On 5/20/2024 7:57 PM, Richard Damon wrote:
On 5/20/24 2:59 PM, olcott wrote:
On 5/19/2024 6:30 PM, Richard Damon wrote:
On 5/19/24 4:12 PM, olcott wrote:
On 5/19/2024 12:17 PM, Richard Damon wrote:
On 5/19/24 9:41 AM, olcott wrote:
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True(L,x) is always a truth bearer.
when x is defined as True(L,x) then x is not a truth bearer.
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So, x being DEFINED to be a certain sentence doesn't make x to have the same meaning as the sentence itself?
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What does it mean to define a name to a given sentence, if not that such a name referes to exactly that sentence?
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p = ~True(L,p) // p is not a truth bearer because its refers to itself
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Then ~True(L,p) can't be a truth beared as they are the SAME STATEMENT, just using different "names".
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Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))
p = ~True(L,p) Truthbearer(L,p) is false
q = ~True(L,p) Truthbearer(L,q) is true
Irrelvent.
If Truthbearer(L, p) is FALSE, and since p is just a NAME for the statement ~True(L, p), that means that True(L. p) is not a truth bearer and True has failed to be the required truth predicate.
That is the same thing as saying that
True(English, "this sentence is not true") is false
proves that True(L,x) is not a truthbearer.
If you are defining your "=" symbol to be "is defined as" so the left side is now a name for the right side, you statement above just PROVES that your logic system is inconsistant as the same expression (with just different names) has contradicory values.
You are just showing you utter lack of understanding of the fundamentals of Formal Logic.
ϕ(x) there is a sentence ψ such that S ⊢ ψ ↔ ϕ⟨ψ⟩.
The sentence ψ is of course not self-referential in a strict sense, but mathematically it behaves like one.
https://plato.stanford.edu/entries/self-reference/#ConSemParNo what it shows is that formal logic gets the wrong answer because
formal logic does not evaluate actual self-reference.
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Just like (with context) YOU can be refered to a PO, Peter, Peter Olcott or Olcott, and all the reference get to the exact same entity, so any "name" for the express
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True(L,p) is false
True(L,~p) is false
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So since True(L, p) is false, then ~True(L, p) is true.
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~True(True(L,p)) is true and is referring to the p that refers
to itself it is not referring to its own self.
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*ONE LEVEL OF INDIRECT REFERENCE MAKES ALL THE DIFFERENCE*
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Why add the indirection? p is the NAME of the statement, which means exactly the same thing as the statement itself.
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p = ~True(L,p)
does not mean that same thing as True(L, ~True(L,p))
The above ~True(L, p) has another ~True(L,p) embedded in p.
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Is the definition of an English word one level LESS of indirection than the word itself?
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This sentence is not true("This sentence is not true") is true.
Right, that is a sentence about another sentence (that is part of itself)
Likewise with ~True(L, ~True(L, p)) where p is defined as ~True(L, p)
p defined as ~True(L, p) isn't a sentence refering to ~True(L, p), it is assigning a name to the sentence to allow OTHER sentences to refer to it by name,
Yet when p refers to its own name this creates infinite recursion.
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I don't think you understand what it means to define something.
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x := y means x is defined to be another name for y
https://en.wikipedia.org/wiki/List_of_logic_symbols
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LP := ~True(L, LP)
specifies ~True(~True(~True(~True(~True(...)))))
Nope.
When LP refers to its own name this creates infinite recursion.
It means that LP is defined to be the sentence ~True(L, LP)
replacing the LP in the sentence with a copy of LP IS a level of indirection, so you can get the infinite expansion if you keep or derefencing the reference in the statement.
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"Definition by example" is worse than "Proof by example", at least proof by example can be correct if the assertion is that there exists, and not for all.
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A simpler isomorphism of the same thing is proof by analogy.
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Which isn't a valid proof in a formal system. You seem to think Formal System are a loosy goosy with proofs as Philosophy.
True(English, "this sentence is not true") is false
Is 100% perfectly isomorphic to its formalized version
LP is defined as ~True(L, LP)
True(L, LP) is false
It is merely easier to see that "this sentence is not true"
cannot be true because that makes it false and
can't be false because that makes it true.
LP is defined as ~True(L, LP)
works this same yet yet it is not as intuitive.
So we see that the above is a correct formalization
of the English and that gives us the cognitive leverage
of intuition.
A level of indirection:
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p: "This sentence is true", which is exactly the same as "p is true" since "this sentence" IS p
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p := True(L,p)
specifies True(True(True(True(True(...)))))
Nope, it is equivelent to that, but doesn't SPECIFY that.
LP := ~True(L, LP) means that every instance of LP
in the RHS is the same as the RHS.
Clocksin & Mellish say this same thing.
BEGIN:(Clocksin & Mellish 2003:254)
Finally, a note about how Prolog matching sometimes differs from the
unification used in Resolution. Most Prolog systems will allow you to
satisfy goals like:
equal(X, X).
?- equal(foo(Y), Y).
that is, they will allow you to match a term against an uninstantiated
subterm of itself. In this example, foo(Y) is matched against Y, which
appears within it. As a result, Y will stand for foo(Y), which is
foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))),
and so on. So Y ends up standing for some kind of infinite structure.
END:(Clocksin & Mellish 2003:254)
As I said above that is expanding levels of indirecction.
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*Prolog sees the same infinite recursion and rejects it*
?- TT = true(TT).
TT = true(TT).
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?- unify_with_occurs_check(TT, true(TT)).
false.
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Right, because prolog can't handle any levels of self referencing, and thus is not suitable for logic that can do that.
Nothing can handle "some kind of infinite structure."
You have been told this, but don't seem to understand it. My guess is you can't understand any logic more complicated than what Prolog handles, so don't realize how much it just doesn't handle.
No the whole problem seems to be that you simply don't
bother to pay close enough attention the EXACTLY what I say.
When I prove my point you simply ignore that I proved my point
and baselessly assume that I must be wrong. You will probably
completely "forget" my Clocksin & Mellish quote immediately after
you read it, or skip over it and assume that they are wrong.
-- Copyright 2024 Olcott "Talent hits a target no one else can hit; Geniushits a target no one else can see." Arthur Schopenhauer
Re: True on the basis of meaning --- Good job Richard !
Re: True on the basis of meaning --- Good job Richard !