Talk:Principle of explosion

False Proof?

it appears to me the proof is false. it just seems to demonstrate modus tollendo ponens which is:

       A	B		¬A		A∨B   <-> 	¬A→B
       f	f		t		f		      f
       f	t		t		t		      t
       t	f		f		t		      t
       t	t		f		t		      t

what we want to prove however is:

    A∧¬A  → B 

which we can write as

  ¬(A∧¬A) ∨ B

which, by de morgan's rules, is

   (¬A∨A) ∨ B

checking the truth table:

       A B  (¬A∨A) ∨ B
       f f     t   t 
       f t     t   t
       t f     t   t
       t t     t   t

presto. — Preceding unsigned comment added by 95.91.213.239 (talk) 02:13, 2 November 2021 (UTC)

edit: ugh, my hacked together tables have been eaten. check the edit page for the tables, i can't be arsed to figure out how to do tables in wikipedia — Preceding unsigned comment added by 95.91.213.239 (talk) 02:14, 2 November 2021 (UTC)

I've fixed the tables. — Charles Stewart (talk) 14:23, 8 November 2021 (UTC)

Truth tables provide a particular semantics that assumes (i) negation distributes over all other connectives (i.e., De Morgan's laws) and (ii) conjunction and disjunction distribute over each other. If you can assume both of these things, then explosion follows. Logics without explosion lack either or both of these distributivity properties, and you need a different kind of semantics for them. — Charles Stewart (talk) 14:27, 8 November 2021 (UTC)

Thx for fixing the tables. Your excellent answer seems a bit off tangent, i was mostly talking about the proof on the article's page not proving what it claims to prove,

best, — Preceding unsigned comment added by 95.91.213.239 (talk) 22:58, 13 November 2021 (UTC)

How can the proof be a proof when it is so obviously false?

It begins be asserting two things which cannot both be true. "All lemons are yellow" and "Not all lemons are yellow" must be false.

Having asserted two things which cannot be both true it goes on to use both mutually exclusive statements together in a second appeal to an absurd reasoning.

The statement "'All lemons are yellow' or 'Unicorns exist'" is only true if one of the statements it comprises is true. If "All lemons are yellow" then the statement is true.

It then goes on to say "Not all lemons are yellow". If this is true then we haven't proven the preceding statement about unicorns, we've only proven the half of the statement we'd relied upon previously is false.

I'm all for examining the Law of Excluded Middle, but until you have used a reasonable axiomatic system to prove two contradictory statements then you don't have a starting point for reasoning as in the example presented. The whole point of the negating statements definition is "P and not P" must be false and "P inclusive-or not P" must be true.

Obviously-nonsense reasoning, such as that presented in the example, cannot possibly be reasonably accepted as "proof". Vapourmile (talk) 12:23, 24 November 2020 (UTC)

santa example and paraconsistency

is the santa example really correct? There are many cases in AI and philosophy where both X and ¬X can both be proved true, but don't lead to the explosion. The argument, AFAIK, would be that step (3) doesn't work, because (“All lemons are yellow" OR "Santa Claus exists”) can be satified by the proven "All lemons are yellow". It's doesn't matter that "not all lemons are yellew" has also been proved. AFAIK languages like Prolog work in this way.

   1We know that "All lemons are yellow" as it is defined to be true.
   2 Therefore the statement that (“All lemons are yellow" OR "Santa Claus exists”) must also be true, since the first part is true.
   3However, if "Not all lemons are yellow" (and this is also defined to be true), Santa Claus must exist - otherwise statement 2 would be false. It has thus been "proven" that Santa Claus exists. The same could be applied to any assertion, including the statement "Santa Claus does not exist".  — Preceding unsigned comment added by 188.39.62.164 (talk) 18:07, 28 January 2015 (UTC) 

doubts of origin of term

I've never heard the "principle of explosion" so named before; I know this as "Ex falso quodlibet" and the "absurdity law". Where does "principle of explosion" come from? ---- Charles Stewart 07:27, 26 Aug 2004 (UTC)

