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The info on the page about Gödel's incompleteness theorems is good, but I think it would be good to define consistency in its own right, and explicate it in more detail here. What I put here right now is pretty weak, though. —Preceding unsigned comment added by 140.142.182.182 ( talk)
The list starting "systems proved to be consistent" is simply bad and evidence of confusion. If it was changed to: systems that are complete wrt. a model (in the sense of maximal complete set) this would work, but then wouldn't fit in the category. The leading paragraph isn't great either. I've shifted the page from Consistency to Consistency proof, changed Consistency into a disambig page, and rewritten it. The previous text was:
In mathematics, a formal system is said to be consistent if none of its proven theorems can also be disproven within that system. Or, alternatively, if the formal system does not assign both true and false as the semantics of one given statement. These are definitions in negative terms - they speak about the absence of inconsistency. Formal systems that do admit contradictions suffer a semantic collapse, in the sense that deductions in them cannot truly be assigned any significant content, by schemes that apply across the whole system.
To add:
I'm adding:
I plan to add later:
The title of this entry should perhaps be not just "Consistency" but "Classical deductive consistency." See the remarks below regarding "classical consistency." As for "deductive consistency," this article pertains only to deductive logic, not logic broadly construed as a study of inference in general (including, e.g., statistical inference, or Peirce's and Hanson's abductive inference). -- BurkeFT ( talk) 21:08, 28 July 2013 (UTC)
The only sense of "Consistancy" given in the article is the lack of explicit contradiction. This applies only to Classical Logic systems. (They this doesn't work, for example, for Graham Priest's Paraconsistant logics.)
This sense also fails to work at all in systems without an explict negation. (Since by that definition any such system can't generate a contradiction, since they have no negation).
However, senses that are functionally equivalent to the standard one in systems with negation, and still work in positive propositional logics, have been around for more than 70 years... Hilbert (about the time he was playing with positive propositional logics) gave a number of definitions that worked for his use.
His "Absolute Consistancy" (for example) just says that if a system can prove anything, it is inconsistant. If there are Well Formed Formula in a system that the system can not prove then it is consistant. (For traditional systems, if the system contains a contradiction then it can prove anything. And if there are WFF's in the system that it can't prove then it must not have a contradiction, since a [traditional] system with a contradiction can prove anything.)
As an example, Feys, in his 1965 book "Modal Logic" (published posthumously) established the consistancy of a number of Modal Logic systems by showing that there were WFF's in those systems that those systems could not prove. [Really short slick proofs, by the way].
I would personally find it a better if a more general sense of consistancy were used that applied to more than just classical logics. (Note that Wikipedia has a Paraconsistent logics page, so this page doesn't cover a notion that even applies to all the logics on the Wikipedia pages, much less the logics in the literature.)
Nahaj 02:55, 31 October 2006 (UTC)
I proposed here that Hilbert's second problem should be covered in its own article, separate from this one. Please discuss it there. CMummert 14:36, 5 January 2007 (UTC)
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 03:52, 10 November 2007 (UTC)
Just so that everyone knows, Gregbard recently linked hundreds of occurrences of the word "constistent" to this page. Many of the links had nothing to do with mathematical logic. I am in the process of removing those links which do not fit this context. Another avenue might be to expand the scope of this article to give a more notional definition of consistency, applicable outside the domain of mathematical logic. The current page can then be moved over to consistency (logic) or the like. At any rate, it seems fairly clear to me that, unless there is a formal system close at hand, then linking to an article on consistency in logic flies in the face of WP:CONTEXT. silly rabbit ( talk) 14:27, 1 May 2008 (UTC)
First use of the term 'self-consistent' on fictional universe links to this page, and this is similar to what's discussed above. Although it was a nice surprise to see an article on consistency in a formal context, most people probably expect the ordinary concept of consistency (which, I suppose, is very similar - e.g., "Bob's story about last night was consistent" means that Bob's story didn't entail a contradiction? Probably..) 71.202.36.15 ( talk) 07:42, 7 July 2011 (UTC)
The statement in item no. 2, "Since these theories are satisfactorily described by the model we obtain from the completeness theorem, such systems are complete" is incorrect. Usually, a theory is called complete if for any , either or can be proved from it. This indeed holds for Presburger arithmetic, but it certainly does not hold for any theory that cannot talk about its own provability relation. Take Euclidean geometry without parallel axiom, for example: it is not complete. -- Tillmo ( talk) 12:54, 18 January 2009 (UTC)
I have read the talk page, and I don't understand why an article that was almost completely about consistency proofs was moved to here, and made to serve as a general article about consistency. I think it does both the general concept of consistency, and the technical notion of consistency proof, a disservice. — Charles Stewart (talk) 13:13, 6 May 2009 (UTC)
The article says:
This seems clearly nonsense, because supposing that "φ is a theorem of Φ" (which is not defined in the article) is intended to mean "Φ ⊢ φ", every formula in the set Φ is trivially a theorem of Φ. I think what was meant here was that Φ is absolutely consistent iff there is any formula that is not a theorem of Φ. In conventional logics this is equivalent to the definition of "simply consistent" given previously, by the explosion principle. But I thought I would bring the matter up here since I am not actually familiar with the term "absolutely consistent". If nobody corrects me soon, I will fix the article. — Mark Dominus ( talk) 17:21, 30 January 2011 (UTC)
The article contains the following claim:
There are different notions of logical completeness; the footnote to Tarski's definition shows that here syntactic completeness is intended.
