Talk:Paraconsistent logic

2005 comments

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)

What might we say about the choice between the man's utterance of "I am currently in the room" as opposed to its negation "I am not currently in the room" (1998)? Perhaps that the actual negation should be "It is not true that I am currently in the room". This is a false choice like that between "P is true" and "The negation of P is true": It could be that P is in fact unprovable, and the actual choice should be between "P is true" and "It is not the case that P is true". 203.116.59.23 13:55, 20 September 2005 (UTC)Joel Tay

Removed some content from the "Motivation" section

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. dbtfz^talk 05:47, 25 February 2006 (UTC)

Recent addition of material on direct logic

I removed the recently added content on "direct logic" since no evidence is provided indicating that it is more notable than any of the many other systems of paraconsistent logic that have been proposed over the years. We can't include all of them, so only a few of the most notable and important ones are discussed. (Remember: this article is aimed primarily at non-specialists.) For example, LP is discussed briefly in the article because it is easily the most influential and well-known paraconsistent logic. And dual-intuitionistic logic is discussed (very briefly) because it illustrates the relation of paraconsistent logic with another notable type of logic, intuitionistic logic. If someone wants to add a brief characterization of da Costa's C-systems, that would be appropriate.

Quite frankly, the addition of the material on direct logic looks like a blatant case of a researcher trying to promote his or her own work with little or no regard for whether the material is actually appropriate for the article. If anyone really believes that the material I removed should be in Wikipedia, I recommend creating a new article, Direct logic (cf. Relevance logic), which could be linked to from this article. Also, a very brief mention of direct logic in this article might be approprite, if done right. dbtfz^talk 18:40, 1 April 2006 (UTC)

• I removed the following:

Classical mathematical logic was devised for mathematical theories which are consistent and consequently make heavy use of the principle of indirect proof which can expressed formally as

$(B\vdash A,\lnot A)\vdash \lnot B$

which expresses the intuitive idea that if a contradiction follows from $B$, then $B$ must be false. The principle "from a contradiction, anything follows" can be derived from the principle of indirect proof. Some paraconsistent logics (such as Direct Logic) allow a more limited form of indirect proof called direct indirect proof which can be expressed formally as

$(B\vdash \lnot B)\vdash \lnot B$

which expressed the intuitive idea that if $B$ infers its own negation, then it cannot hold. Unlike full indirect proof, the principle of direct indirect proof can be used without causing explosion.

This section is basically out of synch with what the rest of the article says. It is also, I think, not quite correct. Ex contradictione quodlibet has as much to do with the rule of weakening as with reductio ad absurdum. Relevant logic rejects the rule of weakening, for instance. -Dan 14:31, 5 June 2006 (UTC)

Article lacks motivation for Ex contradictione quodlibet

The article currently lacks motivation for Ex contradictione quodlibet, so it proposed to add the following (which is the heart of the matter):

The above principle of "from a contradiction, anything follows" may not seem very intuitive. So why is it included in classical logic? Classical mathematical logic was devised for mathematical theories which are consistent and consequently make heavy use of the principle of indirect proof which can expressed formally as

$(B\vdash A,\lnot A)\vdash \lnot B$

which expresses that if a contradiction follows from $B$, then $B$ must be false which is reasonable for consistent theories.

The principle of indirect proof can be used to derive the principle that "from a contradiction anything follows" in the following way in classical logic:

$A,\lnot A\vdash (\lnot B\vdash A,\lnot A)\vdash \lnot \lnot B\vdash B$

Now if the above is added, then the article ought to also mention direct indirect proof as well because it is a closely related principle that does not cause explosion.