I've also learned this as "Ex falso quod libet," I don't really think "principle of explosion" is a good name for this article; it sounds like something about stoichiometric combustion ratios. Shiggity (talk) 21:00, 23 October 2008 (UTC)

"Ex falso quodlibet" is certainly the formal name, "Explosion Principle" is a more informal, colloquial term thrown around by philosophers. The idea is that if a contradiction is true, then the world explodes, and so anything can be true. —Preceding unsigned comment added by 24.188.68.9 (talk) 04:18, 5 March 2009 (UTC)

While I'm not sure of the origin of "principle of explosion", it was the term used in my undergraduate logic course and is used, at least colloquially, by logicians. To clarify the comment above, the thought behind the term has little to do with an actual explosion so much as an explosion in what is provable once a contradiction is introduced. While this term may not be familiar or intuitive to you, it seems to be trending upwards in use, currently garners far more google results than "Ex falso quodlibet", and has appeared in popular culture references (XKCD). It seems most likely then that "Ex falso quodlibet" is the less common term, and the current title should be left as is. 137.22.171.16 (talk) 20:10, 19 May 2012 (UTC)R.J. Connor

Google search confirms that the name "principle of explosion" was introduced into academic publications in (or around) 1998, either by Graham Priest in the Stanford Encyclopedia of Philosophy, in the article entitled Dialetheism, where it is used without any explanation of what it is, or by Walter Carnielli and Joao Marcos in "A Taxonomy of C-systems", where it is explained as being the same as the Principle of Pseudo-Scotus and abbreviated as (PPS).Flosfa (talk) 16:47, 29 October 2017 (UTC)

According to the Google Books Ngram viewer, the term "principle of explosion" goes back at least to 1800, and since that is as early as Google Books covers, it may well be older. Dezaxa (talk) 21:10, 26 May 2023 (UTC)

Contrary view

There are many people, myself included, that are convinced that the logic necessary for "The Principle of Explosion" (or "ex contradictione (sequitur) quodlibet") is inherently flawed.

Each such argument makes use of some logical rule which, itself, depends upon the assumption that contradictions cannot occur.

For example, any proof of "disjunctive syllogism" depends upon an assumption that all statements of the form "Both A and (NOT A)" are false. Likewise, every other "proof" of this sort depends upon a logic rule whose truth requires this.

If one constructs any argument that includes a statement of the form "Both A and (NOT A)" as a premise and then makes use of a rule which depends upon statements of that form being false in all cases, then the argument is inherently flawed.

So, the fact of the assumption of a contradiction necessarily invalidates all arguments that could demonstrate arbitrary conclusions. Not to mention that defining a means for categorizing statements as "true" or "false" becomes equally problematic.

In general, any presumption of a contradiction invalidates logic itself.

One of the motivations for investigating paraconsistency is that people often have inconsistent beliefs but are still able to reason. Paraconsistent logics are interesting to philosophers and in AI. Take a look at the SEP article on paraconsistent logic to find out more about them. Wikipedia could do with more of this information being written up here. --- Charles Stewart 21:14, 18 May 2005 (UTC)

I would like to add to the first comment that, yes, "any presumption of a contradiction invalidates logic itself." That's the Principle of Explosion. Once you assume a contradiction, you break your logical system. From the "Tolerating the impossible" section of the logic article, "... the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction." [1] Perhaps a similar statement should be made in this article. -- Ben-Arba 07:57, 4 December 2006 (UTC)

The anonymous original poster writes: "If one constructs any argument that includes a statement of the form 'Both A and (NOT A)' as a premise and then makes use of a rule which depends upon statements of that form being false in all cases, then the argument is inherently flawed."

