# Talk:Principle of bivalence

WikiProject Philosophy (Rated Start-class)
This article is within the scope of WikiProject Philosophy, a collaborative effort to improve the coverage of content related to philosophy on Wikipedia. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the general discussion about philosophy content on Wikipedia.
Start  This article has been rated as Start-Class on the project's quality scale.
WikiProject Mathematics (Rated Start-class, Low-priority)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Low Priority
Field: Foundations, logic, and set theory
WikiProject Electronics (Rated Start-class)
This article is part of WikiProject Electronics, an attempt to provide a standard approach to writing articles about electronics on Wikipedia. If you would like to participate, you can choose to edit the article attached to this page, or visit the project page, where you can join the project and see a list of open tasks. Leave messages at the project talk page
Start  This article has been rated as Start-Class on the project's quality scale.
WikiProject Computing (Rated Start-class)
This article is within the scope of WikiProject Computing, a collaborative effort to improve the coverage of computers, computing, and information technology on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Start  This article has been rated as Start-Class on the project's quality scale.
WikiProject Computer science (Rated Start-class)
This article is within the scope of WikiProject Computer science, a collaborative effort to improve the coverage of Computer science related articles on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Start  This article has been rated as Start-Class on the project's quality scale.

## Comment by Brighterorange

I removed the following text:

The principle of bivalence is intuitionistically provable.
Define ¬A as (A contradiction). I.e., a false statement is one from which one can derive a contradiction. This is the standard intuitionistic definition of what it is for a statement to be false.
So using this definition, if we have (A ¬A) this can be written as (A (A contradiction)) contradiction.
So ¬ (A ¬A)

.. because I believe that it is factually incorrect.

• The text before it is contradictory (if a proposition can be truth-valueless in intuitionistic logic, how can the principle be provable?)
• I disagree that ¬A is the same as "A is false." At least from the Martin-Löf perspective, intuitionistic logic is concerned only with judging that propositions are true. Thus, a proof of ¬A must be thought of as a proof that "the negation of A is true". (I don't believe the distinction is pedantic here, since this article is about the meaning of true and false!)
• The article on Bivalence and related laws states that the law of bivalence can not be stated as a proposition
• The proposition given is exactly the statement of the law of noncontradiction, which bivalence is "not to be confused with."
• The statement that is proved does not appear to be, even informally, a propositionized version of the principle of bivalence,
• it is a negative statement and the law of bivalence is positive
• it states a conjunction, while the law of bivalence states a disjunction (Brighterorange 17:17, 14 Apr 2005 (UTC))

OTOH The principle of bivalence is intuitionistically contradicted: This statement is false.
—The preceding unsigned comment was added by 20040302 (talkcontribs) 22:47, March 3, 2006 (UTC).

## Confusing article

The claim is made, also in various other articles, that the Principle must not be confused with the Law of excluded middle, and for deeper insight we are referred to here, so one should hope that the exposition here would clarify things. Unfortunately it doesn't.

What are we to make of this sentence: "A proposition P that is neither true nor false is undecidable." A proposition is not a decision problem, how can it be undecidable? What does it mean that a proposition is neither true nor false?

Then: "In intuitionistic logic, sometimes the truth-value of a proposition P cannot be determined (i.e. P cannot be proved nor disproved)." That happens all the time, doesn't it? What about Goldbach, GRH, and so on? If this sentence should be here at all, it needs to be formulated in a way that the intended meaning shines through.

Then, next sentence: "In such a case, P simply does not have a truth-value. Other logics, e.g. multi-valued logic, may assign P an indeterminate truth-value." This confuses the notion of logic in the sense of a system with proof principles and such that can be used to prove theorems with something that is essentially "just" an algebra (a set with operations).

Two formulations of the Principle are given. The intro has: either P is true or P is false. Later we see: "P is either true or false." Both agree that this is "for any proposition P", but the latter mysteriously adds: "at a given time, in a given respect", making clear (but not to the unsuspecting reader) that we are working in the Aristotelian philosophical tradition here. Apart from that, are these two formulations equivalent? In normal parlance they are, but here we are clearly out of the comfort of normal parlance, and nothing means what it appears to mean anyway.

The subsection entitled Bivalence is deepest claims that it is "not possible to state the principle of bivalence in such a way [i.e., as a propositional formula], as the traditional propositional calculus just assumes sentences are true or false." But why is it not just the conjunction of (the propositional formulas for) Excluded middle and Non-contradiction?

Concerning the status of ⊧P (by which I mean truth instead of provability) we can distinguish three possibilities: (1) we know for certain (by whatever means) that this is the case, (2) we know equally certainly that this is not the case, and (3) we don't know (yet?) one way or another. Likewise for ⊧¬P and ⊧P ∨ ¬P. Altogether 3×3×3 cases. From the article I cannot figure out which of these are compatible with the Principle, and which are excluded by it.

Help. --LambiamTalk 12:44, 3 June 2006 (UTC)

Unfortunately it seems this confusion is not unique to Wikipedia. ${\displaystyle (P\vee \neg P)\wedge \neg (P\wedge \neg P)}$ ("exclusive disjunction of contradictories" per Suber) sort of looks like "bivalence", but it seems to me, now that we mention it, that we could have a situation where P or not-P is always true, P and not-P is always false, but P and not-P might not be individually true or false. So if the "exclusive disjunction of contradictories" is not "bivalence", then "bivalence" is not formalizable in this way at all. In any case, even if we're not clear on this point, it's pretty clear that some of this article is just wrong. 192.75.48.150 19:11, 29 June 2006 (UTC)
I'm a bit confused.

If I said "this statement is false" then it can be neither true nor false, how does this work with the law of bivalence?

This is Kleene's (1952) 3-valued logic for the cases when algorithms involving partial recursive functions may not return values, but rather end up with circumstances "u" = undecided. He lets "t" = "true", "f" = "false", "u" = "undecided" and redesigns all the "propositional connectives". Our friend the LoEM appears on queue:

"We were justified intuitionistically in using the classical 2-valued logic, when where were using the connectives in building primitive and general recursive predicates,since there is a decision procedure for each general recursive predicate; i.e. the law of the excluded middle is proved intuitionistically to apply to general recursive predicates.
"Now if Q(x) is a partial recursive predicate, there is a decision procedure for Q(x) on its range of definition, so the law of the excluded middle or excluded "third" (saying that, Q(x) is either t or f) applies intuitionistically on the range of definition. But there may be no algorithm for deciding, given x, whether Q(x) is defined or not (e.g. there is none when Q(x) is μyT1( x, x, y )=0 ). Hence it is only classically and not intuitionistically that we have a law of the excluded fourth (saying that, for each x, Q(x) is either t, f, or u).
"The third "truth value" u is thus not on par with the other two t and f in our theory. Consideration of its status will show that we are limited to a special kind of truth table.

It will take some reading and some time to get the jist of this. Kleene is definitely trying to accommodate the intuitionist viewpoint. The following are his "strong tables" that he says is "not the same as the original 3-valued logic of Lukasiewicz (1920)." (§64, pp. 332-340).

 ~Q QVR R T f u Q&R R T f u Q→R R T f u Q=R R T f u Q T f Q T T T t Q T T f u Q T T f u Q T T f u f T f T f u f f f f f T T T f f T u u u u T u u u u f u u T u u u u u u

wvbaileyWvbailey 22:20, 7 November 2006 (UTC)

## A story about Clever Teacher

The story custed by the contingent. A boy caught a butterfly and asked the teacher to answer whether it is alife keeping the insect between his hands. If teacher tells "It is dead", he will let the butterfly to fly away. If teacher tells "It is alife" he will imperceptibly squash the poor insect to fool the teacher otherwise. The wise teacher got to the core of the trap and answered: "Everything is in your hands". Excuse me if it is inappropriate for bivalence. Peahaps it links to the liar paradox, since present state manipulates the future. --Javalenok 11:03, 15 September 2006 (UTC)

The law of bivalence itself has no analogue in either of these logics: on pain of contradiction, it can be stated only in the metalanguage used to study the aforementioned formal logics.

I thought it was unformalizable, and I'm not surprised that it is so on pain of contradiction, but I'm curious how this is a consequence of the liar paradox. 192.75.48.150 15:09, 15 September 2006 (UTC)

I think that claim is only good if you accept the Tarski-kind of reasoning that holds that liar-like sentences are meaningful where they occur and so a language containing them can't be a tool for formal study of a language: i.e., that you can't do formal semantics for a natural language. But the principle clearly can be formulated formally, like this: (For all S)(val(S) = T or val(S) = F and not(val(S) = T and val(S)=F). Formulated in certain languages--paradigmatically where "S" ranges over the sentences of the language being used--then a language where you can say this will also be a language where you can formulate a liar paradox. But it's hardly universally agreed that liar paradoxes show something unacceptable about the langauges they are formulated in. There are various ways of arguing that (1) They are grammatical accidents that cannot really be legitimately formed in those languages, or (2) That even of formable as sentences they don't mean anything (not even what they appear to mean), so they don't pose a problem, (the "ungrounded" approach) or that (3) Even if a language has such problematic sentnecs, and they're meaningful, they don't undermine the functioning of the rest of the language (the "relevance" approach).