• Except it looks like the above derivation also makes use of the rule of weakening (and in fact double-negation elimination). Your paragraph makes it sound like indirect proof is the motivation for ex contradictione quodlibet, and the only way to avoid it is to reject or restrict indirect proof. This goes against what the rest of the article about disjunctive syllogism. I also think it is flat out wrong. Now I don't mind a brief mention of direct logic, but maybe we can do better than this? -Dan 20:58, 5 June 2006 (UTC)

• Of course, both weakening and double negation elimination are part of classical logic. The question being addressed is the motivation for including Ex contradictione quodlibet in classical logic. Indirect proof is certainly one of the most important motivations for Ex contradictione quodlibet.

• It turns out that rejecting (or restricting it as in Direct Logic) indirect proof by itself is not sufficient to prevent explosion. As the article points out, many paraconsistent logics (including Direct Logic) reject disjunction expansion as well. It turns out that restricting indirect proof and rejecting disjunction expansion is sufficient to prevent explosion. Imposing both of these seems to be one of the fundamental ideas behind Direct Logic.

• But it certainly can be improved!

Well, we'll have to leave this until we get some clarification on the system in question. If you haven't already, you might want to pop over to Talk:Direct logic. -Dan 15:09, 7 June 2006 (UTC)

Both weakening and double negation elimination produce explosion?

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)

How exactly is Paraconsistent Logic weaker?

The article says that paraconsistent logics deem fewer inferences valid, and rarely deem inferences valid that classical logic doesn't. I always thought that standard logics restricted themselves to making claims about inferences concerning statements whose atoms were true or false, but not both or neither, but that paraconsistent logics validate inferences over a far broader range of statements. That means they validate lots and lots of inferences about which standard logics say nothing one way or the other right? This means that the logical truths of LP are not "precisely those of classical propositional logic" because they include truths of the form p triple bar p, where p is an atom which is both true and false. This is a logical truth of LP, but it is not even in the domain of consideration for standard logic. By broadening the truth assignments we're are broadening the possible worlds under consideration, but that means we get to endorse truths about which we previously said nothing, even though we are endorsing them by using the same propositional variable we used before, right? Am I misunderstanding something? Bmorton3 18:05, 27 July 2006 (UTC)

In paraconsistent logics, one may use a set of inconsistent premises which would explode in classical logic without suffering such an explosion. So to allow more (inconsistent) assumptions, one is sacrificing some of the deductive power of classical logic. If one did not, then the explosion would occur and the theory would become trivial, i.e. every proposition would be a "theorem" in the theory. Is that clear? JRSpriggs 05:36, 28 July 2006 (UTC)

One way of looking at it is that propositions actually mean less than they seem to mean in paraconsistent logics. JRSpriggs 05:44, 28 July 2006 (UTC)

One is sacrificing some deductive power by giving up some inferences you could make in standard logic, in return for gaining deductive power of a different kind, being able to make inferences at all over domains you couldn't touch safely before. We are weaker in some ways but stronger in others. Even inferentially, we give up some inferences, but can also make inferences where previously we had to keep silent. We can cash out what we've given up in lots of different ways (and it often varies on our approach), maybe we weakened the power of adjunction, or of negation, like in the Polish or South American stuff. Maybe we weakened the conditional, like in the relevance stuff, or we weakened the assertion stroke by going non-monotonic and allowing retractions like in the Batens stuff. You could say we have weakened the meanings of the propositions (although even here, we weakened the meanings of one or more of the connectives, but in return bought ability to have broader meanings in our atoms!). You could say instead that we have retained the same meanings for the connectives but have changed our valuation system. In that case the propositions haven't changed their meaning at all, its just that the meaning of true and false has change. But on any other these we've weakened some things, but gained strength in others. It looks to me like Classicist POV to say that we have fewer inferences in a paraconsistent logic; classical logic endorses inferential formulae that paraconsistent logic does not, but paraconsistent logic in turn endorses application of inferential formulae to particular inferences that classical logic does not because of the domain changes (and not just a few, but lots). We have to weaken something to escape explosion, but that doesn't mean we wind up being overall weaker (even in the strictly inferential sense) at the end of the day. This is still an issue of trade-offs not of being "invariably weaker" Right? Am I being partisan here, or misunderstanding something? 'Cause I've almost convinced myself to demote the "invariably weaker" paragraph and integrate some of it into the trade-off stuff Bmorton3 13:35, 28 July 2006 (UTC)

