

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:
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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:
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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.

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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.
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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.
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When we define a system that cannot possibly be inconsistent
then a proof of consistency not needed.

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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
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A system entirely comprised of Basic Facts and Semantic logical entailment cannot possibly be inconsistent.
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Sure it can.
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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.
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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.

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No, but any logic system that can support them

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Can be defined in screwy that has undecidability
or not defined in this screwy way.

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And you can't define it otherwise.
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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.
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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.

Unless the full semantics is formalized syntactically
using something like Montague Grammar of natural language
semantics.

In order to be formal the system must
define "proof" as any string that satiisfies the syntactic rules
that the system specifies for proofs.
 

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