Talk:Many-valued logic

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(Added) PEM[edit]

The fact that Aristotle didn't fully accept the law of the excluded middle doesn't seem to be mentioned in the Laws of thought article nor in the article on Lotfi Askar Zadeh. The difference between this article and those two are slightly confusing. The best I can gather is that Aristotle put forth the law of noncontradiction and the law of the excluded middle, but expressed in De Interpretatione that the law of the excluded middle could produce some problems. Perhaps somebody that knows this subject could tweak the wording in these articles to clarify things a tad. -Chira 21:39, 11 August 2005 (UTC)

I've tried to answer Chira's question (again.) Moved from article: (the law may originate from one of them, Chrysippus). Meaning what? He lived after Aristotle, and Aristotle's laws imply the law in question. Dan 23:05, 6 April 2006 (UTC)


Could someone write an example who understands this? Or maybe a link to a tutorial? Thank you --Gaborgulya 23:06, 15 May 2007 (UTC)


"Many-valued logic" is somewhat more common, according to Google Scholar than "multi-valued" logic, by a margin of about 50%. Additionally, "multi-" seems to be mostly used in the specific subfield of the design of ALUs in digital circuits - relevant, but not the core of the topic.

Is it worth changing the name? I lean to saying it is. — Charles Stewart (talk) 08:21, 29 April 2009 (UTC)

I lean to agreeing with you. At least, insofar as my experience points to the greater frequency of many v. multi in this particular arrangement.—αrgumziω ϝ 19:52, 21 August 2009 (UTC)

Merging with Multi-valued logic[edit]

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.
Result of discussion was merge Many-valued logic, do not merge Three-valued logic or Four-valued logic.

We have a problem here. We have one article about many-valued logic and another article about multi-valued logic. Should these be consolidated?

Yes. Under the name trinary logic (the most common kind, with 3 values) and more-than-3-values cases treated as side cases with redirects pointing to it. Also the entry for Null says that Nil=Null which is just not true in some versions of Lambda Calculus, it's a four valued logic (T, F, Nil but with Null as well - used for indeterminate states that can't be resolved, i.e. there may be a value, it's a 'don't know').
It seems that the people writing these articles haven't worked with these concepts much.
Don't wait, then, fix it! --Robert Merkel
Actually, the entry for Null doesn't say what you think it says. It says in LISP, null is called nil, which is correct AFAIK. LISP != lambda calculus. Chadloder 03:28 Jan 24, 2003 (UTC)
I disagree that trinary logic should be the main article name. Trinary is a special case of multi-valued logic, not the other way around, right? Sure trinary logic was developed first (or second, rather) but ultimately that's less important. Chadloder 03:29 Jan 24, 2003 (UTC)
I also disagree with making trinary logic the main article name. Jason Quinn (talk) 06:44, 24 August 2013 (UTC)

I suggest both articles be consolidated under multi-valued logic (which sounds better than 'many-valued logic'. A 4 valued logic is no more a side case of a 3 valued logic than is the trinary of the bivalent. The major conceptual split is between bivalent and multi-valued logics. The aspects relevent to computer science are subordinate to the logical issues.

Although I fully agree that many-valued logic should be merged with multi-valued logic (as both names are just synonyms referring to the same class of logics), I'd strongly oppose merging them (as suggested by the templates in this article) with Łukasiewicz logic or three-valued logic (and probably four valued logic as well), since the latter are very specific subareas of the former, and meaningful full-length articles can be written about each of them. Łukasiewicz logic has been deeply studied, with several monographs devoted to it; a full encyclopedic description of the area will definitely be long enough for a separate article. The many kinds of three-valued logic (strong and weak Kleene, McCarthy, Łukasiewicz, Post, etc.) plus the general results on all 3-valued logics (e.g., on functionally complete sets of connectives) make 3-valued logics worth a separate article as well. The case for four-valued logics is less clear, but still the area (with, e.g., Dunn–Belnap FOUR, special properties of 4-valued variants of 3-valued logics, etc.) seems broad enough to justify a separate article. Therefore I suggest to remove the three templates from the article (even though moving some portions between these articles and a significant expansion of all of them will of course be necessary). -- LBehounek (talk) 14:29, 26 January 2011 (UTC)
I removed the suggestion to merge Łukasiewicz logic. I added it as I reminder to create a subsection on Łukasiewicz logic in this article, but it definitely warrants a separate article as well. I would still suggest to merge both three- and four-valued logic into this article. I like the overview-style articles at the German-language Wikipedia ( and the Stanford Encyclopedia of Philosophy (, as several of the issues and systems for three-valued logics generalize to n-valued logics. Spreading this among multiple article may make it difficult to find for many readers. —Ruud 16:06, 27 January 2011 (UTC)
Okay, this sounds as a reasonable solution. Three- and four-valued logics should definitely be covered in the article on many-valued logics in some level of detail anyway, so the current versions of these articles can be merged. If there is enough material to justify separate articles for 3- or 4-valued logics at some later point, there will always be an option to re-create them with expanded contents, and link them via the Main page template from the appropriate sections in the general article on many-valued logics. -- LBehounek (talk) 15:58, 31 January 2011 (UTC)

