## The Principle of Bivalence and the Law of the Excluded Middle. Please help me understand

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The following is my understanding of the two concepts:

Principle of Bivalence (PB): A proposition is either true or false
Law of the Excluded Middle (LEM): Either a proposition is true or its negation is true = P v ~P

PB limits possibilities of truth values to two viz true or false.

LEM states that every proposition is either true or its negation is true.

So, for example take Fuzzy Logic (FL)

FL violates the PB in that a proposition can be partly true or partly false etc.

However take a statement in FL e.g. S = It is sunny today. in FL, S can be partly true but it can't be that S is partly true AND S is not partly true. In short LEM is not violated - no contradiction is derived.

Have I understood it correctly? Thanks.
• 194
Yes, that's it. On the other direction, note that intuitionists generally uphold bivalence, but they reject excluded middle.
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(Y) thanks a ton.
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Is that right? If it's partly true and partly false, and if false isn't true, then it's partly true and partly not true.

And from here, "in a logic based on fuzzy sets, the principle of the excluded middle is therefore invalid".
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Did you understand PB and LEM distinction? Can you explain it to me please.

If it's partly true and partly false, and if false isn't true, then it's partly true and partly not true.

Partly true AND partly false = Partly true and partly not true. So this is a contradiction.
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Why would partly true and partly not true be contradictory? "Part" refers to less than the whole, and there is nothing to indicate which particular part is being referred to. So as long as the part which is true is not the same particular part as the part which is not true, there is no contradiction. "Partly" implies division such that true and not true are not said of the same thing, they are said of different parts, the parts being different parts of the same thing.
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"Partly" implies division such that true and not true are not said of the same thing, they are said of different parts, the parts being different parts of the same thing.

To the extent that I ''understood'' that's the gist of LEM. The partly true and the partly not true must refer to different things. If they're about the same thing then we have a contradiction which violates LEM.
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The logical problem here is in the relationship between a whole and the parts which the whole is assumed to be composed of. If you talk about the parts of a whole, you have logically divided the whole into parts, such that the whole no longer exists. It cannot exist if the parts are thus separated. So to have a premise referring to the parts, and a premise referring to the whole is to have inconsistent, contradictory premises because the premise which refers to the parts assumes the parts as individual things, and these have no individual existence unless the whole has been divided. And this dissolves the whole.

Therefore it is false to assume that a whole is composed of parts. The whole is divided potentially, not actually. So to assume that the whole actually is parts is a category mistake. One may quite readily commit this mistake with the use of set theory.
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The whole is divided potentially, not actually. So to assume that the whole actually is parts is a category mistake.

I think you're committing a category mistake here by conceptualizing truth value in a materialistic sense.

When someone says ''it is raining'' is partly true it doesn't mean raining is decomposed into parts. All it means there's another possibility in truth viz. partly true.
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intuitionists generally uphold bivalence, but they reject excluded middle.

Can you give an example?

The only intuitionist I've read much is Dummett, who rejects both: he takes the principle of bivalence as the semantic correlate of the law of the excluded middle, which is a syntactic rule.

He does uphold tertium non datur :
• ¬¬(P ∨ ¬P) (syntax)
• No proposition is neither true nor false (semantics)

What does it look like to uphold bivalence but not the law of the excluded middle?
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He does uphold tertium non datur

That's Latin for the law of excluded middle (lit. "no third (possibility) is given").
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Yeah, I know. I should have said, Dummett upholds what he calls "tertium non datur". Anyway, it's a different principle.
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Do you guys understand the difference between PB and LEM? If you do kindly explain it to me.

My understanding is this:

PB seems to limit the truth value of propositions to true or false. It doesn't permit another interepretation re truth. This is all good when we have clear-cut divisions as in math e.g. x = 2 is either false or true. However, there are lot of areas in which this clarity is missing e.g. heights and weights of people as relates to concepts such as tall and fat respectively. A person may be neither fat nor thin and neither tall nor short. Basically, PB fails to capture the fuzzy cases and thus, I guess, fuzzy logic.

LEM on the other hand doesn't restrict possible truth values. Rather it simply states: for a given propsition P, either P is true or ~P is true. From here my comprehension is an indirect one. From the law of noncontradiction (LNC) we can see that for any given proposition P it is impossible that P & ~P. In other words ~(P & ~P). Using DeMorgan's rule ~(P & ~P) is logically equivalent to P v ~P which is LEM. Basically the ''middle'' that is ''excluded'' in LEM is the contradiction P & ~P.

