XPost: sci.logic, comp.theory
On 8/20/23 12:42 PM, Richard Damon wrote:
On 8/20/23 10:41 AM, olcott wrote:
I redefined the foundation of formal logic that is just as expressive
and powerful as Higher Order Logic (HOL) and eliminates Gödel
Incompleteness and Tarski Undefinability.
And what have you actually proved you can do with this?
Remember having changed the foundation, you need to TOTALLY rebuild the
logic
We simply extend the notion of a syllogism and require that
*All conclusions must be a semantically necessary*
*consequence of all of their premises*
otherwise the argument is invalid.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
And what exactly do you mean by that?
From everything you seem to have said, you have had to remove the
ability to do "abstract" logic, and can only work with
Noticed I lost the rest of the thought:
Your logic seems to only be able to handle "Concrete" ideas, as you need
to know the "Semantic Meaning" of the terms. Thus you can't use things
like the prepositional logic that:
If A -> B has been proven and A has been proven then B can be proven
as a rule in your logic, as that rule doesn't "Semantically Necessary"
by your definitions.
Thus, for instance, Rules like "Induction" can't be expressed in your
system of logic.
Since you have wiped the slate clean of rules by changing the
foundation, you have a LOT of work, especially since by this you can't
do anything "generically" but need to show the "semantic necessity" for
every individual case.
This is best accomplished by merging the notion of model theory directly
into higher order logic.
We get rid of Gödel Incompleteness in that every case where a conclusion
cannot be proven from all of its premises determines that the logic
sentence is invalid.
And, as Godel has proven, this means that you logic system is either inconsistant or can not support the needed basic principles or the
natural numbers.
We get rid of Tarski Undefinability in that True(L, x) is defined as
logic sentences where all the premises are known to be true.
No, you haven't. You explainations just show that your logic system is limited to FINITE systems,
We know that they are true because they are established facts such as
Haskell Curry’s elementary theorems. Expressions of language L within
formal system T that are stipulated to have the semantic value of
Boolean True. https://www.liarparadox.org/Haskell_Curry_45.pdf
So, the only things that are True in your system are what you initially established as Truths?
Sounds like a VERY limited logic system.
And how does this statement mesh with your rule above. Here you say
something is true only if it is an "Elementary Theorem", which are BY DEFINITION only statements which have POSTULATED as True, i.e. the
initial truth makers of the system.
This seems to imply that your "necessarily true" from earlier means was
already postulated to be true.
Or, have you just been caught in a fundamental misunderstanding of what
you are talking about?
Or do you not understand that when Curry says these statements are
"True", he isn't being exclusive about it, but just using these to
establish that these form an initial set of true statements, and we can
find more, but repeated (even infinitely) applying of truth perserving
rules to get new truths, and thus the set of elementary theorems do not "define" the Truth predicate, and thus you still need to figure out how
to determine, in finte work (to be a predicate) how to determine if a
statement is actually true.
These facts could also be formalized natural language specifying facts
of the verbal model of the actual world. CycL seems to be the most
robust knowledge ontology language is anchored in Higher order logic.
https://en.wikipedia.org/wiki/CycL
You seem to have a fundamental problem of distinguishing between
"Knowledge" and "Truth".
You have defined a logic system that can not do logic, since the only
"True" statements are those that were initially defined to be True.
Sounds like a very useful system (NOT!).
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