Talk:Paraconsistent logic

message: "Now consider a valuation V\, such that V(A,1)\, and V(A,0)\, but it is not the case that V(B,0)\,. It is easy to check that this valuation constitutes a counterexample to both explosion and disjunctive syllogism."
It really isn't that easy to check. It is not clear what the point is.
I don't really see how the statement in question differs from FDE - it seems to accept that a formula can be both true and false, or, if it does not, what precisely does it entail?
"A formula must be assigned at least one truth value, but there is no requirement that it be assigned at most one truth value." Does this not entail that LP is FDE?
I find it quite unclear. Could you at least draw out how you relate the logical statement cited above to the disjunctive syllogism?
Thanks  —Preceding unsigned comment added by 89.100.19.185 (talk) 13:50, 26 March 2010 (UTC)
thread_title:  Don't follow 

message: The article seems to suggest that the liar doesn't "explode" in paraconsistent logics. that's not true for most PLs. Kim Stebel 23:57, 5 September 2005 (UTC)
thread_title: 2005 comments

message: I removed some content from the "Motivation" section on the ground that it seemed to me not particularly well-written, and involved material that is (or ought to be) covered more thoroughly in Liar paradox and Dialetheism, both of which are prominently linked to.  Feel free to integrate this material back into the article if you feel it would be an improvement.  dbtfztalk 05:47, 25 February 2006 (UTC)
thread_title:  Removed some content from the "Motivation" section 

message: Why do you think that both weakening and double negation elimination produce explosion as stated in the article?
:Oh. I really meant all three together. I hope it is better now. -Dan 20:34, 14 June 2006 (UTC)
thread_title:  Both weakening and double negation elimination produce explosion?  

message: I'm a rookie here, that's why I decided to comment on the article instead of changing it.
1) Stating that in Aristotle's wrtings one can find some ideas concerning inconsistency-tollerant logic is ambiguous. In my opinion this statement needs at least some justification (outlining the disscusion might be too much).
2) "Not all advocates of paraconsistent logic are dialetheists" i would say that only few of them are. The picture you get depends on the references. in the references section I saw many books/articles by the representees of Australian school of paraconsistency. :)
3) As far as I know, G. Priest has two affiliations -- the Univeristy of Melbourne and St. Andrews.
4) In my opinion LP is not the most well-known paraconsistent logic. What is the criterion? What about da Costa's C?
5) "One important type of paraconsistent logic is relevance logic". According to my references, those two classes overlap. And even if we assume that this is not the case, it is paraconsistent logic which is a subset of relevance logic and not the other way.
5) Notable Figures section. Why {\L}ukasiewicz? His {\L}_{3} is paraconsistent but this wasn't intended, so to say. Even if we agree that he should be mentioned, there are others who fulfill the same criterion and are not on the list: Orlov, Kolmogorov, Lewis. The view that Vasiliev system(s) is(are) paraconsistent is widely criticized. As a forerunner should count also Carnot, shouldn't he?
6) In my opinion, Resources as well as External links section should be extended. They are not representative.Glaukon 16:23, 9 September 2006 (UTC)
replies: text: 1) the cite is in {\L}ukasiewicz if you want to add it.  This page isn't doing much in-line yet. 2) No one has stats, but its to prevent the common misconception that advocating PL requires Dialethism 3)sure, as it says on his page 4) fine re-phrase. 5) Pl and RL DO overlap, but there are PLs which are not RLs, and RLs which are not PLs, so neither is a subset of the other, nor is that implied by calling RL one type of PL. 6) Why do you think {\L}ukasiewicz didn't intend to be paraconsistent? He wrote about the priority of the law of syllogism and the law of non-contradiction in Aristotle, right?  I don't know the details of Kolmogorov or Orlov, maybe they should be added, Lewis is a weird case, look at what Rescher and Brandom argue about him.  Maybe add him, but also the Rescher and Brandom cite. 6) They are a bunch of the more recent stuff. Add more if you want.  Be BOLD! Bmorton3 14:59, 11 September 2006 (UTC)
replies: text:  Two comments... In one year Priest will no longer be affiliated with Melbourne, but will be moving to the graduate faculty at CUNY (and maintain his affiliation with the Arche research center at St. Andrew's).  Second, L3 is not paraconsistent.  What is your source on this? Paraconsistent (talk) 10:31, 21 July 2008 (UTC)