In fuzzy logic we saw that PB is false and LEM is true. In paraconsistent logic (I'm guessing here) PB is true but LEM is false.

What do you think?
• 194
Here's an easier example than fuzzy logic. Suppose some statements are true (T), some statements are false (F), and some statements are both true and false (B, also called a truth-value glut)---so bivalence does not work. If statements have their usual truth-values, then negation and disjunction are defined as in the classical case. If a statement is both true and false, then its negation is also both true and false. If both disjuncts of a disjunction are both true and false, then the disjunction is both true and false; if one is true and the other both true and false, the disjunction is true; and if one is false and the other both true and false, then the disjunction is both true and false. Also, suppose that we are less interested in false statements than in not true statements (so both true and false is a designated value). This is a paraconsistent system, by the way, and some who defende dialetheism employ it.

In that case, we have a failure of bivalence but we still have excluded middle: if a statement has a classic truth-value, then the disjunction is classically valid and hence true. If the statement is both true and false, then the disjunction will be both true and false, so true (or, at least, valid).
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LEM on the other hand doesn't restrict possible truth values. Rather it simply states: for a given propsition P, either P is true or ~P is true.

(I'll continue to speak for Dummett as best I can ...)

You can also look at this as an inference rule, or an introduction rule: it says that ⌜P ∨ ¬P⌝ is a theorem in your system, for any P.

Bivalence is not an inference rule, but a semantic principle: ⌜P ∨ ¬P⌝ is always true, which means for any statement either it is true or its negation is.

(The whole point of logic is to tie together truth and theoremhood, but that depends on your system.)

As you say, we have the law of (non)contradiction: ⌜¬(P & ¬P)⌝ for any P. Or its semantic version: No statement is both true and false.

If you expand "¬(P & ¬P)", what you get is "¬P ∨ ¬¬P". To get from there to "¬P ∨ P", you need another rule, something like ⌜¬¬P → P⌝ for any P. Intuitionists do not do this, which is why they end up keeping the law of (non)contradiction but not the law of the excluded middle.

Why wouldn't you accept ⌜¬¬P → P⌝? If you interpret "¬ ..." as "It has not been shown that ..." or something to that effect, then it becomes pretty reasonable. If it hasn't been shown that it hasn't been shown that P, that's not quite the same thing as having shown P, is it?

And thus, the intuitionist rejects bivalence because truth isn't just something a statement has or doesn't, it's something that it could be shown to have or not (whether that's happened yet or not). Through verification if it's an empirical claim, or proof if it's not.

I don't know about other intuitionists, but Dummett also upholds ⌜¬¬(P ∨ ¬P)⌝, which is "No statement is neither true nor false" semantically. He does not allow "truth-value gaps", defended by Frege and Strawson. If you also reject bivalence, the idea is that you cannot show that a statement is neither true nor false, that such a conclusion cannot be reached, that for no P will you be able to produce ⌜¬(P ∨ ¬P)⌝. So not only do you never introduce "P ∨ ¬P", you also never introduce "¬(P ∨ ¬P)".

Which, if you think about it, is not such a tragedy.

I have trouble enough keeping a grip on intuitionistic logic, so I'm not going to try to address fuzzy and paraconsistent logics.
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Just to clarify, is bivalence employing the exclusive or? I've always thought of a non-bivalent logic as one that has a distinct truth-value (e.g. null) rather than one that allows for a proposition to have two truth values as in your example (which just seems to be denying the law of non-contradiction).
• 194
Yes, as far as I know, most readings of bivalence include an explicit clause such as "but not both". Also, note that, in my formalization, B can be (formally) treated as a third truth-value, though its interpretation (for a dialetheist) is the one I gave.
• 194

It may be useful to introduce some distinctions. Let's call the principle of weak bivalence the idea that there are only two truth-values, and the principle of strong bivalence the idea that every sentence must have exactly one truth-value (so it is either true or false). Then most intuitionists accept weak bivalence (there is no third truth-value), but reject strong bivalence. Note that rejecting strong bivalence is not the same as accepting that there is a sentence which doesn't have a truth-value (or that has both, or a third one), since intuitionists don't accept the equivalence between "not every" and "there is an x such that not...".
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The "strong" and "weak" thing works for me.
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Excluded middle is just a law, a rule - a tool - of logical antithesis. It is subject to all the constraints, restraints, and limitations thereby implied. Good for activities with certain features, but not always, and otherwise often not so good at all. Because I had to look up tertium non datur, I found this, that seems worth adding to the conversation (slightly edited):

"Tertium non datur. The reconciling “third,” not logically foreseeable, characteristic of a resolution in a conflict situation....