I do not understand what extra power you think paraconsistent logics have (other than tolerating inconsistent assumptions, as I said). Please give a specific example of an inference which you could do in a paraconsistent logic which you could not do in classical logic. JRSpriggs 04:10, 29 July 2006 (UTC)

It includes atomic claims which are at once both true and false in its domain, which is forbidden by the valuation rules in a classical logic. See my example below. Bmorton3 16:42, 31 July 2006 (UTC)

The article as it stands reads:

It should be emphasized that paraconsistent logics are in general weaker than classical logic; that is, they deem fewer inferences valid...

So I think it is clear weaker is to be understood in a strictly technical sense. Classical logic does, in fact, allow you to work with incoherent assumptions, and it allows you to derive any conclusion. Now, in the above discussion, there seems to be some implication that classical logic does not make claims about what happens in these situations. Technically, it does! True, it might then be noticed that the classical conclusions are useless. But, this sort of judgement is not in itself a rule of classical logic. For one thing, it is not always obvious when assumptions are incoherent.

Classical logic makes claims about what happens when multiple formulae conflict with each other, but each is consistently evaluatable, like the explosion claims, (A&~A)horseshoe B. Both classical and paraconsistent logics deal with this claim. Part of the value of paraconsistent logic is the weakening of explosion. Classical logic does make claims about what happens when contradictory WFFs are conjoined, namely we get triviality. But it makes no claims about what happens when an ATOM is both true and false. In that sense, it does not allow one to work with incoherent assumptions. Part of the value of paraconsistent logics is the loosening of the restrictions on valuation. Consider some very basic inference, like &E, A&B therefore B. Classical logic endorses &E, only over A's and B's which are true or false but not both. Most paraconsistent logics will endorse it over A's and B's that are true, false or both (and some probably endorse it for the neither case as well). So consider the inference "This clause is false; and it is snowing outside, therefore it is snowing outside." Classical logic says nothing about this inference because the first clause isn't in it's domain of discourse. Paraconsistent logics endorse the inference. For each inference schema that classical logic endorses, and paraconsistent logic doesn't reject, we can create an inferential instance that paraconsistent logic will endorse, but classical logic will stay silent on by using an atom that is both true and false (if any such atoms exist). So if the dialethists are right, there are lots of inference instances that paraconsistent logics endorse, but classical logics stay silent on. Paraconsistent Logics are not weaker full stop, they are inferentially weaker in someways and stronger in others. Bmorton3 16:42, 31 July 2006 (UTC)

I think I see where you're coming from now. In the article, PL is presented in terms of semantics and not syntax. A paraconsistent system of logic will have a broader class of semantic models. But in fact, weaker inference rules should always result in a broader class of semantic models, and vice versa. Inferences are made with formulas though, not valuations. So, "This clause is false; and it is snowing outside, therefore it is snowing outside" is classically valid, assuming we expand it to allow self-reference in formulas. Of course allowing self-reference makes classical logic, and many systems of paraconsistent logic, explode. 192.75.48.150 17:23, 31 July 2006 (UTC)

PS: I also sense you might also find the word weaker derogatory in this context. But I'm pretty sure that even advocates of paraconsistent logic use it, exactly the way we are using it. 192.75.48.150 17:42, 31 July 2006 (UTC)

First, no, folk who study Paraconsistent logics use stronger and weaker in lots of different senses, but the main one seems to be the distinction between strong paraconsistency and weak paraconsistency. See Bremer 2005, chapter 1. The worry isn't that you're being derogatory, but that your making "strong" paraconsistency look less "strong" than classical logic, when it isn't. In this sense, classical logic is significantly weaker than weak paraconsistency. Strong paraconsistency is strong because it endorses claims about inferences in even very non-normal models (such as nes where contradictions are finally provable). Now you are right that there is a sense in which strong paraconsistency is inferentially weaker than classical, but that's not the main sense in which the terms strong and weak are used in paraconsistency, and even inferentially, there is a sense in which strong paraconsistent logics can make claims about inferences that classical cannot.