Not so. In classical logic, a line of reasoning may be correct even if the premises are false (or contradictory, which amounts to the same thing). In fact this is a common thing to do. A proof by contradiction starts by assuming something that you know is untrue. --Jorend 18:22, 7 February 2007 (UTC)

Using an example from classical logic to argue against him holds no value because he doesn't accept classical logic (classical logic affirms the principle of explosion); I take the original poster's point to be about what should be the case in a logic, not what is currently done; you need to argue from what, intuitively, follows. As a minor side note, the claim that false and contradictory premises are the same seems rather dubious. 137.22.171.16 (talk) 20:19, 19 May 2012 (UTC)R.J. Connor

I agree with the poster that the principle of explosion is wrong.

1) It is both blue and not blue

2) It is blue

3) It is not blue

4) It is either blue or red

5) Use disjunctive syllogism

6) Uses LNC, it cannot be both blue and not blue

7) 6) contradicts 1), invalid proof

Also, it can flash between blue and off within the time period of the proof to satisfy 1-3) and render 4) false. Any proof here must define for itself, an instantaneous time period. Such an impossible period is identical to nothing and cannot be defined otherwise it would be something. There can never be a proof of the principle of explosion. The law of non-contradiction also needs to define the meaning of "at the same time". Should it not be "over the same time period". John Middlemas (talk) 23:25, 18 June 2014 (UTC)

Although there are some forms of logic in which the principle is incorrect, that argument is not one of them. However, if it can be found in a published work which qualifies as a reliable source, it could be included in the article, even though apparently meaningless. — Arthur Rubin (talk) 11:36, 5 September 2014 (UTC)

Vicious Circle

The second and third "Proof theoretic" arguments lead to a vicious circle. In the second argument, the circle comes from assumptions 1. 2. and the conclusion. The same kind of problem arises in the third proof. Please fix or remove. —The preceding unsigned comment was added by 76.211.229.34 (talk) 19:57, 2 May 2007 (UTC).

Accessibility

There are accessibility issues for this article. A WP article should be informative but self-contained. In other words, while it makes good sense to me, I can see how it would be incomprehensible to some, and there are no links to relevant theoretical material to make the contents of the article express any understandable meaning. Either the article should contain more plain-language explanation, or at the very least should make proper use of segue to other articles. - May 19, 2007

I agree. I can make absolutely no sense out of this article as it contains far too much symbolism and jargon. This article needs to be flagged as it doesnt conform to Wikipedia standards. - Tiwaking Augus 11, 2007

I second all this. I don't consider myself a genius, nor an idiot, by any standards, and reading this, and re-reading this, I had no idea what this theory purports. Perhaps one of the Simple English wiki writers should add a summary paragraph? —Preceding unsigned comment added by 24.140.106.178 (talk) 02:41, 18 December 2007 (UTC)

I think I have improved the intro. However, more work could be done on the body of the article; perhaps with an entire "lay" paragraph under each of the two arguments. So I have left the "confusing" tag in place. — Epastore (talk) 04:37, 20 December 2007 (UTC)

Agreed. Thanks for your work on this; we do still need quite a bit of work in the form of prose to explain the logic, though. Chris Cunningham (talk) 09:58, 20 December 2007 (UTC)

Deleted Proof

#$\phi \wedge \neg \phi \,$

1. assumption

2. $\neg \psi \,$

   assumption

3. $\phi \,$

   from (1) by conjunction elimination

4. $\neg \phi \,$

   from (1) by conjunction elimination

5. $\neg \neg \psi \,$

   from (3) and (4) by reductio ad absurdum (discharging assumption 2)

6. $\psi \,$

   from (5) by double negation elimination

7. $(\phi \wedge \neg \phi )\to \psi$

   from (6) by conditional proof (discharging assumption 1)

I deleted this proof (shown above) on the grounds that it begs the question when it uses a reductio ad absurdium (RAA). Since a reductio ad absurdium is (almost invariably, to my knowledge) justified on the grounds that it leads to the principle of explosion, the principle of explosion can't be justified from it. That argument would be summed up like so: Contradictions allow one to justify everything, so we shouldn't assert them (this last bit is the RAA); We shouldn't assert them (i.e., the RAA), and so contradictions allow one to justify everything. Since the proof begs the principle of explosion, it isn't explanatory or helpful in justifying the principle of explosion.

