Groups | Search | Server Info | Login | Register

Groups > comp.theory > #21431

Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism)

Subject Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism)
Newsgroups comp.theory, comp.ai.philosophy, comp.ai.nat-lang, sci.lang.semantics
References (19 earlier) <rdmgi6$r2m$1@dont-email.me> <bI2dnR28Eult8mLDnZ2dnUU7-LvNnZ2d@giganews.com> <rdnv1m$mog$1@dont-email.me> <gcKdnZSGeccdHmLDnZ2dnUU7-IHNnZ2d@giganews.com> <rdo1ft$6rl$1@dont-email.me>
From olcott <NoOne@NoWhere.com>
Date 2020-07-03 15:59 -0500
Message-ID <vbSdnTJVX8fMAWLDnZ2dnUU7-XnNnZ2d@giganews.com> (permalink)

Cross-posted to 4 groups.

Show all headers | View raw

On 7/3/2020 2:38 PM, André G. Isaak wrote:
> On 2020-07-03 13:14, olcott wrote:
>> On 7/3/2020 1:56 PM, André G. Isaak wrote:
>>> On 2020-07-03 11:50, olcott wrote:
>>>
>>>> Valid in model theory and valid in deductive inference have two 
>>>> distinctly different meanings.
>>>>      Satisfiability
>>>>      A formula is satisfiable if it is possible to find an 
>>>> interpretation
>>>>      (model) that makes the formula true.
>>>>      https://en.wikipedia.org/wiki/Satisfiability
>>>>
>>>>      Interpretation (logic)
>>>>      An interpretation is an assignment of meaning to the
>>>>      [non-logical] symbols of a formal language.
>>>>      https://en.wikipedia.org/wiki/Interpretation_(logic)
>>>>
>>>>      Model theory
>>>>      A model of a theory is a structure (e.g. an interpretation)
>>>>      that satisfies the sentences of that theory.
>>>>      https://en.wikipedia.org/wiki/Model_theory
>>>>
>>>>      A wf B is said to be logically valid if and only if B is true
>>>>      for every interpretation. (Mendelson 2015:65)
>>>>
>>>> Mendelson, Elliott 2015. Introduction To Mathematical Logic. Boca 
>>>> Raton: CRC Press Taylor & Francis Group.
>>>>
>>>> A theory T is incomplete if and only if there is some sentence φ 
>>>> such that (T ⊬ φ) and (T ⊬ ¬φ).
>>>>
>>>> This is the first time that I have every applied logical validity 
>>>> within the context of symbolic logic so I may not have it correctly:
>>>>
>>>> It seems that within the context of: (T ⊬ φ) and (T ⊬ ¬φ) that
>>>> those instances of φ in T would be defined as logically_invalid:
>>>>
>>>> ∀φ ∈ theory T (T ⊢ φ)  otherwise logically_invalid(T, φ)
>>>> ∀φ ∈ theory T (T ⊢ ¬φ) otherwise logically_invalid(T, ¬φ)
>>>>
>>>> I may have this incorrectly because this is the very first time that 
>>>> I applied the concept of validity within symbolic logic, but it 
>>>> looks like a formal system is defined as incomplete whenever it has 
>>>> a WFF φ such that Logically-Invalid(T, φ) and Logically-Invalid(T, ¬φ).
>>>
>>> I have no idea what you think 'invalid' means, but what you are 
>>> saying above is essentially nothing unless you have some special 
>>> meaning in mind.
>>>
>>> (A ∨ ¬A) is valid because it true on all possible interpretations.
>>>
>>> (A ∨ B) is not valid because it is not true on all possible 
>>> interpretations. It is therefore invalid.
>>>
>>> You seem to want invalid to mean not-well-formed, but that isn't what 
>>> it means. It simply means that there are interpretations under which 
>>> something is false.
>>>
>>> André
>>>
>>
>> Semantically ill-formed.
>> ("2 + 3 = 5", "5 > 3"} therefore the Sun has spots.
> 
> So according to you (A ∨ B) is 'semantically ill-formed'? In what way? 
> What well-formedness rule is being violated by this?
> 
> What about x + y = y + x ? That's not provable in Q, but how exactly is 
> that 'semantically ill-formed'?
> 
> What about ¬(¬A ∧ A)? That's provably true, but it involves negating a 
> contradiction. According to you (¬A ∧ A) is 'semantically ill-formed' 
> since it isn't provable (at least not in any consistent system given 
> that it is a contradiction). But if (¬A ∧ A) is 'semantically 
> ill-formed' how can we possibly negate it to get the perfectly valid 
> statement ¬(¬A ∧ A)?
> 
> André
> 

Here is a superb example of semantic well-formedness:

In 1847 George Boole changed this arrangement. For example, to validate 
the argument

All monarchs are human beings. No human beings are infallible. Therefore 
no infallible beings are monarchs.
Boole would interpret the symbols P,Q,R as names of classes:

P is the class of all monarchs.
Q is the class of all human beings.
R is the class of all infallible beings.