Second, inferences are not made with formulae or valuations, they are made with premises in language, but understanding how formulae link up with premises in a language is an issue of syntax and semantics working together. Formal formulae can be thought of in 2 distinct ways, as themselves being premises in a formal language, or as attempts to represent formally premises in an informal language. If you think of a formulae as simply being elements of the formal language alone, then its hard to make any claims about stronger and weaker, because every formal langauge is going to use its own set of abstract atoms, otherwise we could create WFFs with elements from multiple logical systems. Thus A&B therefore A, in classical logic and A&B therefore A in paraconsistent logic are incommensurable, because the A's and B's are elements of different languages and should really be written A(CL)&B(CL) therefore A(CL), and A(PL)&B(PL) therefore A(PL). If instead we look at A&B as stand-in's for elements of a natural language, then we can't divorce the syntax and semantics so quickly. The semantics is part of how we decide how to translate from the one langauge to the other.

Third, classical logic does not endorse the inference "This clause is false, and its snowing outside, therefore its snowing" for 2 distinct but relevant reasons. First, the clause is self-referential and there is no good way to represent that in any n-order logic, and there is no good way to "expand classical logic to allow self-reference in formulae" and this is a key motivation for paraconsisent logics. But second, if you attempt to run a valuation of a set of claims including these claims using the classical rules for valuation, the expression "this clause is false" cannot be evaluated "suitably" (and on most def's this will mean it can't even be represented with a WFF, and that will mean it CAN'T explode CL). Most, well structured classical logics treat "This clause is false, and its snowing outside" the same way tehy treat &&&A, as a non-WFF, and therefore trivial in the other sense (implying nothing). Removing inference schema does in general mean broadening the possible models, but this just IS a species of strengthening the inferences allowed; that's the heart of my "derogatory" worry, I guess, if we don't treat broading the range of statements that are allowed to be represented with WFFs as a form of increasing the inferences allowed, then the main point of most non-classical projects is lost.

Fourth, The fact that the syntax-semantics distinction just isn't that clear cut for natural languages is a huge part of the Dialethists agenda and motivation, See Bremer chapter 2, or Priest 1979.Bmorton3 19:24, 31 July 2006 (UTC)

Erm, hmmm. I don't have the book you cite at hand, but "weak" as an adverb, or adjective applied to a property of a formal system, as in "X is weakly P" and "X is strongly P" (or "X has weak P-ity" and "X has strong P-ity") is something quite different from when the adjective is applied directly as a comparative between formal systems ("X is weaker than Y"), where I think it is understood the way I've been saying. In the context of substructural logics, for instance, I don't think it's controversial to say that linear logic is weaker than system R which is weaker than classical logic. Despite the fact that conjunction "splits", which is what I think (but I'm not sure) that you are referring to in point 2, we can still make good sense of the statement. A web-acccessible example is Restall's relevant and substructural logics, where in section 2.1 he talks about Church's "weak implication", further down "Meredith and Prior were... looking for logics weaker than classical propositional logic...", in 2.2 "So, a logic of entailment must be weaker than R. However, it need not be too much weaker", and so on. He never defines what is meant, but it is perfectly clear, and Restall is hardly pushing a Classicist POV (his POV is succinctly declared in section 2.4.2). As to allowing self-reference, this may be a motivation for some, but certainly not system R, for instance. I did oversimplify with the syntax/semantics bit, and I hope I have not insulted your intelligence by doing so, but still when we're talking about what inference rules are valid or admissible in a formal system, we are saying something specific about its consequence relation. In this respect we are perfectly in accord with the practice in, say, set theory, where we can assert that CZF is weaker than ZF, in turn being weaker than ZFC. This assertion is not confusing or ambiguous, and does not offend constructivists in any way, notwithstanding the fact that obviously CZF would therefore have a broader class of models than ZF, in turn than ZFC (assuming ZFC is consistent, blah blah blah) 192.75.48.150 20:10, 2 August 2006 (UTC)

