The Principle of Bivalence and the Law of the Excluded Middle. Please help me understand
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.
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.
Comments (64)
And from here, "in a logic based on fuzzy sets, the principle of the excluded middle is therefore invalid".
Quoting Michael
Partly true AND partly false = Partly true and partly not true. So this is a contradiction.
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.
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.
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.
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.
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 :
What does it look like to uphold bivalence but not the law of the excluded middle?
That's Latin for the law of excluded middle (lit. "no third (possibility) is given").
Yeah, I know. I should have said, Dummett upholds what he calls "tertium non datur". Anyway, it's a different principle.
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?
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).
(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.
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...".
The "strong" and "weak" thing works for me.
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.
Quoting TheMadFool
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..
That sounds like Hegel -- we were talking about logic.
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.
Quoting Metaphysician Undercover
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.
Is this a bad example?:s
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.
Quoting TheMadFool
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.
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.
Ok. Where's the contradiction?
Quoting Metaphysician Undercover
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?
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?
Quoting TheMadFool
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.
Quoting TheMadFool
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.
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.
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.
: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.
Quoting Metaphysician Undercover
It's exactly the opposite. Violating PB is admitting a multivalued logic that I described. Violating LEM is a contradiction.
Dude, I may be a little further along than you, but don't think I don't struggle to understand this stuff! Totally worth it though, so keep at it.
What I'm trying to get at, is that there is an important ontological issue here with respect to how we look at the relationship between parts and wholes. When we talk about something, we identify an object which is referred to by the word. The object itself may be apprehended as a an independent whole, or it may be apprehended as a part of a larger whole. Whichever of these two is the case may be explicitly stated, implied, or left ambiguous. So when you say "the woodwork's lovely", it's implied by the context (the preceding question), that this object is a part of a larger whole, the chair.
The point which I would like to bring to your attention, is the act of dividing the chair, by identifying the different parts as individual objects. It's not that we actually cut the chair into pieces by identifying the different parts, but we do this in principle, logically. So when we have identified the different parts, and are speaking about the different parts, logically we no longer have a whole which is "the chair". The chair has been divided logically. This is because there is no law of logic which properly establishes the relationship between the parts and the whole. Each is identified simply as an object. But since the role of a part, in relation to a whole, varies according to the particular part, and the particular whole, there is no logical principle which states how what is true of the part relates to what is true of the whole, and vise versa.
Quoting TheMadFool
You asked me "where's the contradiction", and so I answered that the contradiction is in thinking that when one is referring to the part (the leg), one is referring to the whole (the chair). I was surprised that you did not see this as a contradiction, and that you asked, "where's the contradiction".
Quoting TheMadFool
Well I think you have this backward. Violating LEM is not contradiction, that's why they have the law against contradiction as well as the LEM, it's two different things. Violating the LNC is to say that of the subject both P and ~P are applicable. That would be contradiction. Violating LEM would be to say that something else applies, which is neither P nor ~P. According to the Wikipedia article on PB, to violate PB is to violate the LNC. And this is what you said when you said that the thing is partly P and partly ~P, that it is both P and ~P. To say that it is neither P nor ~P (violate LEM) is something completely different.
Why is it a bad example?
In any case, it helps to distinguish between: semantics, which is how we interpret our system, from syntax, which is concerned with which expressions are allowable in the system. Bivalence is a matter of semantics (how are going to interpret the system's formulas? Are going to allow only two truth-values? Is every formula required to have a truth-value?), whereas excluded middle is a matter of syntax (can we show that, given an initial set of sentences---the axioms---, plus certain rules of transformations---the rules of inference---, we are able, for every formula p, to reach "p v ~p"?). Of course, these two dimensions are not wholly independent, in the sense that we usually want there to be a certain parallelism between the two, but there is a certain amount of freedom in how to enact this parallelism.
1. I more or less understand that PB allows either true or false. What would a TRIvalent system look like?
2. In LEM what is the ''middle'' that is ''excluded''?
Thanks for the clarification. I think I have some grasp of the idea now.
Can you have a look at the above questions. Thanks.