Now consider the following deductive system. Its formulas are formed from two variables A and B, and the unary operator ¬. It has one axiom, A, and one inference schema, "from Φ, infer ¬¬Φ". So the theory of the system consists of A, ¬¬A, ¬¬¬¬A, and so on.
Is this theory semantically consistent? Under an interpretation that assigns true to A, with the standard interpretation of the logical constant ¬, all formulas of the theory are true.
Is this theory syntactically consistent? Clearly, there is no formula Φ such that both Φ and ¬Φ are members of the theory.
Is this theory complete? No, since neither B nor ¬B is in the theory. -- Lambiam 01:50, 29 May 2012 (UTC)
what is soil — Preceding unsigned comment added by 220.227.97.146 ( talk) 04:38, 19 September 2012 (UTC)
Why do we need the detailed paragraph about completeness in the lead? It seems to violate WP:LEAD, because the history of various completeness results/proofs is not the topic of this article. Tijfo098 ( talk) 09:27, 8 November 2012 (UTC)
I would have expected instead something like Russell's paradox to be mentioned because it's classic example of inconsistency. Tijfo098 ( talk) 09:29, 8 November 2012 (UTC)
Also most of the section on FOL initially titled "Formulas" is just the completeness theorem for FOL ( sequent calculus) using Henkin's proof method instead of Gödel's cf. [1]. Tijfo098 ( talk) 09:37, 8 November 2012 (UTC)
I'm deleting the following for the same reason Travatore deleted it from Gödel's incompleteness theorems:
This is just the professional equivalent of spamming. Bill Wvbailey ( talk) 22:13, 23 November 2012 (UTC)
References
Why does consonancy redirect here? Two alternative places it would be better redirected to are Consonance and dissonance and Literary consonance. -- 184.23.18.191 ( talk) 17:38, 10 February 2016 (UTC)
In the opening sentences (emphasis mine):
Surely that can't be right. All formulas would include negation pairs. Perhaps "all tautologies" was intended?
-- phil ( talk) 06:34, 28 February 2017 (UTC)
Internal logic redirects to this article.
However, internal logic as an abstract concept refers to a whole range of things we experience in the real world such as the laws that govern everyday life: don't cross that light at red or the cop will give you a ticket! -- or the most basic instructions of a computer chip: Windows will crash because only an omnipotent being can completely understand all the possible outcomes in a set of instructions in chip architecture -- or interpersonal relationships: your life partner will be seriously ticked off if you forget to buy the cat's favourite food yet again when shopping. That plus astrophysics.