I hope I'm being clear, but I wouldn't be surprised if I'm not. In any case, I thought I'd explain my actions in more detail here so as to avoid a revert war or whatever, and because I felt bad, since the proof probably took some time to write out, and because someone might come up with a good reason for putting the proof back in! :)--Heyitspeter (talk) 08:37, 23 June 2008 (UTC)

Why must reductio ad absurdum be justified by explosion? How about: contradictions are impossible (law of non-contradiction), so we shouldn't assert them (RAA); we shouldn't assert them (RAA), so contradictions allow one to justify everything (explosion). The first is allegedly one of the "laws of thought", so this wouldn't be circular. (I have also used the law of the excluded middle, as explained in the reductio article. This is also allegedly a law of thought.) --Unzerlegbarkeit (talk) 14:55, 23 June 2008 (UTC)

Yeah, it was late and I was hasty. I'm with C.I. Lewis in believing that no possible future world is a priori impossible; that the a priori doesn't limit reality; however you want to word it - so I often don't jump to that axiom explicitly. It's been re-added.--Heyitspeter (talk) 19:12, 23 June 2008 (UTC)

Well, if it matters, I don't accept that justification myself either. I'm with L.E.J. Brouwer in believing that mathematics precedes logic and that the law of the excluded middle is not generally valid. But of course most people accept it. There is an alternate justification which is alluded to in the BHK interpretation article: the principle of explosion is formally valid because in most formal systems, even if you leave out negation entirely, you can still come up with a proposition BOT that already implies everything expressible in that system (e.g. 0=1 in arithmetic) - so define "not A" as "if A, then BOT". This may or may not be worth mentioning here. --Unzerlegbarkeit (talk) 21:02, 24 June 2008 (UTC)

Ahhh. Thanks for the leads on the BHK interpretation. I just read the article and really liked the ideas contained!--Heyitspeter (talk) 22:12, 24 June 2008 (UTC)

Yes, I believe that the title in not the right one!

I agree with Charles Stewart, I've never heard the "principle of explosion" so named before; I know this as "Ex falso (sequitur) quodlibet".--Popopp (talk) 12:55, 14 September 2008 (UTC)

Hewitt has propsed the name IGOR for "Inconsistency in Garbage Out Redux"

In place of the Latin that nobody knows these days, Carl Hewitt has proposed the principle be named IGOR for "Inconsistency in Garbage Out Redux". --98.210.241.92 (talk) 20:01, 5 March 2009 (UTC)

xkcd warning

Just a friendly warning about today's xkcd topic, which is based on the principle of explosion. I might suggest limiting editing ability. 68.231.22.246 (talk) 16:09, 19 February 2010 (UTC)

Geez, you must be like 15. You don't have to give an XKCD warning on every topic. Get over it. —Preceding unsigned comment added by 68.63.212.186 (talk) 20:02, 19 February 2010 (UTC)

Yes, it is absolutely necessary to give an XKCD warning on something like this. This page hadn't been edited since September of 2009, and suddenly it's already been through nine edits today. Coincidence? No. Any time an obscure topic/person/idea is mentioned in the comic, it is pretty much inevitable that the corresponding wikipedia page will come under some heat. 72.219.56.68 (talk) 20:16, 19 February 2010 (UTC)

xkcd links do not belong here. See Talk:Tautology_(rhetoric)#XKCD_link_does_not_belong for a recent discussion about this problem. Snied (talk) 21:40, 19 February 2010 (UTC)

Clarity

I moved the so-called "informal proof" to the top so readers can see what it's about, after which 99.9% of them will realise it's not of interest to them. The mathematical or pseudo-mathematical proofs can't be made clearer to a non-expert. IMO the subject is a bit like the algebraic "proof" that 2=1, following a division by 0, i.e. that formal methods can be applied to a meaningless statement or operation, to yield a startling but spurious "result". But who cares, it amuses philosophers and keeps them out of mischief Chrismorey (talk) 21:17, 23 August 2013 (UTC)