You're right about "X is weakly P," being different from "X is weaker than Y," and further "X is weaker than Y in respect R" is different from either. Er the web location you cite seems to be a citation record, but lack the actual article (or I couldn't find it), but you are certainly right that "x is weaker than Y" is a normal locution when comparing odd formal logics. Sigh. I guess I am failing to articulate my point, and it is probably too subtle for the purposes of WP anyway (or is wrong). There is a perfectly good mathematical logic sense in which everything you say is right, I just worry that it obscures a point in philosophical logic. CL must endorse sequent schema which any PL does not in order to escape explosion (although which ones can vary). If that is all we mean by "X is weaker than Y" everything is rosy. But a sequent isn't an inference. An inference in a pragmatic notion that requires claims being used in a particular way, and this kind of pragmatics has to have feet in syntax and semantics. PL endorses inferences that CL does not even though it does not endorse sequents that CL doesn't (or rarely does). If you say ZF is weaker than ZFC, we might just let it go as a summary, or we might say "in what sense?" If you say ZF is weaker than ZFC in that claims are provable in ZFC that are undecided in ZF, well right, but there are also claims about choice-less worlds which are expressible and provable in ZF, but inexpressible in ZFC, at least if a claim can be about a world or a model. There is a real sense in which ZFC is weaker than ZF, even though there is also a real sense in which ZF is weaker than ZFC. If you say ZF is weaker than ZFC in terms of the claims it makes, well yes and no. But we're just going in circles now, and I'm no longer convinced my point can contribute to this article, sigh. Bmorton3 21:09, 3 August 2006 (UTC)

Feel free to clarify if you like, but please don't demote it too much, because I think the point is important: paraconsistent logic works in the domains for which it is intended precisely because it is weaker. 192.75.48.150 15:46, 31 July 2006 (UTC)

PS doesn't the article give the "or" rules for PL twice rather than the or and the and rules? Bmorton3 16:42, 31 July 2006 (UTC)

I think it is correct. There are supposed to be two "or" and two "not" lines. "And" is to be defined in terms of these. 192.75.48.150 17:23, 31 July 2006 (UTC)

You're right, I miss understood what you meant by the "in other words" just after that, I thought you were going one to spell out how to do so, rather than re-stating the same point again. my error. Bmorton3 17:57, 31 July 2006 (UTC)

Some general remarks

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)

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)

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)

OK, I'll became bold soon -- I just need to think how to put things in the most appropriate order. Now -- just a reply. 1) you mean his "On the Principle of Contradiction"? I thought that you are rather referring to G. Priest interpretation of Aristotle's standpoint. Anyway, that's a fishy subject. I'll check the references. 2) OK, but, as far as I know, most of paraconsistent logicians are not "happy" with contradictions. They just want to explore the subject, to invent a tool for dealing with it and eventually get rid of the contradictions (Brazilian, Polish, Flemish schools) -- most of them do not connect their works with any ontological convictions concerning contradictions. That's is why this passage should be rewritten -- it suggests that dialetheism is some leading option among p. logicians. 5) Right. But, again, the article says different things. 6) I said that his system was not intended as paraconsistent. It was Jaśkoski who showed that in Ł3 the explosion law is not valid. Again, I'll consult the original text, but I think that Łukasiewicz was merely criticizing the principle of contradiction and he was not stating that it should be invalidated. Anyway, I do consider him as a forerunner of paraconsistency but others that I've mentioned should also be put on the list. Unfortunately I don't know much about Rescher's work. All I know that Lewis' system of strict implication is paraconsistent (the law is not valid there). It was not intended (as in the case of Łukasiewicz).Glaukon 16:58, 11 September 2006 (UTC)