There is sufficient material for 3-valued [1] JSTOR 2274919 and possibly for 4-valued logics [2] [3] to have separate Wikipedia articles for those. I'm even considering splitting Logic of Paradox to a separate article because there are multiple angles for that one (like there are for Lukasiewicz logic). I agree (post factum) with the merger of multi-valued logic. Tijfo098 (talk) 17:26, 13 April 2011 (UTC)

I could be convinced to split the four-valued logic article in two because there are only two 4-valued logics that seem to be of significant interest. Tijfo098 (talk) 01:10, 14 April 2011 (UTC)

There's been a merge to Many-valued logic template on Three-valued logic page since January 2011. As per the suggestion above and my own thinking, that can be removed. I will do so now as the merge discussion never really happened anyway. Jason Quinn (talk) 06:44, 24 August 2013 (UTC)

The above discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.

Boolean algebra?[edit]

Could anyone mention which of the presented logics satisfy the laws of Boolean algebra? Jochen Burghardt (talk) 10:04, 3 May 2013 (UTC)

Boolean algebra is an algebraic structure, not a logic. The corresponding logic is classical propositional logic, and it is not included in any of the logics listed (P3 has the same tautologies as classical logic, but does not satisfy the rule of modus ponens). That should not come as a surprise, as classical logic has no proper consistent extension.—Emil J. 10:56, 3 May 2013 (UTC)
What I meant is: since there is a truth table for each logic given on this page (i.e. Priest's P3, Bochvar's B3, and Belnap's B4), it can also be viewed as an algebra (like G.Boole did for the usual 2-valued logic). It may then be asked whether e.g. "∧" is commutative, associative, and so on. Birkhoff [1] gave 9 laws (about "¬", "∧", "∨", but not involving "→" and "↔") to be satisfied in order to be called a Boolean algebra.
Meanwhile I achieved to write a little C program to check these laws on the truth tables given on this page. None of them satisfies (x ∧ ¬x) = F or (x ∨ ¬x) = T (btw: this tautology is satisfied by classical logic; so P3 seems to me not to have "the same tautologies as classical logic" - ?). Bochvar's B3 in addition violates the absorption laws, viz. (x ∧ (x ∨ y)) = x and its dual. All other laws are satisfied. In particular, each logic on this page leads to a distributive lattice. If inB4 the truth table for negation is modified such that (¬ N) = B and (¬ B) = N, then it leads even to a Boolean algebra, according to my C program. Jochen Burghardt (talk) 11:39, 4 May 2013 (UTC)
  1. ^ Lattice Theory, Am. Math. Soc., Providence, 1967
In P3, I is also a "designated truth value". So x ∨ ¬x would be a tautology. —Ruud 13:29, 4 May 2013 (UTC)
Ok, thanks, I see now. I added a sentence clarifying the meaning of "designated" to section "Examples" of the article, since the connection between "tautology" and "designated" was not explicit there, nor is it in the Tautology (logic) article. Jochen Burghardt (talk) 06:30, 5 May 2013 (UTC)

Laws of Form[edit]

wow, amazing this document didn't mention spencer G Brown and Laws of form (there is a 1979 edition), where an indication is made that an imaginary logic value is needed to help to resolve Goedels incompleteness theorem type paradoxes. In this case, a third logic value would actually be an oscillation between true and false as i appears to be an oscillation between 1 and -1. The famous sentence "This sentence is false" resolves with an oscillating logic value. (Contributed by Matthew Scott)

We already have a rather lengthy article on the Laws of Form. Given its somewhat fringe nature I'm note sure if we should discuss it here as well. —Ruud 14:52, 14 July 2015 (UTC)