The difference not well-stated in the article--and which I won't undertake to fix--seems to be that Non-contradiction and Excluded middle are sentences, or patterns of sentences, holding or not holding in a language. Bivalence is a claim about the semantics of a language. They will of course, correspond in certain ways. But the claim at issue takes for granted some more controversial and complex claims about the ability of languages to contain semantic apparatus applying to themselves.

That should have been fixed 5 years ago, but it is done now. As for the other comment above this, I don't think it can be done. It's an issue of metalanguage. I've removed unsourced parts of the article which claimed otherwise per WP:REDFLAG and WP:OR. Tijfo098 (talk) 10:54, 30 March 2011 (UTC)

## Problems

I think there are problems with the laws of logic in this article. For example:

The statement "This statement is true" is both true and false. The statement "This statement is false" is neither true nor false. —Preceding unsigned comment added by 144.173.6.67 (talkcontribs) 16:20, 18 October 2006

## Vagueness

I've done much fixing of the article, but the vagueness section is still much in need of repair. — Charles Stewart (talk) 19:58, 30 June 2009 (UTC)

## Some issues with semantics of bivalence & gappiness

From my recent revert:

• There is no n-ary generalisation of bivalence worth noting. Bivalence is not about truth tables, it is about truth. To apply bivalence to a many-valued logic, one identifies some set of propositional values as true, and the others as not true.
• "Gap theory" means Gap creationism (20 000 ghits vs 46). I don't think it can be right to talk of a particular theory or family of theories identified only by rejection of bivalence. But the content here is worth rescuing:
A theory that rejects bivalence is sometimes called a gap theory'. Gap theories allow that some sentences may lack a truth value, or that there are more than two semantic values. Some motivations for rejecting bivalence are:
* Intuitionism: If truth is grounded in verification or proof conditions, then since some sentences are undecidable or undecided they are neither true (provable) nor false (refutable).
* Irreferential (non-denoting) singular terms: If a sentence involves an irreferential singular term like 'Pegasus', it is truth-valueless ("gappy");
* Future contingents (see next section);
* Paradoxes: E.g. Kripke's theory of truth deems the liar sentence gappy;
* Vagueness (see below).

• The discussion in "Bivalance and non-contradiction" seems useful to me. I don't see why nearly all of it should be deleted.

The point about Boolean subalgebras is a good one. It may be worth having a section on algebraic semantics that explains why Stone duality has led some to reject bivalence. — Charles Stewart (talk) 08:06, 3 July 2009 (UTC)

## Bivalent logic

Bivalent logic redirects here - what does it mean? Is it correct to say bivalent logic describes the set of logic systems that have only two values, and binary logic would be an exemplary member of this set? Are bivalent logic and two-valued logic synonymous? --Abdull (talk) 21:55, 8 February 2010 (UTC)

Yes, bivalent and two-valued are synonymous. Binary is also a synonym, but "binary logic" is usually used in digital circuit engineering, so it usually means propositional two-value logic. Perhaps that one should be redirected elsewhere, like to logic gate. Tijfo098 (talk) 10:12, 28 March 2011 (UTC)
I have made binary logic a dab. Tijfo098 (talk) 10:50, 28 March 2011 (UTC)

## Rename request

{{Requested move/dated|Two-valued logic}}

Principle of bivalenceTwo-valued logic — Occurs twice as often in google books compared to the current title. Tijfo098 (talk) 06:32, 28 March 2011 (UTC)

• Comment what about "binary logic"? 65.93.12.101 (talk) 06:36, 30 March 2011 (UTC)
• That is too ambiguous, see section above, and the binary logic page itself. Tijfo098 (talk) 09:30, 30 March 2011 (UTC)

I don't doubt that the term "two valued logic" appears more, but I think "principle of bivalence" is a better title for an article that deals with the philosophical aspects. The issues with formal logic are a secondary issue for this article; the real meat is the philosophical issues, such as whether a hypothetical event in the future must have a determinate truth value in the present. Doing a search on Google books is going to catch a lot of sentences like "this is a two valued logic" that are not very relevant to this article. — Carl (CBM · talk) 10:34, 30 March 2011 (UTC)

You're right, my proposal would inadvertently narrow the scope of the article to the more technical issues. So I'm withdrawing the move request. Tijfo098 (talk) 10:46, 30 March 2011 (UTC)

## New request

{{Requested move/dated|Bivalence}}` Principle of bivalenceBivalence

• There are quite a few Google books hits which talk of this topic without "principle of" or using "law of" instead etc. So it seems more natural unprefixed. (Compare with extensionality, which also has "principle of" appearances). Tijfo098 (talk) 11:06, 30 March 2011 (UTC) Relisted; I also pinged the two editors who had commented at previous recent req-move discussion. DMacks (talk) 09:34, 6 May 2011 (UTC)
• To many people "bivalence" first means an atom having 2 covalent bonds. Anthony Appleyard (talk) 14:26, 9 May 2011 (UTC)
• Are you sure? I get that in Google Books if I search for bivalent, but bivalence returns only logic and philosophy books on the first 5 pages (50 hits)—and I am logged out, so no personal prefs. And I just got tired there, no chem/physics books on the 6th page either. Tijfo098 (talk) 14:38, 9 May 2011 (UTC)
My dictionary (Merriam-Webster's 9th Collegiate) had only "bivalent" -- the chemical defintion above. If anyone has access to the full Oxford English Dictionary (I don't) we'd probably have to accept their findings as definitive. A quick look at the on-line available version agrees with Tijfo098. Bill 23:38, 9 May 2011 (UTC)
It's not hard to derive the noun with that meaning, and one scientific dictionary does that. So, it's perhaps best to leave the article at the current name which is unambiguous, even if "bivalence" is seldom used with the chem meaning. Tijfo098 (talk) 02:55, 10 May 2011 (UTC)

## Confused about Boolean valued logics vs. classical logic

Takeuti says that any complete Boolean algebra is a good model (presumably as in model theory) for classical logic.

Bell et al. on the other hand say that "Boolean valued logic is classical logic", but they seem to refer to only propositional calculus and any Boolean algebra via Lindenbaum–Tarski algebra.

I'm missing something here; why is Bell not talking about complete Boolean algebras? I suspect they mean different things by "classical logic", perhaps propositional vs. ??? Or perhaps Bell is assuming the "Boolean valued logic" being over a finite Boolean algebra (which is complete)? Tijfo098 (talk) 09:44, 28 March 2011 (UTC)

I don't have the full context of those excerpts, but I think Bell is referring to these theorems:
• A sentence S is provable in classical propositional logic if and only if, for every Boolean algebra and every assignment of the letters of S to elements of the algebra, the corresponding value of S is the maximum element 1 of the algebra.
• A sentence S is provable in intuitionistic propositional logic if and only if, for every Heyting algebra and every assignment of the letters of S to elements of the algebra, the corresponding value of S is the maximum element 1 of the algebra.
No completeness is needed in the algebra. Takeuti is also referring to the theorem of the first bullet there, I believe. — Carl (CBM · talk) 13:58, 28 March 2011 (UTC)

Actually, no, they are not talking about the same thing! You can give boolean valuations to predicate logic. This requires inf for ∀ and sup for ∃. That's why you need a complete Boolean algebra in general for classical predicate calculus. See [1] and verso. I'm in hurry now, so I'll fix the article later... Tijfo098 (talk) 18:48, 29 March 2011 (UTC)

For predicate logic, you can use complete Boolean algebras, but you can also use cylindric algebras, which is how people who want to do algebraic logic seem to do it. I'm not very familiar with cylindric algebras, though. The theorem I wrote above was intentionally stated just for propositional logic, where completeness is not needed. — Carl (CBM · talk) 18:57, 29 March 2011 (UTC)
By "fix" I meant add to the article the info about classical FOL. What you wrote is obviously correct for propositional. Tijfo098 (talk) 19:13, 29 March 2011 (UTC)

## Extensionality / Compositionality

This edit removed the text on that, claiming the source doesn't talk about it, but note that extensionality is given in italics in the source, and if you google around, there are other sources talking about it as "principle of extensionality" in the same context. [2] [3] etc. In other words, engage the google before you press the delete button. (And there is such non-trivial thing as axiom of extensionality which expresses that in ZF.) Tijfo098 (talk) 12:37, 29 March 2011 (UTC)