text: 1)Lukasiewicz, J. "On the Principle of Contradiction in Aristotle" reprinted in Review of Metaphysics 24 (1971), 2) No one knows the "most" claim one way or the other.  There are plenty in both camps.  Lots of PL logicians are exploring it for reasons you mention, lots are exploring it for more dialetheist reasons.  How many Jain paraconsistent logicians are there left in the Indian traditions?  I've got no clue and neither do you.  I would love to see more discussion of Indian, Brazilian, Polish, and Flemish styles, but was afraid that anything I wrote would be POV and OR to boot. 6)The cite I was thinking of was, Rescher, Nicholas and Brandom, Robert.  The Logic of Inconsistency.  New York: Basil Blackwell, 1980.  C. I. Lewis' position on paraconsistency is not entirely unrelated to his pragmatism, according to Rescher and Brandom, if I recall correctly.  Like Lukasiewicz, even if he wasn't quite aiming at PL, concerns relating to PL were guiding his thinking.  I think the argument that either were not intended to be PL would be hard to make, better to remain agnostic on it, unless you have good evidence.  Good luck! Bmorton3 19:38, 11 September 2006 (UTC)

text: Eesh Not all P are D implies that most are? I would never have thought so, but certainly phrase it, "Some P are D, and some are not", if you prefer which is neutral and in line with availible data.  Bmorton3 19:34, 12 September 2006 (UTC)
thread_title:  Some general remarks 

message: The second subsection has the headline "Paraconsistent logics are invariably weaker than classical logic" and starts with the sentence "It should be emphasized that paraconsistent logics are in general weaker than classical logic". I know nothing of paraconsistent logic, but these statements contradict each other. Perhaps a better headline would be "Paraconsistent logic vs Classical logic", or "The relative strengths of...". -Ashley Pomeroy 22:56, 9 April 2007 (UTC)
replies: text: They say the same thing, no? Paraconsistent (talk) 10:28, 21 July 2008 (UTC)
thread_title: Weakness

message: I would say that the defining feature of a paraconsistent logic should be inconsistency tolerance and not the fact that it avoids explosion. There are some proof systems around able to deal with certain inconsistent sets of premisses that do explode in the presence of a contradiction. That is, they go beyond consistency in the sense that they can get relevant information out of certain inconsistent knowledge bases and I do believe they deserve to be called paraconsistent systems even though they cannot deal with contradictory premisses {of the form 'p', 'not p'}. David--Davidpm888 (talk) 14:24, 25 November 2007 (UTC)
replies: text: The definition in terms of the invalidity of explosion is simply, technically how paraconsistent logics are classified.  See here (http://plato.stanford.edu/entries/logic-paraconsistent/) Paraconsistent (talk) 10:28, 21 July 2008 (UTC)
thread_title:  General remark: The defining feature of paraconsistency 

message: A friend of mine once showed a book about Paraconsistent logic, where the listed applications were not math-related. He was a psychologist, and he specifically wanted to study Paraconsistent logic to understand the reasoning of psychopaths - one of its applications. Other applications were the analysis of marketing, politics and religion :-) Albmont (talk) 19:56, 11 December 2008 (UTC)
replies: text: Prof. Hewitt, while I am waiting for Hewitt et.al. 2015 to arrive, I was trying out Gardenfors' idea to use topological reasoning, rather than restricting our concepts to logical reasoning --(formulation due to Norman Foo 1994). My vehicle is market failure. In particular, the segmentation by economists to rivalrous vs. nonrivalrous goods in one dimension, and to exclusive vs. nonexclusive goods in another dimension. The segmentation is logical, but Foo teaches us that there is something overlooked in this economic formulation (Foo 1994 figures 3 and 4). That something is to be determined, as far as I know. Following Foo 1994, if we examine the convexity of the concepts in market failure, we can see that the concepts hinge not only on human wants, but also on their utility. Thus market failure is not a universal, stable condition, but rather a time and situation-dependent condition. The domain of definition for market failure is not just economic, but also political, psychological, and social. It appears that the very names of the things I am seeking are currently to be determined. Might this search for domains of definition be an interest of yours? I can see how the robustness of the domains of definition might qualify as an interest of yours. --Ancheta Wis    (talk  | contribs) 08:08, 10 September 2016 (UTC)
thread_title:  Applications 