According to Wikipedia a trivalent logic has three truth values, true, false, and an indeterminate third value. The third value appears to be best described as "unknown".
Quoting TheMadFool
The excluded middle is anything other than "is" or "is not". Either the apple is red, or the apple is not red, and the LEM insists that there is no "middle", between being red and being not red.
Ok. This is understandable.
Quoting Metaphysician Undercover
Is ''the apple is red'' AND ''the apple is not red'' also excluded?
1. If it is then why? Also raises another issue viz. why have the law of noncontradiction? It seems to be a corollary of LEM.
2. If it isn't then it leads us to a contradiction and also, why?
No, since disjunction is inclusive. A few posts ago I described a system in which you have a truth-value glut (i.e. a proposition being true-and-false), but in which LEM was upheld. Notice that, strictly speaking, from a contemporary point of view, the so-called law of excluded middle just says that, for any formula p, the string "p v ~p" is acceptable in the system. Note also that the disjunction relates p to not p, and not p to its falsity or whatever. This gap can be exploited to give a different semantics to negation, in such a way that (what I have called) weak bivalence is upheld.
Quoting TheMadFool
Actually, the situation is probably the reverse. Assuming classical principles, such as double negation and reductio ad absurdum, it's possible to prove LEM from LNC. Of course, those are precisely the principles questioned by intuitionists...
Quoting TheMadFool
Well, dialetheists can live well with contradictions, since they also drop ex falso quod libet, so contradictions don't trivialize the system.
So, if LEM doesn't exclude [P & ~P] what is this ''middle'' that's being ''excluded''?
Quoting Nagase
Can you give me a short proof from LNC to LEM?
Let me try:
.....................~(P & ~P) > (P v ~P)
1. ~(P & ~P).............assume for conditional proof
2. ~P v ~~P..............1 DeMorgan
3. ~P v P...................2 Double Negation
4. P v ~P...................3 Commutation
5. ~(P & ~P) > (P v ~P).....1 to 4 Conditional proof
Now the other way round:
....................(P v ~P) > ~(P & ~P)
1. P v ~P...........assume for conditional proof
2. ~~P v ~P......1 Double Negation
3. ~(~P & P).....2 DeMorgan
4. ~(P & ~P).....3 Commutation
5. (P v ~P) > ~(P & ~P)....1 to 4 Conditional proof
So (P v ~P) <=> ~(P & ~P)
That is to say LEM and LNC are logically equivalent.
So, what I can't get is what you mean when to my question ''Is ''the apple is red'' AND ''the apple is not red'' also excluded [by LEM]?'' you said ''No, since disjunction is inclusive.''???!!!
I don't think it's helpful to concentrate so much on the name of the principle, in this case. The name comes from a time when the separation of syntax from semantics was not so clear, so it is inevitable that there will be some confusion attached to it. In any case, historically, the idea has been that, for any statement p, either p is true or ~p is true, and there is no third or middle option. You could perhaps think of p and ~p as "poles" and LEM saying that there is no intermediary position between them (indeed, we still refer to the polarity of a proposition, i.e. whether it is an assertion or denial).
Quoting TheMadFool
Assuming classical logic. Note that you used both double negation and DeMorgan in your proofs; intuitionists, for instance, deny both these principles (well, they accept weaker forms of them which will not by themselves be able to prove this equivalence). As an exercise, try proving the equivalence without using these or r.a.a (which is also rejected by intuitionists).
EDIT: Note that, if you assume classical logic, any tautology is equivalent to any other (they are always true), so, assuming classical logic, of course LEM is equivalent to LNC. It is also equivalent to p -> (q -> p), and infinitely other formulas. The question is if we can prove them equivalent without assuming classical logic.
Quoting TheMadFool
I'm not sure what is the problem. LEM literally says "either p or ~p", not "either p or ~p, but not both". That is, the disjunction there is inclusive, not exclusive.
[Quote=Wikipedia] For example, if P is the proposition:
Socrates is mortal.
then the law of excluded middle holds that thelogical disjunction:
Either Socrates is mortal, or it is not the case that Socrates is mortal.
is true by virtue of its form alone. [B]That is, the "middle" position, that Socrates is neither mortal nor not-mortal, is excluded by logic[/b], and therefore either the first possibility (Socrates is mortal) or its negation (it is not the case that Socrates is mortal) must be true.[/quote]
Bold emphasis mine.