The axiom of extensionality in ZF is quite different: it says that two sets are equal if they have the same elements. The "principle of extensionality" is usually described by just saying that classical logic is "truth functional" - that the truth values of a compound sentence depend only on the truth values of the parts. But, it is possible to use truth-functional logics that study set theory without the axiom of extensionality (this is actually common in type theory, less so in set theory). So the principles are not the same, although they are both about some distinction between extensional and intensional systems.
I also was unsure whether the extensionality remark was appropriate for this article. But really that whole paragraph is a little unclear (the "characterized" claim justifiably has a "vague" tag on it), so I thought I wouldn't change it unless I rewrote the whole paragraph. — Carl (CBM · talk) 12:54, 29 March 2011 (UTC)
I did rewrite the paragraph a little while ago, what do you think? — Carl (CBM · talk) 13:39, 29 March 2011 (UTC)
To Tijfo098: If you had given pointers to [4] and [5], I would have known what you was referring to. Nevertheless, there are two issues. First, this notion of "extensionality" is not very common, and compositionality is indeed most common in this context. Extensionality usually means that pointwise (or element-wise in set theory, or observational in computer science) equality implies equality (try "principle of extensionality" in google, even though the most common name is axiom of extensionality). Secondly, compositionality is a property shared by many logics, so I don't think it is relevant to put a focus on it specially for the case of classical logic. But maybe, I missed the point? --Hugo Herbelin (talk) 14:15, 29 March 2011 (UTC)
For some reason several authors think it's worth mentioning in connection to classical logic when they review it in their introduction to many-valued logics. I agree that it's a somewhat trivial point to make, which is why I had put in parentheses. Tijfo098 (talk) 14:31, 29 March 2011 (UTC)
(edit conflict) Thanks. I've linked extensionality as "see also" instead (that article does talk of the axiom of extensionality however). On the other issue (of Boolean-valued semantics/models), isn't the "Algebraic semantics" section now redundant to what you wrote? (Before you moved & rewrote it that paragraph was in the lead). Tijfo098 (talk) 14:20, 29 March 2011 (UTC)
One more point: Some people refers to the principle (p ↔ q) → (p = q) as propositional extensionality (see e.g. [6]) but it is still different from what the references you are giving say (afaiu, what they express is the property that the value of φ(p) when p is an arbitrary formula is the same as the value of φ(x) when x is an atom whose value is given to be the one of p). --Hugo Herbelin (talk) 14:34, 29 March 2011 (UTC)
That is more of the type of extensionality that people study in type theory or set theory. If you replace the "propositions" in that link with sets, what you get is exactly the set theoretic axiom of extensionality. — Carl (CBM · talk) 14:41, 29 March 2011 (UTC)
I don't really understand what the second paragraph of the "Algebraic semantics" section is trying to say. So I guess I wouldn't mind removing it, unless someone else can rephrase it to bring out the main point. I agree it's somewhat redundant to the text I wrote this morning.
The link to Boolean-valued model is not so great because that article focuses mostly on models of set theory. That's why I left a red link to Boolean-valued semantics in my text. It would take me a lot of work to write a decent article on that, because I would have to read up on algebraic semantics and because I don't know of good references for things like the two theorems I wrote higher on this talk page. So I just made a link for now. — Carl (CBM · talk) 14:27, 29 March 2011 (UTC)
Ok, thanks. I'll remove the section then; you can't say much of the algebraic semantics for an arbitrary two-valued logic methinks. I'll leave the B-v semantics a red link. Tijfo098 (talk) 14:31, 29 March 2011 (UTC)

Hmm, if you guys are right, then Extensionality should be split in two articles. But I don't think you are! In PL terms extensionality is deep equality, and intensionality is shallow equality [7]. Tijfo098 (talk) 15:24, 29 March 2011 (UTC)

I'm just saying that "extensionality" in type theory or set theory is not the same as "extensionality" of truth values. In type theory, extensionality of a functional means that it gives the same results on extensionally equal inputs. In these philosophical books, "extensionality" of a logic means that if subformulas of a formula are replaced with equivalent subformulas, the overall formula has the same truth value. But subformulas are not "inputs" to the larger formulas. For example the subformulas themselves may have subformulas, which are still subformulas of the original formula, and just those sub-subformulas could be substituted instead. That has no analogue in substituting arguments of a functional in type theory. — Carl (CBM · talk) 11:02, 30 March 2011 (UTC)
"But subformulas are not "inputs" to the larger formulas." Aren't they? What about functional PLs like Haskell, where "subprograms" are exactly subformulas (Haskel programs are executed by expanding a graph of formulas with subformulas until one reaches the terminal nodes), and in general the Curry-Howard isomorphism. I think it depends what "are" means in your statement. I think [8] hits the nail here: "These principles [bivalence & exntensionality] form only an idealization of the actual relationships." Tijfo098 (talk) 11:58, 30 March 2011 (UTC)
No, I think they are not. In a formula like ${\displaystyle (\exists x)(\forall y)(R(x,y)\to S(y))}$, there are no "inputs" whatsoever (no free variables). The subformula R(x,y) is not an "input" in any sense; R is not a free variable that can be replaced. And even if it was able to be replaced somehow, the entire subformula ${\displaystyle R(x,y)\to S(y)}$ would also be an "input" subject to replacement. But then it is no longer possible to change one "input" independently of changing another "input". So it's true that classical logic is truth functional, but the sense in which this is a form of extensionality isn't the same as the sense in which equality of sets is extensional. — Carl (CBM · talk) 13:16, 30 March 2011 (UTC)

I managed to find sources that call the extensionality of equality the replacement property [9] [10]. I think this terminology makes more sense in the context of syntactic replacements, which is what we are discussing here. See also Kleene (gives both terms) and footnote 92 defines extension of a predicate by means of a set. Tijfo098 (talk) 14:37, 30 March 2011 (UTC)

Fitting and Kleene are indeed talking about extensionality of equality, not about replacing arbitrary subformulas. That's not the same as replacing subformulas, though.
Gibbay et al. state a weak form of replacement for subformulas on p. 124, but they don't state a general principle that when two formulas are logically equivalent (that is, their universal closures are equivalent) then one can always be substituted for the other in a larger formula. — Carl (CBM · talk) 16:01, 30 March 2011 (UTC)
Well, p. 124 is for FOL. But it seems you missed page 8, [11] which is where they formalize their statement from the introduction as Lemma 1.1.3, which is about certain kinds of propositional calculus (not necessarily classic, but including it). Tijfo098 (talk) 16:32, 30 March 2011 (UTC)
I can't see that page. Frankly, it's very difficult to tell what your argument is when it relies on me having to guess what you see in links to google books. Could you just come out and say what you mean, so that it isn't necessary for me to follow links? That would make the conversation much more efficient, and I would appreciate it. — Carl (CBM · talk) 16:37, 30 March 2011 (UTC)

Frankly, I'm not quite sure what exactly is your point either. I have been assuming you are objecting to calling "replacement for subformulas" (the principle of) extensionality. This is a terminology issue. I think I've shown above that there are sources calling it that way, and there also sources calling it "replacement rule/principle" as well (which seems more natural to me). You appear to me to think to think otherwise, and say no such rule/principle is mentioned anywhere in those sources, i.e. that I'm misinterpeting all of them. It's not really an important issue with respect to this article, so let's just end this futile conversation. Tijfo098 (talk) 17:50, 30 March 2011 (UTC)

## Whole article is based on a false premise

There have been so many comments on this page in the past seven days that I thought I would just start a separate section rather than try to answer them individually.

The article starts out by defining the principle in question to be that "every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.[1]"

It then goes on to say that "A logic satisfying this principle is called a two-valued logic[2] or bivalent logic.[3]"

Now the problem with this is that Boolean algebras are traditionally distinguished from Heyting algebras as being the algebra of two-valued logic. This is not the same thing as the "principle of bivalence" if we are to believe the article's definition of the concept. Picture a universe consisting of two places, Paris and London, each with two states, raining (1) or not raining (0). Then the proposition "it is raining" has four truth values, namely 00, 01, 10, and 11, because it can be interpreted in either Paris or London, and for each such interpretation the proposition further has two interpretations, that it is raining or not.

Nevertheless this proposition is interpretable in a two-valued logic, because there is no question that "either it is raining or it is not raining" (unless you want to quibble over drizzling or snowing). Regardless of whether we are interpreting this proposition in Paris or London (for definiteness, the centre of each say), one or the other possibility must be the case.

Until the article comes to grips with this issue it must be considered confused. Attempts to fix it are equally confused as long as they don't come to grips with it either.

The interesting question for Wikipedia is where it wants to position this article. If as a contribution to fuzzy thinking (and Wikipedia is full of such articles, for example the one on the greenhouse effect) then it is an ideal, even exemplary, candidate. If as a contribution to what Ernst Schröder called "exakte Logik" however then the sooner it is deleted the better, as there is no obvious logic article to redirect it to. --Vaughan Pratt (talk)

When you start speaking of interpreting a proposition in Paris or London you are already talking of possible worlds, thus modal logic (of some kind). The article could use further contrasts and discussion; it has some wrt. future contingents, which exhibit a related problem, but the lead is not wrong as written, and its terminology is sourced. Believe me, this article was far worse not so long ago [12]. Tijfo098 (talk) 08:09, 4 April 2011 (UTC)
It's not very clear to me what you're trying to say, but sentences like "Alice knows it's raining in Paris" and "Bob knows it's not raining in London" can be expressed in epistemic modal logic. Tijfo098 (talk) 08:34, 4 April 2011 (UTC)
If you can come up with a few respectable references that use "two-valued logic" with your meaning (i.e. algebraic semantics over arbitrary BAs), then we can turn two-valued logic into a dab, and a clarification here. But I'm not convinced that the usage with that meaning is remotely common. I've only seen Boolean-valued used for that. Tijfo098 (talk) 08:14, 4 April 2011 (UTC)

I think there are two issues. First, the principle of bivalence is primarily of interest in philosophy, as part of the theory of truth. It's particularly troubling when temporal aspects of language are taken into effect, because accepting the principle seems akin to accepting a sort of determinism. If "The Libyan rebels will win the 2011 civil war there" has a determined truth value, then it seems like the outcome of the war is already determined at the present time. This obviously has philosophical ramifications.

Some philosophically-motivated logic texts use the phrase "principle of bivalence" in relation to formal logic, but the real interest is for theories of truth, not in relation to propositional logic.

