Talk:Principle of bivalence

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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) → 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 (talk • contribs) 22:47, March 3, 2006 (UTC).

Contents 1 Confusing article 2 A story about Clever Teacher 3 On pain of contradiction 4 Problems

[edit] Confusing article

This article is quite confusing.

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. --Lambiam^Talk 12:44, 3 June 2006 (UTC)

Unfortunately it seems this confusion is not unique to Wikipedia. ("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)

[edit] 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)

[edit] On pain of contradiction

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.

[edit] 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.