Then he would point out that the original argument paraphrases into a 
set-theoretic consequence:

(P ⊆ Q),(Q ∩ R = 0) ⊨ ( R ∩ P = 0)

https://plato.stanford.edu/entries/model-theory/#Cons

Here is the same example adapted to be ill-formed.
(P ⊆ Q),(Q ⊇ P) ⊨ (R ∩ P = 0)

X = P ⊆ Q
Y = Q ⊇ P
Z = R ∩ P = 0

{X,Y} ⊬ Z and {X,Y} ⊬ ¬Z



-- 
Copyright 2020 Pete Olcott

Back to comp.theory | Previous | Next — Previous in thread | Next in thread | Find similar

Thread

Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-26 20:03 -0500
  Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-26 20:53 -0700
    Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-26 23:46 -0500
      Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-26 22:54 -0700
        Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 11:04 -0500
    Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-06-27 10:24 +0100
      Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 11:14 -0500
        Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-27 09:38 -0700
          Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 11:58 -0500
            Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-27 11:54 -0700
              Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 15:26 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-27 14:52 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 17:09 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-27 17:43 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 19:46 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-27 20:15 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-27 22:11 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-28 16:32 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 David Kleinecke <dkleinecke@gmail.com> - 2020-06-28 14:43 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 olcott <NoOne@NoWhere.com> - 2020-06-28 16:59 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 André G. Isaak <agisaak@gm.invalid> - 2020-06-28 20:18 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-28 21:45 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-28 21:27 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-28 21:59 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-29 00:48 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-28 23:57 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-29 13:28 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-29 12:33 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-29 14:33 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-29 13:52 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-29 15:02 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-29 14:14 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-29 15:39 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-29 15:13 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-29 17:09 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-29 21:21 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 01:09 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-30 00:24 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 16:46 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-30 16:46 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 18:24 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-30 18:15 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 19:33 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-30 19:18 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 20:34 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-30 20:06 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 21:31 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-06-30 20:54 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-01 11:03 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-01 10:35 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-01 17:35 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) Chris Buckley <alan@sabir.com> - 2020-07-01 22:47 +0000
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 07:01 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 10:51 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 10:46 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 13:59 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 13:37 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-07-02 08:53 +0100
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 09:52 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) Chris Buckley <alan@sabir.com> - 2020-07-02 15:06 +0000
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 10:56 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) Chris Buckley <alan@sabir.com> - 2020-07-02 22:52 +0000
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 18:35 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) Chris Buckley <alan@sabir.com> - 2020-07-02 23:48 +0000
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 18:57 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) David Kleinecke <dkleinecke@gmail.com> - 2020-07-02 19:22 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-02 22:03 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 23:13 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 00:36 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-02 23:43 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 12:50 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 12:19 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 13:21 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 12:33 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 13:48 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 13:55 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 13:06 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 13:07 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 12:56 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 14:14 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 13:38 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-03 15:59 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-03 22:54 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-04 08:56 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-04 08:12 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-07-04 10:06 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) André G. Isaak <agisaak@gm.invalid> - 2020-07-04 09:31 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Text Analysis) olcott <NoOne@NoWhere.com> - 2020-07-04 14:31 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Text Analysis) André G. Isaak <agisaak@gm.invalid> - 2020-07-04 15:17 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V20 (Text Analysis) olcott <NoOne@NoWhere.com> - 2020-07-04 17:29 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-05 13:01 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-05 16:40 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-05 23:52 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-06 00:53 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-06 10:15 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) David Kleinecke <dkleinecke@gmail.com> - 2020-07-06 08:33 -0700
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-06 11:49 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) André G. Isaak <agisaak@gm.invalid> - 2020-07-06 11:27 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-07 22:22 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) André G. Isaak <agisaak@gm.invalid> - 2020-07-06 10:48 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) André G. Isaak <agisaak@gm.invalid> - 2020-07-06 13:55 -0600
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-07 22:13 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V23 (Newspeak) olcott <NoOne@NoWhere.com> - 2020-07-07 22:44 -0500
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) Alan Smaill <smaill@SPAMinf.ed.ac.uk> - 2020-06-30 11:17 +0100
                Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V13 (Isomorphism) olcott <NoOne@NoWhere.com> - 2020-06-30 08:44 -0500

csiph-web