Second, the sentence "it is raining" lacks a specific truth value in English because of vagueness. The lede sentence is clear that it is talking about sentences that express propositions. If no context of time and place is taken into account, "it is raining" doesn't express a proposition, and so nobody would expect it to have a fixed truth value.

In my opinion this article should be "positioned" to focus on the philosphical issues. But we have a general lack of philosophers, and so many articles about philosophical topics are underdeveloped. — Carl (CBM · talk) 11:21, 4 April 2011 (UTC)

Yes, Vaughan Pratt is trying to capture (i.e. give formal semantics) to a natural language sentence like "it is raining". Even Boolean-valued logic is nowhere sufficient for that. Formal semantics of natural languages is an unsolved problem; the current state of the art seems to be type-logical semantics, which rely on type logics, an offshoot from categorical logic. This is another area with poor coverage in Wikipedia (we have something about the ancient Montague grammar, but that's about it). See for instance ISBN 0792330951, ISBN 0262220539, ISBN 1402039042 for what is missing. Maybe Vaughan Pratt can convince his neighbors from CSLI to help Wikipedia with that. :-) Tijfo098 (talk) 16:30, 4 April 2011 (UTC)
Ok, maybe I underestimated Wikipedia, there is a brain dump on categorial grammar. Tijfo098 (talk) 16:32, 4 April 2011 (UTC)

I also find that the current article is confusing in its scope. If it is about mathematical logic and a formal meaning of bivalence (in that the semantics of the formal language under study is two-valued), then the paragraphs about natural language should go to another article. Conversely, if some focus is to be put on non-formal languages (natural languages), then the lede should be reformulated into something like "In philosophical logic, ..." or "In natural language semantics, ...". (so, I guess I'm on the same line as CBM here.) --Hugo Herbelin (talk) 13:08, 6 April 2011 (UTC)

PS: Several articles (Two-element Boolean algebra, Bivalent, Special:WhatLinksHere/Two-valued_logic) point to the current article via Bivalent logic or Two-valued logic. Since most of these pages intend a formal meaning, at least one of the pages Bivalent logic or Two-valued logic should be kept distinct so as to preserve a page on the formal meaning? I guess it should be (at least) Two-valued logic. --Hugo Herbelin (talk) 13:18, 6 April 2011 (UTC)

I think I understand better what bothers me in the current version of the article. While the mathematical part is simply descriptive (it states the principle and its consequences), the criticism part seems to address the question of the applicability or no of the principle to "real life". Is this the understanding of other editors too? Could it be made clearer that this section is not about the principle considered as a mathematical property that a logic may have or not, but about discussing the principle considered as a "physical law"?

Anyway, I also attempted in parallel a clarification of the definition of the principe of bivalence in the mathematical part. I hope it is OK for other editors. --Hugo Herbelin (talk) 19:02, 9 April 2011 (UTC)

What bothers me about the topic (as opposed to this article) is the topic's obscurity, that in all my books I can't find any good (historical) discussion of it. This apparent obscurity makes me wonder if indeed it may be a harmless kind of latter-day obfuscation, aka " learned bullshit ". For instance, who first called it "the Law (Principle, whatever . . .) of Bivalence"? I found a flimsey definition of it in Bart Kosko's 1993 Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, NY, ISBN 0-7868-8021-X: "bivalent logic. The logic most people mean when they say logic. Every statement or sentence is true or false or has the truth value 1 or 0. The simplest form is propositional logic, where each statement aserts a simple black-white description or 'atomic fact' as in 'The avocado is green." . . . In predicate logic each statement asserts a set of black-white descriptions as in 'All avocados are green.' Aristotle first codified bivalent logic 300 years before the birth of Jesus. At the turn of the twentient centurey Bertrand Russell and Alfred NorthWhitehead showed that most math reduces to bivalent logic." (p. 288).
While he may have "codified" it, I doubt Aristotle framed the LoEM in terms of "bivalence" (if so I'd like to see the quote). And I'm skeptical (of the truth of) the last sentence. By the end of the 1800's the so-called "Laws of Thought" were worked to exhaustion by the logicians, notably in Boole's "algebra of logic" with its 1 and 0. (For a huge historical search into "The Laws of Thought", derived from what I could cull from my own references and the available googlebooks see: User:Wvbailey/Law of Excluded Middle. I'm sure there's plenty more to be found. But what puzzles me is the lack of philosophy surrounding the notion of "bivalence" per se, but rather all the philosophy is around truth and falsity until Boole . . .)
My point is this: "Logic" by the end of the 1800's was Boolean 1 and 0 with Boole/De Morgan/Venn weird algrebra of in particular xn = x, and then Jevons' 1 + 1 = 1. Here's probably the source of the "bi-value" notion. Again, who first used the phrase? By 1879 Frege is opening his Begrisffsschrift with a congenial discussion of the problem: "In apprehending a scientific truth we pass, as a rule, through various degrees of certitude. . . . I found the inadequacy of language to be an obstacle . . . This deficiency led me to the idea of the present ideography." We know that Russell derived his Principles of Mathematics from Frege and Peano, so it is along this pathway that we find ourselves with the modern 2-valued logic.
I'll keep digging . . . Bill Wvbailey (talk) 21:50, 9 April 2011 (UTC)
Isn't L. E. J. Brouwer's intuitionism one of the first attempt to question the 2-valued semantics of logic? --Hugo Herbelin (talk) 23:22, 9 April 2011 (UTC)
I dunno. I've looked in the indices of both van Heijenoort 1967 and Paolo Mancuso 1998 From Brouwer to Hilbert Osford University Press NY ISBN 0-19-509632-0(paper) and found no mention of "bivalence", but the Mancuso index is so lousy it's not a good indicator (and the book is translations of original papers + discussion, a tough read I've not done). For sure Brouwer completely rejected the LoEM when used in any generalized manner that involved the infinite. But I don't think it was in context of "bivalence", but rather in terms of whether or not, in general, it's permissible to move from the assertion of A [ "This sea serpent lives in the Atlantic" ] through a denial of such objects A ie. [~A] ("This sea serpent isn't living in the Atlantic"), to then saying "it's not true that 'it's not the case that "A" ' "[~(~A), it's not true that "this sea serpent isn't living in the Atlantic "] and then finally arrive at the conclusion that in fact " A " is true after all, after denying its existence ( ! ) [ ~(~A) --> A, ... well then, show me the not-non-existing damn thing, please. While it's easy to believe that "It's not true that the switch is not closed" is equivalent to "the switch is closed", if there were a 3rd state, e.g. in transition from open to closed or closed to open, I dunno what the assertion "It's not true that the switch is not closed" would mean)]. So in my mind the jury's out w.r.t. Brouwer and the notion of "bivalence". I moved the next part re Post 1922 below Tijfo098's response . . .

I suspect that the notion/principle was named/realized when many-valued alternatives to classical logic were seriously considered. Do you grok Polish? (Similarly [the principle of] compositionality became evident when substructural logics became fancy and useful. [13]) Tijfo098 (talk) 23:40, 9 April 2011 (UTC)

You may be right. I found a clue. (RE Polish: I only speak golumpke/gwumpke, (yum!). I think I could learn the word "beer" real fast, if I had to.). The first historical treatment of multi-value logic of any import that I know of is the (more or less first usage of) truth-tables by Emil Post in his 1922 (cf Post 1922 1921 Theory of Elementary Propositions section " m valued truth systems" in van Heijenoort 1967:279ff). Post provides a footnoted reference " 18 " on van Heijenoort 1967:279 : " 18 See Lewis 1918, p. 222, for the term 'two-valued algebra' ". Well, well . . . this is getting close. (RE Wittgenstein and his 16-row truth table . . . I don't see anywhere that he ventured beyond notions of truth and falsity). Bill Wvbailey (talk) 05:07, 10 April 2011 (UTC)
Kleene 1952:140 uses the phrase "two-valued propositional calculus" and states "These for n > 2 were discussed by Lukasiewicz 1920 (n = 3) and Post 1921 (any n). They are calculi which can be treated on the basis of truth tables in an arithmetic of n objects, as the classical system is treated on the basis of two (Cf. e.g. Lukasiewicz and Tarski 1930, Rosser and Turguette 1945, 1949, 1952.)"
The Post 1921 reference to Lewis 1918 can be found at van Heijenoort 1967:645 i.e. A survey of symbolic logic (University of California Press, Berrkeley); reprinted 1960 with teh omission of chaps. V and VI (Dover, New York)."
My trusty Websters Ninth New Collegiate dictionary (1990) has two definitions, one from 1869: "(i) having a valence of two, (ii) associated in pairs in synapsis", and another from 1943 "a pair of synaptic chromosomes"; here the word "valence" comes from LL [valentia power, capacity, from valere strong] and is used only by this dictionary w.r.t. to chemistry (1884), biological systems, behaviorism.
So far this little search is not leaving me warm and fuzzy. The search continues . . .. Bill Wvbailey (talk) 14:49, 10 April 2011 (UTC)
Googlebooks won't display the 1918 Lewis because it's still in print ( ! ). But it did a search in the 1918 edition and coughed up 18 instances of the use of "Two-Valued Algebra". Only part of the page-222 snippet was available, but it is the one as referenced by Post 1921 (capitalizations appear in the original):
"The first procedure takes the Boole-Schroeder Algebra as its foundations, and adds to it a postulate which holds for propositions but not for logical classes. The result is what has been called the "Two-Valued Algebra", because the additional postulate results in the law: For any x, if x ≠ 1, then x = 0 [ . . . snippet is cut off here] (Lewis 1918:222).
"We have now seen that the calculus of propositions in Principia Mathematica avoids both the defects of the Two-Valued Algebra. The further comparison of the two systems . . . [includes] the absense of the entities 0 and 1 [ . . . snippet cuts off here]." (Lewis 1918:286).
So we (sort of) wonder (not sure if it matters, really), who first used the phrase "Two-Valued Algebra"? But at least NOW we know that the usage "two valued" derives from Boole's "algebra". And yet the transition of usage from two-valued to bivalent is still as obscure as before (since the word "valent/valence" has been misappropriated from chemistry and biology when applied to the context of logic). A possiblility is a citation in the full Oxford Dictionary of the English Language. Bill Wvbailey (talk) 15:25, 10 April 2011 (UTC)
About compositionality, I just discovered the page Principle of compositionality which reports credits to Frege for the modern formulation. --Hugo Herbelin (talk) 00:05, 10 April 2011 (UTC)