Then there is internal logic in fiction, such as Asimov's Three Laws, that a robot (1) may not injure a human... (2) must obey the orders given it by human beings except where such orders would conflict with the First Law (3) must protect its own existence as long as such protection does not conflict with the First or Second Laws. Essentially, all fiction from Gilgamesh to Groundhog Day follows some kind of internal logic. Yet Wikipedia doesn't seem to cover this concept other than in maths. As I'm a gnome, I don't dare touch this other than on a talk page. == Peter NYC ( talk) 09:51, 28 March 2021 (UTC)
This is the
talk page for discussing improvements to the
Consistency article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
![]() |
Daily pageviews of this article
A graph should have been displayed here but
graphs are temporarily disabled. Until they are enabled again, visit the interactive graph at
pageviews.wmcloud.org |
The info on the page about Gödel's incompleteness theorems is good, but I think it would be good to define consistency in its own right, and explicate it in more detail here. What I put here right now is pretty weak, though. —Preceding unsigned comment added by 140.142.182.182 ( talk)
The list starting "systems proved to be consistent" is simply bad and evidence of confusion. If it was changed to: systems that are complete wrt. a model (in the sense of maximal complete set) this would work, but then wouldn't fit in the category. The leading paragraph isn't great either. I've shifted the page from Consistency to Consistency proof, changed Consistency into a disambig page, and rewritten it. The previous text was:
In mathematics, a formal system is said to be consistent if none of its proven theorems can also be disproven within that system. Or, alternatively, if the formal system does not assign both true and false as the semantics of one given statement. These are definitions in negative terms - they speak about the absence of inconsistency. Formal systems that do admit contradictions suffer a semantic collapse, in the sense that deductions in them cannot truly be assigned any significant content, by schemes that apply across the whole system.
To add:
I'm adding:
I plan to add later:
The title of this entry should perhaps be not just "Consistency" but "Classical deductive consistency." See the remarks below regarding "classical consistency." As for "deductive consistency," this article pertains only to deductive logic, not logic broadly construed as a study of inference in general (including, e.g., statistical inference, or Peirce's and Hanson's abductive inference). -- BurkeFT ( talk) 21:08, 28 July 2013 (UTC)
The only sense of "Consistancy" given in the article is the lack of explicit contradiction. This applies only to Classical Logic systems. (They this doesn't work, for example, for Graham Priest's Paraconsistant logics.)
This sense also fails to work at all in systems without an explict negation. (Since by that definition any such system can't generate a contradiction, since they have no negation).
However, senses that are functionally equivalent to the standard one in systems with negation, and still work in positive propositional logics, have been around for more than 70 years... Hilbert (about the time he was playing with positive propositional logics) gave a number of definitions that worked for his use.
His "Absolute Consistancy" (for example) just says that if a system can prove anything, it is inconsistant. If there are Well Formed Formula in a system that the system can not prove then it is consistant. (For traditional systems, if the system contains a contradiction then it can prove anything. And if there are WFF's in the system that it can't prove then it must not have a contradiction, since a [traditional] system with a contradiction can prove anything.)
As an example, Feys, in his 1965 book "Modal Logic" (published posthumously) established the consistancy of a number of Modal Logic systems by showing that there were WFF's in those systems that those systems could not prove. [Really short slick proofs, by the way].
I would personally find it a better if a more general sense of consistancy were used that applied to more than just classical logics. (Note that Wikipedia has a Paraconsistent logics page, so this page doesn't cover a notion that even applies to all the logics on the Wikipedia pages, much less the logics in the literature.)
Nahaj 02:55, 31 October 2006 (UTC)
I proposed here that Hilbert's second problem should be covered in its own article, separate from this one. Please discuss it there. CMummert 14:36, 5 January 2007 (UTC)
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 03:52, 10 November 2007 (UTC)
Just so that everyone knows, Gregbard recently linked hundreds of occurrences of the word "constistent" to this page. Many of the links had nothing to do with mathematical logic. I am in the process of removing those links which do not fit this context. Another avenue might be to expand the scope of this article to give a more notional definition of consistency, applicable outside the domain of mathematical logic. The current page can then be moved over to consistency (logic) or the like. At any rate, it seems fairly clear to me that, unless there is a formal system close at hand, then linking to an article on consistency in logic flies in the face of WP:CONTEXT. silly rabbit ( talk) 14:27, 1 May 2008 (UTC)
First use of the term 'self-consistent' on fictional universe links to this page, and this is similar to what's discussed above. Although it was a nice surprise to see an article on consistency in a formal context, most people probably expect the ordinary concept of consistency (which, I suppose, is very similar - e.g., "Bob's story about last night was consistent" means that Bob's story didn't entail a contradiction? Probably..) 71.202.36.15 ( talk) 07:42, 7 July 2011 (UTC)
The statement in item no. 2, "Since these theories are satisfactorily described by the model we obtain from the completeness theorem, such systems are complete" is incorrect. Usually, a theory is called complete if for any , either or can be proved from it. This indeed holds for Presburger arithmetic, but it certainly does not hold for any theory that cannot talk about its own provability relation. Take Euclidean geometry without parallel axiom, for example: it is not complete. -- Tillmo ( talk) 12:54, 18 January 2009 (UTC)
I have read the talk page, and I don't understand why an article that was almost completely about consistency proofs was moved to here, and made to serve as a general article about consistency. I think it does both the general concept of consistency, and the technical notion of consistency proof, a disservice. — Charles Stewart (talk) 13:13, 6 May 2009 (UTC)
The article says:
This seems clearly nonsense, because supposing that "φ is a theorem of Φ" (which is not defined in the article) is intended to mean "Φ ⊢ φ", every formula in the set Φ is trivially a theorem of Φ. I think what was meant here was that Φ is absolutely consistent iff there is any formula that is not a theorem of Φ. In conventional logics this is equivalent to the definition of "simply consistent" given previously, by the explosion principle. But I thought I would bring the matter up here since I am not actually familiar with the term "absolutely consistent". If nobody corrects me soon, I will fix the article. — Mark Dominus ( talk) 17:21, 30 January 2011 (UTC)
The article contains the following claim:
There are different notions of logical completeness; the footnote to Tarski's definition shows that here syntactic completeness is intended.