So it is excluded ''that neither Socrates is mortal nor Socrates is not mortal''
Let P = Socrates is mortal
That means it is excluded that (P & ~P). That's the LNC: ~(P & ~P).
So, LEM isn't the inclusive OR at all.
No, this is denied by the LNC. What is denied by LEM is that there is a third option, that the apple is neither red nor not red.
Quoting TheMadFool
I don't see why you say this.
Quoting TheMadFool
Yes, that's the LEM, it cannot be the case that both "Socrates is mortal", and "Socrates is not mortal" are false.
Quoting TheMadFool
Correct, the LNC says that it cannot be the case that both "Socrates is mortal", and "Socrates is not mortal" are true.
Do you see the difference between LEM and LNC? One says that two opposing statements cannot both be true, the other that two opposing statements cannot both be false.
In logic the expression ''neither...nor...'' has a specific translation:
Apple is red = R
Neither the apple is red nor the apple is not red = Apple is not red AND not apple is not red = the apple is not red AND the apple is red = ~R & R = R & ~R
The ''middle'' that is ''excluded'' is the contradiction R & ~R.
Quoting Metaphysician Undercover
Can you expand on this a bit. Sorry for the trouble.
What you have offered here is a specific interpretation which I have never seen. It is not the interpretation of "logic", but perhaps of a specific logical system which I am unfamiliar with. It is an illogical interpretation to me, for the following reason.
"Neither the apple is red nor the apple is not red", refers to a subject, "the apple", and states that it cannot be determined whether the apple is red or not. Perhaps "apple" is not something which has a colour. Whereas, "Apple is not red AND not apple is not red" negates the subject "apple" with "not apple". From here, you cannot proceed to your third statement "the apple is not red AND the apple is red" because you have negated the subject, "the apple" with "not apple".
There is ontological significance to the difference between violating the LNC and violating the LEM. It reflects how we view the existence of the object, which is represented as the subject. Consider quantum physics, and lets say that the subject is a particular electron. If we violate LNC we would say the electron is at X (spatial temporal location), and, the electron is not at X. But the predication "is at X" is the identifying feature of that particular electron, so if we also deny that the electron is at X, we deny our capacity to identify the object, as the subject referred to in the logic.
So the result of denying the LNC is that we assume a subject, with a corresponding object, but we claim with the denial of LNC that it is impossible to identify that subject. Contradiction is inherent within this position because we claim an object (therefore identify the object, as the named subject), yet we deny the possibility that the object has an identity. Ontologically there is no validity to the assumption of an object, it is a name without anything real that the name refers to, as the thing is described by contradictory terms. If there is contradiction within the identity of the object we deny the reality of the object. But then we proceed to talk about the object anyway, ignoring this.
If we violate LEM instead, then with the same example, we have an electron, the object represented as the subject, and we say that the predication "is at X" is false, and the predication "is not at X" is false. Again, we claim an object, represented by the subject, but the contradiction here is not inherent within the claim of an object, it is in the mode of identifying the object, the predication. We have not found the true method to identify the object. So in this case, the contradiction is believed to be within the description of the object, it is not represented as within the reality of the object (which would deny the reality of the object).
The difference then, is that when we violate LNC we allow that contradiction is intrinsic within the identity of the object, the object is inherently contradictory. This negates the reality of the object as contradictory. When we violate LEM the contradiction is within the way that we are attempting to describe the object, this negates the description of the object as contradictory. In the one case, the object is seen as inherently indescribable, while in the other case, the object is indescribable due to faults in the describing method.
Quoting TheMadFool
The difference between assigning "true" to a statement, and assigning "false" to a statement is that when you assign "true" you exclude all other possibilities, and when you assign false you allow all other possibilities. So the one, "true", denies other possibilities while the other, "false", allows other possibilities.
When you violate LNC you say "X is true" and "not X is true", thus denying any other possibilities. When you violate LEM you say "X is false" and "not X is false" allowing for all other possibilities.