As I suspected, Lukasiewicz named it: [14]. The 1930 paper in which L chronicles the early history of the principle has been reprinted in [15] (no ISBN on GB) ISBN 0720422523 (on Open Library) I see it's commonly held by academic libraries, so it shouldn't be too hard to access if you're interested in that kind of details. Tijfo098 (talk) 15:41, 10 April 2011 (UTC)

Bingo, you're correct. Do you speak Polish? Is the word "valent" translated correctly from the Polish? Should it have been "bivalued"?
I had to go elsewhere to discover the history of the paper. Here's the paper's history from a paper by Alessandro Becchi 2002: http://www.philos.unifi.it/upload/sub/Materiali/Documenti/logic_determinism_lukasiewicz.pdf (footnote 1):
"The article in question is a revised version of the address that Lukasiewicz delivered as a Rector during the inauguration of the academic year 1922-23 in the Warsaw University. Later on Lukasiewicz revised the address giving it the form of an article, without changing the essential claims and arguments. It was published for the first time in Polish in 1946. An English version of the paper is available in: Polish Logic 1920-1939, (S. McCall ed.), Oxford, 1967, pp. 19-39, and also in: Jan Lukasiewicz, Selected Works, (L. Borkowski ed.), North-Holland, Amsterdam, 1970, pp. 110-128."
The quote comes from Storrs McCall 1967/2005 Polish Logic 1920-1939, Oxford University Press, NY, ISBN 0-19-824304-9. The essay in question is called "On Determinism":
"Aristotle's reasoning does not undermine so much the prinicple of the excluded middle as one of the basic prinicples of our entire logic, which he himslef was the first to state, namely that every propositon is either true or false. That is, it can assume one and only one of two truth-values: truth or falsity. I call this principle the principle of bivalence " (page 36)."
A googlebooks search indicates that the usage of the word is sparse until the 1990's, and it appears to have been taken over by the fuzzy-logic folks in earnest. Bill Wvbailey (talk) 16:09, 10 April 2011 (UTC)
GB has many gaps. [The attack on] Bivalence was a central point of Michael Dummett's philosophy as well. Tijfo098 (talk) 17:22, 10 April 2011 (UTC)
FWIF, the term appears quite frequently in Susan Haack's 1978 Philosophy of logics, so it's by no means of recent popularity. Tijfo098 (talk) 18:55, 10 April 2011 (UTC)
As is the case with the original Lukasiewicz these usages are in the domain of philosophy as opposed to mathematical logic. But then in his 1952 we find Kleene's reworking of Lukasiewicz's 3-valued logic. (am unclear as to what role Tarski had in this and exactly how this logic came to be -- how that came about is interesting to us historian-types). The translation into mathematical logic doesn't render the philosophy less important, just different in treatment. You've listed two philosophers that should be included, apparently: Dummett and Hayak (I did see Hayak listed by googlebooks). As CBM lamented above, the article's contributors don't seem to be philosophers.
I'd say we've made great progress, but my bullshit Detector is still working overtime. I found this in Hurlimann 2007:42 Chapter 4 Fuzzy Logic: "Lukasiewicz named it the law of the excluded middle or more appropriately the law of bivalence". This is in fact true, as I just found Lukasiewicz's used this phraseology (synonymously) on page 52 of the McCall compilation. So are these ideas (Law of, Principle of) the same? Is this a matter of crappy translation? The problem is the cost of these compilation-books is extravagant i.e. \$150 for the McCall or the other one. So this necessitates a trip to a well-equipped academic library. Bill Wvbailey (talk) 21:28, 10 April 2011 (UTC)
There are several 3-valued logics with different intended semantics. Kleene did not "rework" anything. Tijfo098 (talk)
I beg to differ. As based on the evidence I have before me. See Kleene 1952:335 in particular, but also all of his §63 Partial Recursive Functions, and §64 The 3-valued Logic pages 323-340 for context. He specifically tailored his logic for his usage (partial recursive functions). And coined and/or employed a whole bunch of descriptive words (weak equivalence ≡ versus complete equivalence ≅ (page 328), regular and irregular tables (page 334), strong vs weak tables (p. 334)), to go along with his "logic". Here's what he claims he did:
We give the following three tables [the Lukasiewicz tables for →, ≡, and → and ≅ see the tables reproduced at Talk:Three-valued logic, plus the relevant parts of Three-valued logic and Multi-valued logic where Kleene is mentioned/illustrated] as examples of irregular tables. The present strong 3-valued logic (Kleene 1938) is not the same as the original 3-valued logic of Lukasiewicz 1920; cf. Lewis and Langford 1932 pp. 213ff.) which differs from it by having t instead of u in Row 3 Column 3 of the tables for → and ≡ . . .." (§64 The 3-valued logic page 335).
If his claim is not to be trusted (and why not? Is there evidence?) then only a look at his 1938 On notation for ordinal numbers, Jour. symbolic logic, vol. 3. pp. 150-155, plus a look at all the other papers/books (e.g. Lewis and Langord 1932) written about 3-valued logics between 1920-1952 or other published evidence would be definitive. Bill Wvbailey (talk) 15:28, 11 April 2011 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Please refer to a modern monograph on many-valued logics [16] or [17] and heed WP:PRIMARY. Kleene intended his 3-valued logic, now known as K3 to apply to recursive functions, which aren't contingent on future events; whether a computation finishes or not may not be decidable, but it's not the same thing as being eventually decidable after a point in time. Not all three-valued logics were intended for that. You can even obtain a different logic by keeping all the truth tables from K3 and just changing the designated values [18]; this is the case of LP in the 2nd ref I gave you. (You might understand this better if Wikipedia had an article for logical matrix as it's used in logic, i.e. an algebra with a designated subset, [19] which is entirely different from a Boolean matrix. SEP to the rescue [20]) This discussion is off-topic here anyway. Tijfo098 (talk) 07:03, 13 April 2011 (UTC)

You and I are clearly not on the same page when it comes to using sources -- primary, secondary, tertiary. I'm not persuaded by the argumentation trick of invoking "the canons", the "authority", to win a point. It's like the Bible: You read it your way, I'll read it mine. (As the country song says, we could meet in the middle, but this is about bivalent logic where that doesn't happen). I agree that we move on to something more toothy . . .

### Is the "principle of bivalence" synonymous with the LoEM? Scrap this article?

This is Lambian's complaint ca 2006 (see 2nd entry on this page). From the above quote (found by Tijfo098, by the way), a translation of Lukasiewicz, it appears that Lukasiewicz probably was the first to use the phrase "Principle of bivalence" which has, apparently, come into common parlance. I know from other work that "principle of" and "law of" are often used interchangeably. But the questions I have are two: (1) Is the "Principle of Bivalence" just Lukasiewicz's name for the LoEM? (2) Has it become something more so that it is worthy of its own article?

Question 1. Addressing Tijfo098's complaint about primary sources, this actually came "backwards" from a contemporary text and I was able to confirm it with a view of the original (at least in its English translation):

I found this in Hurlimann 2007:42 Chapter 4 Fuzzy Logic:
"Lukasiewicz named it the law of the excluded middle or more appropriately the law of bivalence".

This is what Lukasiewicz wrote in his Many valued systems of Propositional logic "§6. Modal propositions and the three-valued propositional calculus" to be found on page 52 of McCall. I don't know the date of this, but it is later than ca 1925-1927):

"When I recognized the incompatibility of the traditional theorems on modal propositions, I was occupied with establishing the system of the ordinary 'two-valued' propositional calcuus by means of the matrix method. I satisfied myself at the time that all theses of the ordinary propostional calculus could be proved on the assumtpion that their propositional variables could assume only two values, '0' or 'the false', and '1' or 'the true'.
"To this assumption corresponds the basic theorm that every proposition is either true or false. For short I will term this the "law of bivalence". Although this is occasionally called the law of the excluded middle, I prefer to reserve this name for the familiar principle of classical logic that two contradictory propositions cannot be false simultaneously.
"The law of bivalence is the basis of our entire logic, yet it was already much disputed by the ancients. Known to Aristotle . . ." [et.c A bit later in this paragraph he invokes via footnote 3 his Appendix that he has titled ON THE HISTORY OF THE LAW OF BIVALENCE page 63ff (in McCall).