"As a rule it occurs when the analysis has constellated the opposites so powerfully that a union or synthesis becomes necessary.... Requiring a real solution.., a third thing in which the opposites can unite. Here the logic of the intellect usually fails, for in a logical antithesis there is no third. The “solvent” can only be of an irrational nature. In nature the resolution of opposites is always an energic process: she acts symbolically in the truest sense of the word, doing something that expresses both sides, just as a waterfall visibly mediates between above and below. [“The Conjunction,” CW 14, par. 705.]" (http://frithluton.com/articles/tertium-non-datur/)
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When someone says ''it is raining'' is partly true it doesn't mean raining is decomposed into parts. All it means there's another possibility in truth viz. partly true.

It doesn't mean "raining" is decomposed into parts, but that the world is broken into parts, so that it is raining here, and it is not raining there. Therefore "it is raining" is partly true and partly false by virtue of dividing the world into parts.

If we take the world as a whole, then if it is raining anywhere, "it is raining" is true, and if it is raining nowhere, then "it is raining" is false. But when we divide the world into parts, and refer to one part or another, "it is raining" is both true and false depending on the part of the world being referred to.

The point I made above is that we must state what we are referring to either the whole, or the parts, because it is contradictory to refer to the whole as parts.

All it means there's another possibility in truth viz. partly true.

If there is another possibility, like this, then you deny the LEM. But you wanted to keep the LEM, so we have to find an alternative meaning for "partly true". The way I describe, I believe, is how "partly true" would be commonly used..
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That sounds like Hegel -- we were talking about logic. <ducks>
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The point I made above is that we must state what we are referring to either the whole, or the parts, because it is contradictory to refer to the whole as parts.

Indeed contradiction is dependent on sameness of a given truth. P & ~P is only a contradiction when the two P's refer to the same thing. What I don't see it's contradictory to talk of the whole as parts? Take a chair. I may talk of its seat, its back or its legs without any contradiction. If you're referring to logical entities such as truth and falsity it's clear that truth/falsity can't be divided into parts.

If there is another possibility, like this, then you deny the LEM

No, another possibility of truth value doesn't brrak the LEM. LEM simply puts a restriction on a specific combination of truth values viz. P & ~P.
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Here's an easier example than fuzzy logic. Suppose some statements are true (T), some statements are false (F), and some statements are both true and false (B, also called a truth-value glut)---so bivalence does not work

Is this a bad example?:s
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What I don't see it's contradictory to talk of the whole as parts? Take a chair. I may talk of its seat, its back or its legs without any contradiction.

If you're talking about the seat, you are talking about "the seat", and not "the chair". If you are talking about "the back" you are talking about "the back", not "the chair". Once you divide the chair into parts, such that you are now referring to "the back", or "the seat", or "the legs", each referring to different identified objects, and not "the chair" as a whole, it is contradictory to claim that you are talking about "the chair" when you are referring to "the back"or any one of the other parts. You are not talking about the chair, you are talking about a specific part of the chair.

No, another possibility of truth value doesn't brrak the LEM. LEM simply puts a restriction on a specific combination of truth values viz. P & ~P.

No, LEM explicitly states that there is not any other possibility. It states that of any subject we can predicate either P or ~P, and there is no other possibility. If you insist that there is another possibility of truth value, you break LEM.
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If you're talking about the seat, you are talking about "the seat", and not "the chair". If you are talking about "the back" you are talking about "the back", not "the chair". Once you divide the chair into parts, such that you are now referring to "the back", or "the seat", or "the legs", each referring to different identified objects, and not "the chair" as a whole, it is contradictory to claim that you are talking about "the chair" when you are referring to "the back"or any one of the other parts. You are not talking about the chair, you are talking about a specific part of the chair.

Are you sure about all that?

I could see wanting to get clearer about the logical form of saying "partly ..." but I'd expect some variation there.

If the upholstery of your chair is ugly, doesn't that make the chair ugly? Even if the woodwork is lovely.
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You are not talking about the chair, you are talking about a specific part of the chair.

Ok. Where's the contradiction?

No, LEM explicitly states that there is not any other possibility. It states that of any subject we can predicate either P or ~P, and there is no other possibility. If you insist that there is another possibility of truth value, you break LEM.