message: Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following three principles:See the article on the principle of explosion for more on this.
Disjunction introduction 	A \vdash A \lor B
Disjunctive syllogism    	A \lor B, \neg A \vdash B
Transitivity of Inference	If  A \vdash B and  B \vdash C, then  A \vdash C
Each of these principles has been challenged. 
One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. In this approach, rules of natural deduction hold, except for disjunction introduction and excluded middle; moreover, inference A⊢B does not mean necessarily mean entailment A⇒B. Also, the following usual Boolean properties hold:  double negation as well as associativity, commutativity, distributivity, De Morgan, and idempotence inferences (for conjunction and disjunction).  Furthermore, inconsistency-robust proof by contradiction holds for entailment (A⇒(B∧¬B))⊢¬B.   Carl Hewitt favours this approach, claiming that having the usual Boolean inferences, Natural deduction, Double negation elimination, Weakening for inference (If ⊢A, then B⊢A), and inconsistency robust Proof by Contradiction are huge advantages in software engineering."Carl Hewitt. Formalizing common sense reasoning for scalable inconsistency-robust information coordination using Direct Logic Reasoning and the Actor Model." in Vol. 52 of Studies in Logic. College Publications. ISBN-10: . 2015.
Another approach is to reject disjunctive syllogism. From the perspective of dialetheism, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ¬ A, then A is excluded and B can be inferred from A ∨ B. However, if A may hold as well as ¬ A, then the argument for the inference is weakened.
Yet another approach is to do both simultaneously. In many systems of relevant logic, as well as linear logic, there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism.  Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.
The three principles below, when taken together, also entail explosion, so at least one must be abandoned:
Reductio ad absurdum       	A \to (B \wedge \neg B) \vdash \neg A
Weakening for implication  	A \vdash B \to A
Double negation elimination	\neg \neg A \vdash A
Both reductio ad absurdum and the rule of weakening have been challenged in this respect, but without much success. Double negation elimination is challenged, but for unrelated reasons. By removing it alone, while upholding the other two one may still be able to prove all negative propositions from a contradiction. — Preceding unsigned comment added by Prof. Carl Hewitt (talk • contribs) 20:53, 24 August 2016 (UTC)
thread_title: Tradeoffs

message: Shouldn't the line that reads "Furthermore, inconsistency-robust proof by contradiction holds for entailment (A⇒(B∧¬B))⊢¬B." be "(A⇒(B∧¬B))⊢¬A"? — Preceding unsigned comment added by 2601:602:200:DD1D:45C:20C:ECA5:624B (talk) 02:33, 19 February 2017 (UTC)
thread_title:  I think there is a typo 

message: Suppose we are given a question in the form "Given data Γ, does condition Δ hold?" where Γ and Δ maybe compound propositions composed from atomic propositions using the logical connectives not, implies, or, and and.
In classical logic, we might proceed by testing whether Γ→Δ is a tautology (of classical logic) and whether Γ→~Δ is a tautology. If neither is a tautology, then there is insufficient data. If Γ→Δ is a tautology and Γ→~Δ is not, then the answer is yes. If Γ→Δ is not a tautology but Γ→~Δ is, then the answer is no. If both are tautologies, then the data is inconsistent and no conclusion can be drawn unless the data is modified (perhaps by deleting some parts).
If we have paraconsistent logic available, then we could re-evaluate the inconsistent case by asking the same questions with respect to tautologies of paraconsistent logic. If neither is a tautology of paraconsistent logic, then there is insufficient relevant data. If Γ→Δ is a tautology and Γ→~Δ is not, then the answer is yes. If Γ→Δ is not a tautology but Γ→~Δ is, then the answer is no. If both are tautologies of paraconsistent logic, then even the relevant data is inconsistent and we are still stumped. JRSpriggs (talk) 05:38, 9 September 2018 (UTC)
thread_title:  When and how paraconsistent logic might be used 

message: This article is not helpful to the average user who is not schooled in Logic symbols. We should give an example, or a couple of examples, in plain English without the use of symbols. Ileanadu (talk) 16:59, 1 November 2018 (UTC)
replies: text: The feature of classical logic which supporters of paraconsistent logic are trying to avoid could be illustrated by "You do not have a pet dog. If you have a pet dog, then the Moon is made of green cheese.". JRSpriggs (talk) 17:48, 1 November 2018 (UTC)

text: The point is that from a false statement (such as a contradiction), one can (in classical logic) deduce anything including statements which are irrelevant and absurd. Some people are uncomfortable with this, especially if they want to be able to entertain contradictions, so they invented paraconsistent logic to get around it. JRSpriggs (talk) 19:00, 2 November 2018 (UTC)
thread_title:  Need a plain English example - Badly