First, the disjunction is defined as being inclusive. So any formula with it as its main connective will be inclusive, by definition. So, e.g., p v (q & ~q) is inclusive, even if (classically) both disjuncts can't be true at the same time.
Technically, what is excluded is ~(p v ~p), that is, the negation of the excluded middle, so what you have is ~~(p v ~p). But this is only equivalent to LNC if you assume classical logic (and then it will also be equivalent to infinitely many other tautologies, such as p -> (q -> p), so this equivalence is completely uninteresting). In order to see if they are indeed equivalent, you need to see what happens when you go to other logics. Again, in intuitionist logic, you have LNC, but not LEM, so LNC does not imply LEM. Conversely, in a paraconsistent logic such as the logic of paradox, you can have LEM, but not LNC. So LEM does not imply LNC either.
I think I'm getting there.
PB states that, for a given proposition, either P is true OR P is false (but not both). There is no other truth value e.g. ''uncertain'' or ''x% true'' or (P & ~P).
LEM states that, given a proposition P, either P is true OR ~P is true i.e. (P v ~P). ~(P v ~P) is the ''middle'' that's excluded i.e. ~~(P v ~P).
Since ''v'' in (P v ~P) is an inclusive OR, there's the possibility that (P & ~P) and that's why we need the LNC: ~(P & ~P).
In Fuzzy logic, it's PB that's broken. We have the possibility that a proposition P is neither true nor false, as when P is x% true.
For breaking LEM we need uncertain propositons like P = it'll rain tomorrow. In this case, neither P is true nor ~P is true.
Have I got it now? Thanks for your patience.
There is no partial truth. And partial falseness.
Truth is what is true.
What we normally call truth is not such. It's all Aristotle applying PNC and PEM (don't call them laws please) to contingent identities.
Here's the explanation:
Although it is undeniable of that which is there it is. And it is irrational asserting that it is not.
However here comes the epistemological problem:
What there is?
If one is certain then one is trully aware of what there is. If one is not trully aware, and therefore certain of what there is then one ignores what there is. So one CONTINGENTLY assumes what could there be (axiomatization).
It is applying PNC and PEM to contingent identities what leads to incompleteness and therefore to the ilussion of the absence of truth.
Only when PNC and PEM are applied to concepts that are not contingent is that the mind can grasp truth.
Therefore I introduce you guys to the actual disctinction:
Contingency / Non-Contingency.
The Law of Non-Contradiction (LNC): ~ [X & ~X].
The Law of Excluded Middle (LEM): X V~X.
The Law of Bivalence (LOB): X xor ~X
Comparing & Contrasting:
Non-Contradiction (LNC) vs.
Excluded Middle (LEM) vs.
Bivalence (LOB)!
Four a proposition X, the following options exist:
[i]. X
[ii]. ~X
[iii]. Both X and ~X
[iv]. Neither X nor ~X
Each option can be reformulated as follows:
[i] = 1, [ii] = 2, [iii] = 3, [iv] = 4:
1. X is true
2. ~X is true (i.e. X is false)
3. X is both true and false
4. X is neither true nor false
In classical logic, options (3/iii) and (4/iv) are forbidden, i.e., logically impermissible / excluded by logic.
Options 3 and iii are excluded by the law of non-contradiction.
Options 4 and iv are excluded by the law of excluded middle.
Law of Non-Contradiction (LNC): ~ (X & ~X),
(where “&” is logical conjunction: "and" operator).
The law of non-contradiction (LNC) states the following logically equivalent statements:
It cannot be the case that a X and its negation ~X are true together (at the same time, in the same sense, simultaneously).
Non-contradiction excludes the joint affirmation of X and its negation ~X: that is, it cannot be the case the both X and ~X are true.
If two propositions are direct logical negations of one another (X, ~X), then at least one of them is false, including the option that both are false and excluding the option that both contradictories are true together.
A proposition X and its negation ~X cannot both be true.
Contradictions cannot be (i.e., are excluded or ruled out).
Contradictory propositions cannot both be true.
Nothing can both be and not be. That is, something cannot both be and not be.