So what do we make of this? He seems to be using the name-phrases interchangeably and the name of his Appendix leaves me utterly confused. Why didn't he name it ON THE HISTORY OF THE LAW OF EXCLUDED MIDDLE? And then reserve his new name for a modern treatment.

Question 2. Maybe someone can explain this distinction in some detail (and I mean, by use of quotes from reputable sources, not just flashing a link), but until this happens I move (as in make the motion) that this article be scrapped and its contents moved into the LoEM . . .. Bill Wvbailey (talk) 16:58, 13 April 2011 (UTC)

"Why didn't he name it ON THE HISTORY OF THE LAW OF EXCLUDED MIDDLE?" Because he sees the distinction. You seem not to read what you write: "Although this is occasionally [i.e by others] called the law of the excluded middle, I prefer to reserve this name for the familiar principle of classical logic that two contradictory propositions cannot be false simultaneously." Emphasis and note mine. Tijfo098 (talk) 22:50, 13 April 2011 (UTC)
WHAT distinction? That's what I, Bill, want you to explain. In your own words. And without resort to a link. Maybe a quote. (BTW I know the distinction between syntax and semantics, so you can use those words freely). Thanks. Bill Wvbailey (talk) 22:59, 13 April 2011 (UTC)
Fergodsakes the notions of "two-valued logic" had been established in the 1850's by Boole and discussed in Lewis 1918. The Couturat/Ladd-Franklin quote indicates that there was nothing new w.r.t. "two-valued logic" in 1920, except maybe a moniker. There's something else, though, going on re Brouwer an Intuitionism. See in particular the para. just above the APPENDIX, and the two footnotes that appear before the APPENDIX (I've verified the footnote referencing Post 1921 and the specific in this ref., specific quote appears at van Heijenoort 1967:281). Bill Wvbailey (talk) 23:33, 13 April 2011 (UTC)

. . . or the Principle of contradiction: e.g. from Thompson 1860:248 "The principle of Contradiction. 'The same attribute cannot be at the same time affirmed and denied of the Same subject.' Or ' the same subject cannot have two contradictory attributes.' Or ' the attribute cannot be contradictory of the subject.'•". (For a host of these quotables see User:Wvbailey/Law of Excluded Middle) or perhaps just subsumed as a "modernism" into the Laws of thought.

. . . or per this extremely elegant prose of Couturat 1914:23-24 that introduces two new ideas from Ladd-Franklin (boldface added). Note here the notion of ' 1 ' and ' 0 ' are quite prominent:

r6. The Principles of Contradiction and of Excluded Middle. By definition, a term and its negative verify the two formulas
aa' = 0, a + a' = 1,
which represent respectively the principle of contradiction and the principle of excluded middle.1
C. L: 1. The classes a and a' have nothing in common; in other words, no element can be at the same time both a and not-a. 2. The classes a and a' combined form the whole; In other words, every element is either a or not-a.
P. 1.: I. The simultaneous affirmation of the propositions a and not-a is false; in other words, these two propositions cannot both be true at the same time. 2. The alternative affirmation of the propositions a and not-a is true; in other words, one of these two propositions must be true. Two propositions are said to be contradictory when one is the negative of the other; they cannot both be true or false at the same time. If one is true the other is false; if one is false the other is true. This is in agreement with the fact that the terms 0 and I are the negatives of each other; thus we have
OXI=O, 0+1=1.
Generally speaking, we say that two terms are contradictory when one is the negative of the other.
1As Mrs. LADD•FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse).

Bill Wvbailey (talk) 17:57, 13 April 2011 (UTC)

It is not hard to find "reliable sources" that maintain in increasingly strong terms that there is a strong distinction between the Principle of Bivalence (Po2V) and the Law of the Excluded Middle (LoEM), as for instance here; just take this Google book search and Google scholar search. Basically, these seem to agree that the LoEM is like an axiom schema in a system of formal logic, while the Po2V is "semantic" (truth-theoretic); presumably corresponding to the distinction between ⊢P∨¬P and (⊨P or ⊨¬P). An interesting source discussing this is a 1966 article by van Fraassen; quote: "To throw a clearer light on this, we may distinguish between the logical law of the excluded middle and the semantic law of bivalence. The first says that any proposition of the form P ∨ ∼P is logically true. The second says that every proposition is either true or false, or, equivalently, that one of P and ∼P is true, the other false." Van Fraassen refers to a discussion of this in William and Martha Kneale's 1962 book The Development of Logic, which is available as a 1985 reprint. They write (p. 47) that the Po2V has been distinguished from the LoEM by Łukasiewicz in the appendix to his 1930 article. So all roads seem to lead to Ł.
However, it is also not hard to find some RSs among the search results that in some sense identify the concepts; for example here; quote: "Actually, when we speak here (and below) in traditional terminology of the “Law of the Excluded Middle” we refer to what many writers (e.g., the Kneales) nowadays call the Law of Bivalence."
What to make of this indeed? My take is that Aristotle was not a formal logician, and inasmuch as the LoEM is due to him, I don't think he intended it to be interpreted as an axiom schema, but rather teaching us some philosophical insight about the nature of what it means to ascertain the truth of certain statements. The interpretation of LoEM as "a syntactic expression of the language of a logic" (quote from our article Principle of bivalence) seems to be relatively recent. Some quotes given above appear to suggest that Ł. perhaps came up with the new term in order to have an unambiguous name for his concept, while there are different ways Aristotle's principle can be given expression in a mathematical discussion of formal logic systems. Did Ł. go on from there to redefine the LoEM, so as to make the term unambiguously distinct from his Po2V? A missing piece of the puzzle is what exactly is written in Ł.'s appendix of his 1930 article, cited by the Kneales as "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls", Comptes rendues des séances de la Societé des sciences et des lettres de Varsovie. They also refer to his Aristotle's Syllogistic from the Standpoint of Modern Formal Logic (2nd ed., 1957) pp. 67 ff. I can only see a snippet view of possibly another edition in which Ł. writes on p. 82 that the Po2V "must not be mixed up" with the LoEM.  --Lambiam 19:26, 13 April 2011 (UTC)
I see that what Ł. wrote in 1930 was actually quoted above: "Although this is occasionally called the law of the excluded middle, I prefer to reserve this name for the familiar principle of classical logic that two contradictory propositions cannot be false simultaneously." Note that "I prefer to reserve this name for ..." is much weaker than some modern sources that suggest you're really dumb if you confuse these two so obviously completely different principles for even a single moment. This, as well as some other quotes (like the one above referring to the "traditional terminology of the “Law of the Excluded Middle”") confirm what I thought: in mapping Aristotle's ideas to formal logic, one of several possible mappings was identified with and named as the "LoEM", and so Ł. picked a different name for another mapping. By the way, using a prime to indicate negation in Boolean algebra, I'm inclined to translate Ł.'s formulation ("cannot be false simultaneously") as a·a′ = 0 instead of Couturat's a + a′ = 1. Of course, in Boolean algebra these are equivalent by De Morgan's Laws, but in intuitionistic logic the first is universally valid while the second is not.  --Lambiam 19:58, 13 April 2011 (UTC)
There was no model theory in 1925 (let alone 1913), so I'm not sure what you guys expected from Lukasiewicz. Someone else probably clarified the distinction with LoM later, I presume. Tijfo098 (talk) 20:36, 13 April 2011 (UTC)
Kneale & Kneale is largely considered a well-written early history of mathematical logic. So, I'm more inclined to believe them than sources disagreeing with them. It does appear that they've strengthened the distinction made by Ł in 1930. Tijfo098 (talk) 20:49, 13 April 2011 (UTC)
If you read p. 46 in K & K they say that Aristotle stated both what latter became known as the Law of Contradiction and the Law of Excluded Middle in his Metaphysics, but in distinct passages. So, he made the distinction between these laws according to K & K, from which they concluded he did not see them as one principle. On p. 48 K & K conclude that Aristotle is trying (in DI 9) to assert LEM and deny bivalence. Tijfo098 (talk) 21:18, 13 April 2011 (UTC)
Also on p. 161 K & K write that "According to Boethius [the Stoics] criticized Aristotle for denying [the principle of bivalence]". Tijfo098 (talk) 22:44, 13 April 2011 (UTC)
In ISBN 0198241445 Ł says at p. 82 (referred by Lambiam above):
and cites Cicero, Acad. pr. ii. 95. (By "this method" L refers to trying all possible combinations of variables to solve the dual of the Boolean satisfiability problem, noting that if one gets a result of 0 (false) for a combination of variable assignments, then the proposition is not a tautology.) Tijfo098 (talk) 21:58, 13 April 2011 (UTC)

More interesting, on p. 205 Ł writes (about Aristotle's attempts at a modal logic):

An he adds in a footnote on that page: See J. Łukasiewicz, 'Logika dwuwartościowa' (Two-valued Logic), Przegląd Filozoficzny, 23, Warszawa (1921). A passage of this paper concerning the principle of bivalence was translated into French by W. Sierpinski, 'Algebre des ensembles', Monografie Matematyczne, 23, p. 2, Warszawa-Wroclaw (1951). An appendix of my German paper quoted in n. 1, p. 166, is devoted to the history of this principle in antiquity." And note 1 on p. 166 refers to: Jan Łukasiewicz, 'O logice trójwartościowej', Ruch Filozoficzny, vol. v, Lwów Tijfo098 (talk) 22:13, 13 April 2011 (UTC)