Santa

The example offered as a “demonstration of the principle” proves that Santa Claus exists, not “that Santa Claus both exists and does not exist.” — Preceding unsigned comment added by 76.246.84.113 (talk) 15:26, 28 August 2013 (UTC)

The example stated that,"The same could be applied to any assertion, including the statement "Santa Claus does not exist".", meaning that the example could have read "Santa Claus does not exist", and that the resulting proof would have been "Santa Clause does not exist". Since the figments of imagination are not relevant to the principle it must simply depend on what has been asserted as to what conclusion is drawn. Otr500 (talk) 03:55, 6 September 2013 (UTC)

Re: "in rigor, given that at least one lemon exists"

As far as I can tell, this qualification is not necessary. To be precise, "Not all lemons are yellow" already implies the existence of at least one lemon. Takatiej (talk) 21:59, 16 January 2018 (UTC)

Congratulations, you just discovered the pitfalls of existential import. 😉 Paradoctor (talk) 22:39, 16 January 2018 (UTC)

Hmm, I can not seem to find those pitfalls! The issue seems to be rather simpler than what is implied by the page (section) that you linked to. (Incidentally, its claims of significant ambiguity are not supported by any citations.) Terms like "all", "none", "not all", "some", &c. have uncontroversial meanings that are fairly well-understood by ordinary people. Even if there are differences between formal and informal use (eg. in informal discourse, we may rephrase to avoid use of "all" to refer to empty quantities), there is never any confusion about the meaning. People use such terms correctly and consistently all the time, in formal and informal contexts, without having to call their local philosopher or logician for guidance.

Regarding the original point, "Not all lemons are yellow" implies the existence of at least one counter-example. In contrast, "No lemons are yellow" would allow for the possibility of no lemons existing at all. Since the former is used in the article, and has a well-defined formal meaning, it is not necessary to specify this further. Indeed, the article already describes "All lemons are yellow" and "Not all lemons are yellow" as being two contradictory statements. The parenthesized caveat is thus redundant. It adds nothing. I propose deleting it.

It is possible that I have no idea what I am talking about. Takatiej (talk) 18:52, 24 January 2018 (UTC)

The principle of explosion cannot be proved

William of Soissons appearantly proved the principle. But it was later, in the 15e century, rejected by a school of Cologne. In the 19th century C. I Lewis mathematically seemed to proof the principle. Known logicians like Boole, Frege and Russell have taken this proof for granted but the proof has to be rejected (see the article on William of Soissons). The Principle of Explosion could be taken as an (unprovable) axiom. The priciple is not proved but also not rejected. Aristotle, in his book Metaphysics Gamma, did reject a contradiction but doesn't name the principle of explosion as a reason for it. — Preceding unsigned comment added by Zevensprongen (talk • contribs) 15:09, 8 February 2020 (UTC)

This article seems to be about two completely different concepts

This article seems to me to be about two completely different concepts:

firstly, the fact that, given the propositions P and Not P, then the propositions "if P then Q" and "If not P then Q" will both yield the same outcome

and

Secondly, the idea that an argument that is valid is not necessarily sound

I have no ideas why these two concepts have been combined — Preceding unsigned comment added by 131.106.18.48 (talk) 00:54, 17 May 2020 (UTC)

Ex falso sequitur quodlibet.

This article presupposes known, explained and accepted the principle "ex falso sequitur quodlibet" whose proper explanation remains wanting. HenryLeal (talk) 14:38, 29 October 2020 (UTC)

The example doesn't make sense

The example at the beginning of the article doesn't make sense. The statement "All lemons are yellow OR unicorns exist" is, by itself, significantly more pathological than the contradiction 2600:1002:B0CA:447F:7164:4ADD:5803:EB55 (talk) 18:37, 30 March 2021 (UTC)

Circular reasoning in "formal proof"

The formal proof on the page is suffering from circular reasoning. It peruses the disjunctive syllogism; however, the disjunctive syllogism can only be proved when EFQ is already assumed to be true. Therefore, the "proof" prooves that "EFQ is true given that EFQ is true". 5.18.95.241 (talk) 00:33, 30 September 2022 (UTC)