The law of non-contradiction (LNC) can be reformulated as stating: A proposition X cannot be both true and false!
The law of non-contradiction does not exclude the case that both X is false and ~X is false!
The law of non-contradiction states at least one of X and ~X is false, including the option that both X and ~X are false together, but excluding the option that X and ~X are true together.
Out of two contradictories, at least one of them is false; they can both be false, but they cannot both be true.
Hence, the law of non-contradiction excludes only the joint affirmation of a pair of direct logical negations ("X is true" and "~X is true").
Law of Excluded Middle (LEM): X V ~X,
where V = inclusive disjunction ("or").
LEM states: either a proposition X is true or its negation ~X is true, where "or" is inclusive-or, i.e., LEM includes the conjunction (X & ~X).
LEM states a proposition X is either true or not true (i.e., false), where "or" includes the option that: "X is both true and not true (i.e., false)". Since the inclusive-either-or (inclusive disjunction, "or") of X and ~X can be expressed as the negation (~) of the joint denial (neither-nor, "nor"): inclusive-either-or = not-neither-nor; therefore:
A proposition X and its negation ~X cannot be both false together.
LEM states it cannot be the case that neither X is true nor ~X is true, which can be equivalently stated as follows:
A proposition X cannot be neither true nor false (i.e., not true).
LEM logically excludes the neither-nor option: the option generated from the “nor” operation of the two contradictories X and its negation ~X: [X nor ~X]. That is, the joint denial (i.e., “neither-nor”) of both X and ~X is excluded by the law of excluded middle.
The logical "nor" operation called "joint denial" of contradictories (X, ~X)! The joint denial of {'X is true' and '~X is true'} is the option that says neither X nor ~X is true; that is, (X is false, ~X is false). Denial of X means denying that X is true, and is not mere failing to accept that "X is true" (i.e. reject); quite to the contrary, to deny X is to accept that its logical negation ~X is true, which leads to therefore "X is false".
LEM does not exclude the case that both X is true and ~X is true. LEM does not rule out contradictions!
LEM states at most one of the contradictories X and ~X is false.
LEM states at least one of the contradictories X and ~X is true.
LEM states that at least one of X and ~X is true:
I. {X is true and ~X is true} is excluded by non-contradiction (LNC) & bivalence (LOB)
II. {X is true and ~X is false}
III. {X is false and ~X is true}
IV. {X is false and ~X is false} is excluded by excluded middle (LEM) & bivalence (LOB)
The law of bivalence (henceforth, LOB) states that X is either true or false
LOB includes exactly one of X and ~X is true, and the other false, and vice versa, and moreover excludes both the joint affirmation and the joint denial of contradictories (X, ~X).
Note that LOB does not have a negation operator (~) in its expression (whereas LEM does!)
Further note that the law of bivalence can be expressed as: “X or ~X” where the "or" operator is to be understood as an exclusive-or (i.e., "xor", also denoted as "(+)"); therefore: LOB = X xor ~X.
An exclusive disjunction [“xor”] of X and ~X is also called "The Exclusive Disjunction of Contradictories (X, ~X): [X xor ~X]”: = LOB
LOB excludes both the 'joint affirmation' (i.e., X is true AND ~X is true) as well as excluding 'joint denial' (i.e., X is false AND ~X is false).
A proposition X and its negation ~X form the following permutations
(rows in the truth table)
{X is true and ~X is true} is excluded by non-contradiction (LNC) & bivalence (LOB)
{X is true and ~X is false}
{X is false and ~X is true}
{X is false and ~X is false} is excluded by excluded middle (LEM) & bivalence (LOB)
LOB states, exactly one of (X, ~X) is true, and the other one false.
LOB states {either "X is true" or "~X is true"},
and it cannot be neither [X nor ~X],
and it cannot be both [X and ~X]!
Therefore, the law of bivalence (LOB) can be reformulated as follows:
"Something is not neither or both what it is (X) and what it is not (~X)".
So, the law of bivalence excludes options (3/iii) and (4/iv) because
LOB = LEM & LNC
The law of bivalence is the conjunction of excluded middle and non-contradiction!
LOB = LNC & LEM.