This paper analyzes 'O logice trójwartościowej' and similar early papers of Ł and says on p. 15:

Tarski p. 197 -- Tijfo098 (talk) 00:28, 14 April 2011 (UTC)

## Reverted to a slightly earlier version of the article

Here. My edit summary is not accurate; the old new version was basically correct, it just didn't help explain things at the level of this article. We're already having problems (^^^^^^^^^^^^^^^, also below) with readers understanding what's going on here before we introduced cylindric algebras etc. Confusing the simple case of the classical propositional calculus was not an improvement either. I may he thrown out some useful rewordings along with that; I apologize if I did. Tijfo098 (talk) 04:49, 14 April 2011 (UTC)

Actually there was something wrong in the version I've reverted, but in a different section: "A two-valued logic is necessarily classical and it validates both the law of excluded middle and the law of non-contradiction." This isn't found in the source cited, so fails WP:V, and Béziau (2003) refutes this: "The construction of logics having negations not obeying the principle of non contradiction and/or not obeying the principle of excluded middle, together with the development of a theory of bivaluations showing how to construct bivalent semantics for a wide class of logics, have led clearly to the rejection of the equation:

(B) = (EM)+(NC)

More precisely, they have led to the rejection of the belief that the principle of excluded middle or the principle of non contradiction are consequence of the principle of bivalence. These constructions have been carried out by Newton da Costa and his school (cf. e.g. Loparic & da Costa 1984) and this consequence is an important philosophical aspect of their work." Of course, his refutation involves use of designated values, but so does the LP for instance. Tijfo098 (talk) 21:56, 14 April 2011 (UTC)

I guess you realized that you reverted an edit from me and removed two additions to the page.
For the first addition (a counterexample to having the law of contradiction holding without bivalence being satisfied) looks to me the natural complement to the existing counterexample to having the LoEM holding without bivalence being satisfied. So I don't understand your motivation in removing the content I added.
For the second addition I made (connection between FOL and cylindric algebras), I don't see either what justifies the removal. Could you explain? It is not that cylindric algebra is a very interesting notion per se (they are just an ad hoc abstract characterization of the Lindenbaum algebra of FOL with equality presented without function symbols). Indeed complete Boolean algebras are generally considered more interesting models these days (but it should be clear that the Lindenbaum algebra is not a complete Boolean algebra: quantifiers alone are not powerful enough to express infinite meet and join; a more complex construction is needed, that interprets formula P as the set of contexts Γ such that Γ ⊢ P; this set is a complete Boolean algebra). Note also that complete Boolean algebras are not the only interesting models of classical logic (there are also realizability models, Kripke-style models, phase semantics models, pretopological models, etc.).
Coming to my reverted edit to the lead, I can accept the WP:MTAA criticism but what to put instead? My objective was to clarify in which sense the bivalence was meaningful, but the primary question seems rather to understand if there is a clear definition of what the Po2V and actually the LoEM are. According to the bibliographical investigations you made, there are at least three definitions:
1. It is a property of the semantic of a logic corresponding to a semantic form of the LoEM. This requires the logic to come defined with a specific semantics "M ⊩" for M a model and the Po2V states "M ⊩ P or M ⊩ ¬P".
2. It is a property of the semantics of a logic about values. This requires the logic to come defined with a specific semantics φ assigning values to formula. Then, the principle states that the semantics function takes values in the two-element set {true,false}, i.e. for a model M being given, "φM(P) = true or φM(P) = false" (this definition is not equivalent to the previous one because to turn a relation ⊩ into a function φ, one needs classical reasoning and the axiom of unique choice [21]; the second definition is stronger though, by taking M ⊩ P to be φM(P) = true). This is the definition I thought was the correct one and the point of my edit was to make it clear. [Note: a semantics φ that has this property is what Béziau calls a bivalent semantics]
3. It is a property of the class of semantics of a logic, as discussed and proposed by Béziau, i.e. the ability for a logic to be equipped with a two-valued semantics φ. As shown by Béziau, this definition is uninteresting since any logic can be equipped with a two-valued semantics by taking for φ the semantics that assigns true to the provable/valid formulas and false to the other. Note that satisfying Po2V in Béziau's sense is not the same as being two-valued, which I think can only mean definition 2 above (i.e. two-valued is the property of a semantics, not of a set of formulas). In particular, two-valued implies classical, and I'm sure that this claim can be sourced (of course, this also requires strictly speaking to define what classical logic is and I'm unsure the article classical logic in its current state will be of a great help here).
Of course, this is related to the more fundamental question of what a logic is. In particular, since the principle of bivalence (or at least the notion of being two-valued) is about semantics, it makes sense only when the notion of logic comes defined semantically (by opposition to a proof system-based syntactic definition), i.e., as a set of formula L together with an interpretation φ of formulas into a set of values D, assuming the atoms to be already valued in D (in this definition D has to contain a special value true and φ has to satisfy basic properties such as φ(P∧Q)=true iff φ(P)=true and φ(Q)=true, etc.). In this case the principle of bivalence just says that D = {true,false}.
In short, the main problem of this article in my opinion (and more generally of many articles about logic in WP) is that it talks about notions which are not clearly defined and hence subject to different (sometimes incompatible) claims. The first thing to do would be to summarize the different notions of Po2V found in the literature, together with their historical context. If it turns out that the Po2V is not clearly defined enough to justify an article, then, it will be reasonable to restore the autonomy of the two-valued logic article since the definition of what two-valued means is more universal. --Hugo Herbelin (talk) 08:58, 15 April 2011 (UTC) [Partly rewritten 12:48, 15 April 2011 (UTC)]
A few more things about Béziau:
• When he talks about evaluation relation, it does not mention the interpretation of variables, so he seems to consider that he interprets only closed formulas, and in particular that its study to be applicable say, to propositional calculus, needs to fix a model in advance (which is maybe the standard way to proceed in the philosophical approach to logic but not the standard way to reason in mathematical logic).
• I'm unsure also I finally fully understand Béziau's definition of Po2V and in particular in which sense he says that Kripke semantics is bivalent (in his paragraph about Kripke, it is not the interpretation of formulas which is bivalent, it is the interpretation of formulas at a given world which is bivalent, and this is completely different from the notions discussed up to now).
[Note incidentally that I agree with Béziau on the following point: "Unfortunately most of the introductory books of logics make the confusion between negation as a connective and negation as a truth-function (see Béziau 2002a)" and this confusion also appears on Wikipedia (see e.g. Logical connectives or Propositional formula)]
So, again, first thing to do is to clarify how the notions are defined and if several definitions exist to say to which definition one refers to when a property about this notion is stated. Otherwise everything and its contrary are likely to be said. --Hugo Herbelin (talk) 10:02, 15 April 2011 (UTC) [Edited 12:48, 15 April 2011 (UTC)]

## An attempt to summarize the Principle of bivalence

Pulled from inside the message in a bottle:

At a purely semantic level LoEM does not forbid that a proposition and its negation both be true, it only says that at least one of them is true. This is exactly what happens in LP, and in all paraconsistent logics for that matter. This is already said in the Wikipedia article. I don't know how much clearer than this it can be said. Tijfo098 (talk) 01:38, 14 April 2011 (UTC)

Lambian (above) has addressed some of what I think causes the confusion. It has to do with the distinction of "application" (substitution) of "truth values" (the semantic, or philosophic end of things) into a Logicistic or Formalist (take your pick), purely by-the-rules syntactic construction, i.e. a "logic formula."

(I'm doing my best here to summarize, and may be using the words imprecisely, if so, gentle correction would be appreciated.) What you are saying is actually two things: (1) with regards to syntax, i.e. the symbols and rules concerning their concatenations into "logical formulas", the OR " V " of the LoEM " P V ~P " is (for some historical reason) assumed to be inclusive as opposed to exclusive, and this means that (2) when truth values are applied in a 2-valued semantics, an ambiguous minterm results that corresponds to the "middle" (~P & P).

If what I wrote above is true, here's one source of confusion: that the " OR " is inclusive is never [?] stated when the LoEM is quoted. It's usually cited as one of three laws of thought (e.g. Russell 1912:72 The Problems of Philosophy, Dover reprint 1997), the LoEM and the law of noncontradiction quoted together as a pair, both of which are necessary (cf Ladd-Franklin two sections above, and I do remember also in Socrates) i.e. thus when taken together, (P V ~P ) & ~(~P & P) removes the offending minterm.