Move discussion in progress

There is a move discussion in progress on Talk:Ex Falso (tag editor) which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 17:47, 7 April 2023 (UTC)

Unfinished Proof

The supposed proof of POE leaves its first assumptions undischarged. How many poor souls has it sent down the proverbial rabbit hole of no return? The final result after 2 lines would be P => [~P => Q] from which, regardless of the truth value of P, you cannot infer that Q is true. Danchristensen (talk) 21:15, 5 August 2023 (UTC)

It is a little misleading as written, but those are not assumptions that need to be discharged. They are premises in an argument, or sentences in a theory. I've changed 'Assumption' to 'Premise' to make this clearer. Dezaxa (talk) 11:07, 7 August 2023 (UTC)

Here is a finished proof that P => [~P => Q]

1. P (Assume)

2. ~P (Assume)

3. ~Q (Assume)

4. P & ~P (Join 1, 2)

5. ~~Q (Discharge 3)

6. Q (Elim ~~ 5)

7. ~P => Q (Discharge 2)

8. P => [~P => Q] (Discharge 1)

Danchristensen (talk) 16:12, 3 September 2024 (UTC)

As the article correctly states, the principle of explosion holds that from a contradiction any proposition follows. Symbolically:

$P,\lnot P\vdash Q$

P and ¬P are not assumptions that need to be discharged. When we write $P,\lnot P\vdash Q$ it means that Q is derivable from P together with ¬P. So, the proof given is complete as it stands. While it is true that $P\rightarrow (\neg P\rightarrow Q)$ is a theorem, and also that $(P\wedge \neg P)\rightarrow Q$ is a theorem, neither of these is needed to prove explosion. There are many ways to prove explosion and the one given is simple and straightforward. This is why I reverted the changes made on 3 September 2024. Dezaxa (talk) 19:02, 3 September 2024 (UTC)

Yeah, well, you know, that's just, like, uh, your opinion, man. WP:PROVEIT. Paradoctor (talk) 19:06, 3 September 2024 (UTC)

No it's not just my opinion. It can be found in logic textbooks. I shall add some further citations. Dezaxa (talk) 20:00, 3 September 2024 (UTC)

Your symbolic representation of POE "P, ~P |- Q," is equivalent to both the theorems "(P & ~P) => Q" and "P => (~P => Q)." Essentially, both theorems tell us that an implication is true whenever its antecedent is false. We cannot infer from either theorem, that the consequent is true or that it is false. Danchristensen (talk) 23:44, 3 September 2024 (UTC)

Fair is fair: WP:PROVEIT. Paradoctor (talk) 00:33, 4 September 2024 (UTC)

It is correct to say that from the theorems $P\rightarrow (\neg P\rightarrow Q)$ and $(P\wedge \neg P)\rightarrow Q$ you cannot infer the truth of Q. But that is not the point. The POE states that starting from a contradiction anything can be derived. In fact, if your object language contains the falsum symbol $\bot$ then you could state the POE even more simply as $\bot \vdash Q$. Although it is possible to write a version of the principle in sentential form as $P\rightarrow (\neg P\rightarrow Q)$, doing so is unnecessarily complicated. POE can be stated and proved without introducing the material conditional. And on the basis that simple proofs are superior to complicated ones, the simple four line proof given by C I Lewis should be preferred. Dezaxa (talk) 15:27, 4 September 2024 (UTC)

(clears throat) WP:TALK: Talk pages are for discussing the article, not for general conversation about the article's subject. If you're proposing changes to the article, please clarify which ones (and source them). Paradoctor (talk) 22:42, 4 September 2024 (UTC)

If you actually want to apply POE in a formal proof, the general rule is that P => (~P => Q) for any propositions P and Q. Danchristensen (talk) 15:21, 5 October 2024 (UTC)

Is there a change to the article you're proposing? Paradoctor (talk) 15:42, 5 October 2024 (UTC)