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)

The reason why I've asked for a rationale for the presence of Łukasiewicz on the list was that: "OK, I do agree that this guy contributed a lot and there are strong reasons for treating his ideas as some sort of anticipation of this what is happening within the modern paraconsistent movement, BUT if he is on the list, for the same reasons, there should be a bunch of other guys, and which were NOT there". About dialetheism. Of course there was no research on Plogicians' ontological convinctions. But this argument also applies to the statement that "Not all p. logicians are dialetheists" which implies that most of them are. I can say something about Polish school, and to some extent about Flemish and Brasilian ones. In Poland dialetheism does not win much attention. On a contrary. The same goes, I believe, for the Flemish and Brasilian approaches (this view is confirmed in Takanaka's paper -- Three Schools of Paraconsistency, it's avaliable online, as far as I know). You've mentioned Jaina's logicians. I do agree that this is an extemaly interesting subject and some historical background of paraconsistency is something which is missing in this article too.Glaukon 16:17, 12 September 2006 (UTC)

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)

You're right when it goes about (classical;)) logic. But this is about rhetoric and some kind of impression. If I say "There are Microsoft operating systems which are unstable. Not all Microsoft operating systems are unstable." It depends on the context and many other parameters but one of possible interpretations is that I imply that most of those OSs are unstable, isn't it? All I'm saying is that an article should be reader-oriented and if there is a risk (commonsense assumed:)) of misunderstanding of some fragmet, an author should change it.Glaukon 08:17, 13 September 2006 (UTC)

Weakness

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)

They say the same thing, no? Paraconsistent (talk) 10:28, 21 July 2008 (UTC)

General remark: The defining feature of paraconsistency

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)

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)

Weakness Again

I notice there has not been much discussion lately, so I am going to put this in a new section. On the topic of weakness, some folks had raised some prior complaints. For the most part these complaints are based on misunderstanding. However, there is a real problem with the formulation in the current article. It seems that in response to some worries about how 'weakness' was being interpreted, an addendum was created which states " a paraconsistent logic may validate nonpropositional inferences that are classically invalid... In this way a paraconsistent logic can prove more than classical logic." This is a possibly very misleading way to state the point. What people have in mind is the following phenomenon: given the resources of arithmetic and Godel's technique for arithmetization of syntax, we can define a coding over a formal language which allows sentences of that language to effectively ascribe predicates to the very sentences of that language. This produces something like the natural language phenomenon of 'self-reference' and is the reason why Godel sentences are possible which can roughly be paraphrased as equivalents of ordinary sentences like "I am false", that is we have formal counterparts of the Liar and other such paradoxical sentences. Because these sentences are intrinsically explosive, they cannot even be expressed in a language underwritten by classical logic and as Tarski concluded, this shows that a language underwritten by classical logic cannot contain its own truth (or falsity) predicate. What the paraconsistent language can do is precisely get around this problem. So to my mind, the most perspicuous way to contrast the power of a paraconsistent logic to its classical rival is to emphasis this issue: expressibility. I propose replacing "a paraconsistent logic can prove more than classical logic" with some talk about how language underwritten by paraconsistent logic have greater expressive strength than those underwritten by classical logic. I will make these changes in a few days if there are no objections. Paraconsistent (talk) 10:41, 21 July 2008 (UTC)

You might want to take a look at the following:

• Carl Hewitt. Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency in "Coordination, Organizations, Institutions, and Norms in Agent Systems III" edited by Jaime Sichman, Pablo Noriega, Julian Padget and Sascha Ossowski. Springer. 2008.

--67.169.145.74 (talk) 16:27, 21 July 2008 (UTC)

Could you maybe just summarize what you find to be important about this article?