Why the "Principle of bivalence" remedies this problem is because the two truth values (two-valued semantics) are overtly expressed together with a form of the (syntactic) formula [ (P V ~P ) & ~(~P & P), i.e. the LoEM AND LnC ] to form single a "Principle". Is this on the right track? Bill Wvbailey (talk) 14:45, 14 April 2011 (UTC)

I have to say the confusion with LoEM does appear in some otherwise reliable sources: [22]. Apparently the problem is with the dual use of the term "LoEM" for two different things: [23] [24] [25] [26] [27] [28] [29] [30] [31]. What can be said from this is that the meta-logical (but not plain logical) version of LoEM is the same as the principle of bivalence. Apparently we owe the distinction to Henryk Mehlberg (1958) The Reach of Science, Oxford University Press, and perhaps an earlier 1956 paper of his. It also seems (based esp. on the last source in the list) that others have argued that there is no distinction. So, this is something that should be detailed in the article. Tijfo098 (talk) 17:39, 14 April 2011 (UTC)

In the 1st para above there's some great sources and good stuff. In particular I found Comim et. al. 2008:277 fn 17 [32] very useful, very helpful, at least as representing a POV re an object- versus meta- distinction that allows (if I understand it) a "reduction" from the meta-BoP to the object-LoEM: however, to "get the idea" the reader would need a pretty firm grip on the distinction between an object- and a meta-language. I'd think an explansion of such a notion would be a good addition to the article (but beyond my capability to write it).
Also, along with the last source (1st para) I noticed that Hale and Wright ed. 1997:481 fn 5 [33] adds Russell to the list of those who can be understood as rejecting the BoP. A strange claim, perhaps this came later in Russell's career (I'm not totally surprised, but I wish this were developed more; BTW the supervaluation discussion in the sorities article is interesting).
All of this convinces me that CBM was correct (if I am remembering correctly), that this article should tend more to the philosophic than to the mathematical/logical. As this has moved well beyond my knowlege, library and resources, I doubt I can contribute further to the article. I'll gonna leave it in better hands. Bill Wvbailey (talk) 19:19, 14 April 2011 (UTC)
I'd say it's the other way around: LoEM is the more ambiguous notion. Tijfo098 (talk) 19:44, 14 April 2011 (UTC)
If you liked Comim et al. presentation, you'll probably like Béziau's below as well; it's more detailed. Tijfo098 (talk) 21:06, 14 April 2011 (UTC)
I agree that LoEM is the more ambiguous notion; I think Ł. introduced the term "Po2V" precisely to avoid the ambiguity. Modern logicians make distinctions that traditional philosophers and logicians did not make, and that which was traditionally understood to be Aristotle's LoEM can be mapped in different ways to systems of formal logic. And indeed, as was to be expected, different authors have made different choices. One possible choice is the "semantic" mapping to what is called here the Po2V. Another choice is the "syntactic" mapping to an axiom (or theorem) schema. Among formal logicians the latter appears to have become the more commonly (although not universally) understood meaning of "LoEM", while philosophers and more traditional logicians have not generally followed suit, and continue to use the term "LoEM" for a semantic, truth-theoretic concept. The existence of this ambiguity should be acknowledged in our articles, both here and at Law of the Excluded Middle.  --Lambiam 17:28, 15 April 2011 (UTC)
I was trying to figure out the NPOV angle for the presentation of this issue, and I think [34] is reasonable. Tijfo098 (talk) 00:48, 16 April 2011 (UTC)
I was okay with your source until it got to tertium non datur. Unless I misunderstood (is entirely possible) it does not agree with other sources (most notably Reichenbach; see inside the hatnote in the above section for the reference and the quote). If you look back at Talk:Law of excluded middle section 4.1 where the drawings are you'll find more of the original Aristotle than what is inside the hat-note below. It's not private, I just put it inside because it's long and I don't want to clutter this discussion. Aristotle's proof is, I think, absolutely wonderful. BillWvbailey (talk) 18:32, 16 April 2011 (UTC):

This is in response to Wvbailey's question, "what is the historically-accurate logical symbolism for tertium non datur? Is it as Reichenbach would have it: "For all P: P V ~P " (inclusive-OR). Or is it the ~(P & ~P)" version?"

There is no way history could make it the latter. First, the latter is not expressing "no third possibility", it is expressing "no inconsistency". Second, ~(P & ~P) is intuitionistically valid because it is equivalent to (P & (P → 0)) → 0, which is equivalent to (P → 0) & (P → 0), which is a substitution instance of Q → Q, which is intuitionistically valid. --Vaughan Pratt (talk) 07:17, 17 April 2011 (UTC)

### A few other interesting readings

-- Tijfo098 (talk) 20:37, 14 April 2011 (UTC)

Apparently Peirce also realized some of this, although he did not put it in modern terms (secondary source analysis: [35] read until p. 427 if GB lets you; paper is Jacqueline Brunning, "C. S. Peirce of Future Contigents"). Tijfo098 (talk) 19:44, 14 April 2011 (UTC)

It seems that Susan Haack proved that LoEM + T-schema imply bivalence. (Varzi shows the same thing, [36], except he calls the T-schema "Equivalence Schema"; the two are synonymous [37]) Of course T-schema is mostly of historical interest today, as it's not quite how model theory developed. (See entry for discussion). Tijfo098 (talk) 02:18, 15 April 2011 (UTC)

### A historical approach is best?!

Neither philosophy nor mathematics are fixed in time; as knowledge advances, certain things that had been previously "obvious" and taken for granted, such as "bivalence==excluded middle", are eventually seen to not be the case, and the subtle differences are articulated. The proper approach is probably a historical one: describe what was thought, when, by whom, and why (and why they didn't see that they were wrong). For example, a new, modern 20th century conception is autoepistemic logic, which rejects the law of the excluded middle, but keeping bivalence: a third possibility is that the truth of a proposition is unknown. In such systems, the LoEM can be asserted as a fact, but only if one wishes -- it is not built into the system, a-priori, by default. Aristotle would have understood the arguments for this, but wouldn't have appreciated the significance for computer science and chip design :-) linas (talk) 16:53, 10 June 2011 (UTC)

## AAL to the rescue (or not)

For those that feel that the discussion is not formal enough, there's an abstract algebraic logic framing of many principles related to sentential logics. This are indeed more complicated if one considers non-truth-functional operators. See formalization [38] and turn the page for quite a list of few principles! Tijfo098 (talk) 19:05, 15 April 2011 (UTC)

## Remark on modal logic

In response to my example above of "It is raining" with Paris and London being the possible worlds, User:Tijfo098 responded with "When you start speaking of interpreting a proposition in Paris or London you are already talking of possible worlds, thus modal logic (of some kind)" and asked about incorporating those worlds into the language, using examples like "Alice knows it's raining in Paris" and "Bob knows it's not raining in London". However modal logic does not work that way, it does not name worlds. In a multimodal logic like dynamic logic it may name relations, but by default a model of modal logic is a Kripke structure, meaning a set W of possible worlds and a single binary relation R on W of accessibility between the worlds, which is named implicitly only by the modalities ${\displaystyle \Box }$ and ${\displaystyle \Diamond }$.

A standard translation of intuitionistic logic into modal logic renders conjunction P∧Q and disjunction P∨Q and their units 1 and 0 classically, i.e. without modification, but interprets implication P→Q as ${\displaystyle \Box }$(P→Q), meaning "In all worlds accessible from here, P→Q holds". (Think of accessible worlds as conceivable worlds and read intuitionistic P→Q as "necessarily P→Q.") The intuitionistic meaning of negation ¬P, which merely abbreviates P→0, then becomes ${\displaystyle \Box }$(P→0), that is, "P is inconceivable" (since the P→0 inside the box is interpreted classically).

Now if (Paris,London) is in the relation R but not (London,Paris), this means that Londoners find Paris inconceivable but not conversely. More specifically with regard to rain, Parisians could understand that even though it is not raining in Paris it could be raining in London. Londoners however cannot imagine that it could be raining in Paris but not in London, because Paris is not a world they can contemplate as an alternative to their own.

It follows that in London, either it is raining or it is inconceivable that it is raining. In Paris however it is not the case that either it is raining or it is inconceivable that it is raining, because Parisians can conceive of rain falling on London even though it is (classically) not raining in Paris.

Hence we have found a world in which LoEM fails, namely Paris. LoEM does hold in London assuming this R, but one world is sufficient to rule out a law as a universal truth.

But if Parisians lost their ability to imagine London as an alternative world to their own, so that R became a discrete relation connecting no two distinct worlds (I'm assuming R is at least reflexive, and usually transitivity is assumed too, i.e. an S4 modal logic), then LoEM would become a universal truth, that is, either P holds or P is inconceivable (the meaning of intuitionistic-not-P), because as a "predicate transformer" ${\displaystyle \Box }$ would then be the identity operation on predicates, so that the translation of intuitionistic P→Q to modal ${\displaystyle \Box }$(P→Q) would be semantically equivalent to classical P→Q.

One thing to be aware of is that no finite number of worlds can capture exactly intuitionistic logic in this way, but only something intermediate between classical (Boolean) logic and intuitionistic logic. Only an infinite Kripke structure can give the exact semantics of intuitionistic logic via the translation of intuitionistic P→Q to modal ${\displaystyle \Box }$(P→Q). However two worlds are enough to find counterexamples to LoEM.

This is food for thought about the relationship between LoEM and bivalent logic. With the S4 translation of intuitionistic logic, whether LoEM holds or not can depend purely on the accessibility relation without changing any other detail of the logic. --Vaughan Pratt (talk) 05:09, 17 April 2011 (UTC)

I should add that LoEM holds in classical modal logic, though not in intuitionistic modal logic. The above deals with the translation of intuitionistic logic into classical modal logic. In that translation LoEM does not hold because the translation interprets intuitionistic implication differently from (classical) modal implication, namely by adding ${\displaystyle \Box }$. --Vaughan Pratt (talk) 05:28, 17 April 2011 (UTC)

## Degrees or truth preservation

Making a note here to mention that topi in the vagueness section [39]. Tijfo098 (talk) 07:05, 22 April 2011 (UTC)

## Dummett: Not just tarot

The Logical Foundations of Metaphysics discusses bivalence. 01:49, 15 November 2012 (UTC)