I think the PB and LEM are poorly worded - they sound very similar. I'm confused too - that's why the post.

As far as I understand...

PB restricts truth value possibilities to 2 viz. true or false. A proposition can be either true or false. Nothing else. As an example take an electric light switch. It is either on or off and the light is either on or off respectively. No other possibility exists.

Then there are light intensity modulators (M). If it's a dial you turn it and the intensity of light progresses from dark to moderately bright to full brightness. Here M can have 3 states and the bulb can have three states (dark, moderate, bright).

LEM states that either it's true that the light is in one state (dark, moderate, bright) or it's true that the light is NOT in that particular state (dark, moderate, bright). So, either the light is dark or not dark (moderate/bright). Either the light is moderate or not moderate (dark/bright). Either the light is bright or not bright (dark/moderate).

This is how I understand PB and LEM. Does it make sense?

Can you please have a look at my post above. Am I right?
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If the upholstery of your chair is ugly, doesn't that make the chair ugly?

This is a deductive conclusion which requires the further premise that if the upholstery of a thing is ugly, then so is the thing. Otherwise you have a fallacy of composition. Would one ugly spot underneath the seat of the chair make the chair ugly?

Ok. Where's the contradiction?

If you don't see the contradiction in referring to a "leg", and after this, saying that it is a "chair" you are referring to, without a premise which says a leg is equivalent to a chair, then I can't help you. A leg is not a chair. You don't seem to know what contradiction means.

I think the PB and LEM are poorly worded - they sound very similar. I'm confused too - that's why the post.

If I understand the Wikipedia article correctly, exception to PB is a claim of exception to the law of non-contradiction, instead of claiming exception to the law of excluded middle. So to violate PB is to claim "both P and ~P", whereas an exception to the law of excluded middle would claim "neither P nor ~P".

You can see that it is a matter of interpretation, as one might interpret "both P and ~P" as an instance of "neither P nor ~P". And it is often claimed by those philosophers who study the three fundamental laws of logic, that they are all associated with each other, and to violate one, is to throw them all away. What I believe is that the principal law is the law of identity, and that the following two, NC and EM put restrictions, or rules on to how something is identified.

So to insist on an exception to LEM or to LNC, is to insist on a variance as to how to identify something. And once you get to the finer points of exactly what it is which is being identified, the difference between breaking the LEM and breaking the LNC become significant.
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This is a deductive conclusion which requires the further premise that if the upholstery of a thing is ugly, then so is the thing. Otherwise you have a fallacy of composition

Fair enough.

I'd still say that informally talking about a part may often count as also talking about the whole, that this deduction is in fact made, or expected or implied.

"What do you think of my new chair?"
"Um, the woodwork's lovely."

That's an answer that encourages the fallacious conclusion that the chair is lovely, when it's not, because the upholstery is ugly.

The logical high ground here is yours; I'm just pointing out that the linguistics isn't always so simple.
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Something @Nagase mentioned is helpful here, the idea of a designated value. With the 3-position light switch, there are two obvious ways to do this: entirely on, or one of the others; entirely off, or one of the others.

Dummett uses the comparison of conditional bets to conditional commands. If I tell you not to leave without saying goodbye (i.e., if you leave, ...), not leaving (and not saying goodbye) counts as compliance. But if I bet you that if the Cubs make it to the World Series they'll win, and then they're knocked out in the NLCS, you don't owe me a thing.

Then the question is, which one is assertion like? If you take "truth" as the designated value, you can allow various ways of not being true and lump them together. (More like a bet.)

As for bivalence versus excluded middle: your standard switch can be in one of two positions; whether that also turns on the lights depends on whether there's power. Switch position is syntactic; lights going on or off is like adding an interpretation to your system that assigns truth and falsehood -- semantics. The first is LEM, the second PB.

But the terminology is so confusing, it's best just to be explicit about what you're doing, even making up terms as Nagase did with "strong" and "weak" bivalence. It's the ideas that matter not the terminology.
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A leg is not a chair.

:D Where in the world did I say that or anything that could be interpreted as that. I think your materialistic interpretation is a category error.

If I understand the Wikipedia article correctly, exception to PB is a claim of exception to the law of non-contradiction, instead of claiming exception to the law of excluded middle. So to violate PB is to claim "both P and ~P", whereas an exception to the law of excluded middle would claim "neither P nor ~P".

It's exactly the opposite. Violating PB is admitting a multivalued logic that I described. Violating LEM is a contradiction.
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