Now consider the following deductive system. Its formulas are formed from two variables A and B, and the unary operator ¬. It has one axiom, A, and one inference schema, "from Φ, infer ¬¬Φ". So the theory of the system consists of A, ¬¬A, ¬¬¬¬A, and so on.
Is this theory semantically consistent? Under an interpretation that assigns true to A, with the standard interpretation of the logical constant ¬, all formulas of the theory are true.
Is this theory syntactically consistent? Clearly, there is no formula Φ such that both Φ and ¬Φ are members of the theory.
Is this theory complete? No, since neither B nor ¬B is in the theory. -- Lambiam 01:50, 29 May 2012 (UTC)
what is soil — Preceding unsigned comment added by 220.227.97.146 ( talk) 04:38, 19 September 2012 (UTC)
Why do we need the detailed paragraph about completeness in the lead? It seems to violate WP:LEAD, because the history of various completeness results/proofs is not the topic of this article. Tijfo098 ( talk) 09:27, 8 November 2012 (UTC)
I would have expected instead something like Russell's paradox to be mentioned because it's classic example of inconsistency. Tijfo098 ( talk) 09:29, 8 November 2012 (UTC)
Also most of the section on FOL initially titled "Formulas" is just the completeness theorem for FOL ( sequent calculus) using Henkin's proof method instead of Gödel's cf. [1]. Tijfo098 ( talk) 09:37, 8 November 2012 (UTC)
I'm deleting the following for the same reason Travatore deleted it from Gödel's incompleteness theorems:
This is just the professional equivalent of spamming. Bill Wvbailey ( talk) 22:13, 23 November 2012 (UTC)
References
Why does consonancy redirect here? Two alternative places it would be better redirected to are Consonance and dissonance and Literary consonance. -- 184.23.18.191 ( talk) 17:38, 10 February 2016 (UTC)
In the opening sentences (emphasis mine):
Surely that can't be right. All formulas would include negation pairs. Perhaps "all tautologies" was intended?
-- phil ( talk) 06:34, 28 February 2017 (UTC)
Internal logic redirects to this article.
However, internal logic as an abstract concept refers to a whole range of things we experience in the real world such as the laws that govern everyday life: don't cross that light at red or the cop will give you a ticket! -- or the most basic instructions of a computer chip: Windows will crash because only an omnipotent being can completely understand all the possible outcomes in a set of instructions in chip architecture -- or interpersonal relationships: your life partner will be seriously ticked off if you forget to buy the cat's favourite food yet again when shopping. That plus astrophysics.
Then there is internal logic in fiction, such as Asimov's Three Laws, that a robot (1) may not injure a human... (2) must obey the orders given it by human beings except where such orders would conflict with the First Law (3) must protect its own existence as long as such protection does not conflict with the First or Second Laws. Essentially, all fiction from Gilgamesh to Groundhog Day follows some kind of internal logic. Yet Wikipedia doesn't seem to cover this concept other than in maths. As I'm a gnome, I don't dare touch this other than on a talk page. == Peter NYC ( talk) 09:51, 28 March 2021 (UTC)