-Paraconsistent (talk) 00:15, 22 July 2008 (UTC)

The article points out that Gödel stopped short in the following sense: in the context of unstratified reflection, his argument leads to an actual inconsistency in mathematics. (Of course, Gödel assumed stratified metatheories.) However, the other paradoxes are apparently blocked, e.g., Russell, Curry, Liar, Kleene-Rosser, etc. Also a generalization of Löb's theorem is proved.--67.169.145.74 (talk) 07:20, 22 July 2008 (UTC)

Okay so how does this bear on the current discussion?

-Paraconsistent (talk) 13:02, 22 July 2008 (UTC)

The section is even more misleading in that paraconsistency is motivated by tolerance of inconsistency, not necessarily by tolerance of self-reference. In fact I think the example systems on this page would all fall to paradoxical sentences, yet still quailfy as paraconsistent. --EmbraceParadox (talk) 16:11, 21 July 2008 (UTC)

Point taken, we could just remove the discussion of the 'strengths' of paraconsistent logic, as you say it not clear that self-reference is the most important issue in this respect.

-Paraconsistent (talk) 00:15, 22 July 2008 (UTC)

Incompleteness theorem

Set theory and the foundations of mathematics (see paraconsistent mathematics). Some believe that paraconsistent logic has significant ramifications with respect to the significance of Russell's paradox and Gödel's incompleteness theorems.

We're going to need details here, because I find this very difficult to believe, and the article on paraconsistent mathematics makes no such claim. The incompleteness theorem can be expressed without logic at all. --EmbraceParadox (talk) 16:22, 21 July 2008 (UTC)

There is a good discussion at this SEP article. The basic work on the usefulness of paraconsistent logic to formulations of naive set theory (set theory which can tolerate the existence of the Rusell Set amongst other things) and to Godel's Theorems has been done in the context of relevant logics. Look for work by Meyers and Brady for starters.

-Paraconsistent (talk) 00:18, 22 July 2008 (UTC)

The SEP article is unhelpful. It says

The heart of Gödel's theorem is, in fact, a paradox that concerns the sentence, G, ‘This sentence is not provable’ ... If an underlying paraconsistent logic is used to formalise the arithmetic, and the theory therefore allowed to be inconsistent, the Gödel sentence may well be provable in the theory ... So a paraconsistent approach to arithmetic overcomes the limitations of arithmetic that are supposed (by many) to follow from Gödel's theorem.

But as I say, the incompleteness theorem does not need logic. To make my point I think it will be enough to arithmetize the "not" in the Gödel sentence: "If this sentence is provable, then 0=1." If 0=1, then wouldn't all numbers be equal? x = 1*x = 0*x = 0 = 0*y = 1*y = y. This is arithmetic, and not logic, so it seems to me that paraconsistent logic will be of no help against 0=1. So how could a paraconsistent approach to arithmetic prove that Gödel sentence and not be trivial? --EmbraceParadox (talk) 15:07, 22 July 2008 (UTC)

Sorry you are right the SEP article is too short. I should have looked at it more closely. But you clearly see the thrust of the interest in paraconsistent arithmetic. Put it this way: the paradigm Godel sentence "this sentence is unprovable" can be proven in an inconsistent arithmetic underwritten by a paraconsistent logic without trivializing the theory. That is an interesting result, no? Your concern seems to be (if I follow) that paraconsistent arithmetic might still be faced with incompleteness even if it makes some advance over classical arithmetic. But I don't think I understand your example. The sentence "If this sentence is provable, then 0=1" should not be provable in any context, right? Even in an inconsistent-but-non-trivial arithemtic underwritten by paraconsistent logic, this sentence could simply be unprovable given that the consequent of the sentence is false. You seem to be suggesting that we want an arithmetic theory which proves this sentence as a theorem, but that strikes me as wrong. Am I missing something here?

-Paraconsistent (talk) 15:25, 22 July 2008 (UTC)