Bivalence states that a truth variable X, i.e., a proposition ("truth-bearer"), can only carry one truth value at a time, that (single) truth value being either "true" or "false"; where or is to be understood as an exclusive disjunction, which logically excludes the conjunction of the contradictory disjuncts X and its negation ~X.
The Law of Bivalence is the Conjunction of the Laws of Excluded Middle and Non-Contradiction!
Not really, I think. While you've understood PB rightly, your version of LEM is too strong, for it's simply another way to state PB. That's because falsehood is the same as truth of the negation. In truth, LEM only states that A OR NOT(A) is always true, while it is PB which states that either A is true or NOT(A) is true. In probability theory,
- PB is the statement that each event is either certain or impossible, which is false except when the probability space is trivial,
- while LEM is the statement that the union of any event with its complement is certain, which is always true.
Correct me if I'm wrong, LEM/Bivalence does not apply to things like natural phenomena and Being. Dialectic (both/and v. either/or) reasoning made that distinction.
:kiss:
You know it's funny you spotted that in this necro-thread because lately I've started warming to the general idea of dialectic -- though I don't know that I'll live long enough to develop an interest in Hegel. Don't tell apo.
Nasty. You might be able to get an ointment for it.
I used Popper's critique of historicism recently, which is perhaps why I noticed you comment in this little zombie. Perhaps is becoming the fashion again.
As I understand it, PB states that there are only two (bivalence) truth values viz. true and false and the LEM states that (p v ~p) which simply means that given a proposition, either the proposition itself is true or its negation is true.
Quoting TheMadFool
Right.
Quoting TheMadFool
Also right, and since it is a law, it states that this is true.
Quoting TheMadFool
Actually no, if my understanding is right. The proposition (p v ~p) is weaker than the proposition (TRUE(p) v TRUE(~p)) since stating the truth TRUE(A) of a proposition A is strogner than just stating the proposition A itself. The same is true in probability theory; (E UNION COMPLEMENT(E)) always has a probability of 1 (LEM holds), but ((E has probability 1) OR (COMPLEMENT(E) has likelihood 1)) is sometimes false (PB fails).
Indeed, you're assuming PB when you regard A and TRUE(A) as equivalent, which you're doing in your interpretation of LEM.
Actually, I think not. If p = (it will rain tomorrow), then (p v ~p) is true today because of LEM, but (TRUE(p) v TRUE(~p)) is not true and even untrue (false) today since today, neither p nor ~p is true because the future isn't forechosen. This counter-example shows that the truth-operator doesn't in general distribute over disjunction. Indeed, (TRUE(p) v TRUE(~p)) is a stronger proposition than TRUE(p v ~p). This parallels probability theory, where (P(A)=1 v P(COMPLEMENT(A))=1) (which is only true for rather boring events A) is stronger than (P(A UNION COMPLEMENT(A)) = 1) (which always holds true).
I think that one should give up truth-functionality, the (imho false) principle that the truth-values of individual propositions always set the truth-value of compound propositions "built up" from them. In fact, in any truth-functional three-valued logic where LEM, the Law of the Idempotence of Disjunction (LID, (A v A = A)), and the Law of the Evenness of Undeterminedness with respect to Negation (LEUN, (UNDETERMINED(A) = UNDETERMINED(~A))) hold true, PB can be derived like so:
For every propostion A with truth-value U (undetermined), we have:
1. UNDETERMINED(A) by premise
2. UNDETERMINED(~A) from (1.) by LEUN
3. TRUE(A v ~A) by LEM
4. UNDETERMINED(A v A) from (1.) by LID.
Thus, if we set B := ~A, C := A, and D := A, we have that A and B are both undetermined but their disjunction is true, whereas C and D are also both undetermined but their disjunction is undetermined rather than true. Thus, the logic isn't truth-functional if there is a proposition A with truth-value U, that is to say, if it is truth-functional, then each proposition is either true or false (PB holds).
Instead of LID and LEM, we can also use the Law of the Idempotence of Conjunction (LIC, (A AND A = A)) and the Law of Not-Contradiction (LNC, ~(A AND ~A)).
That's why I'm in favor of a three-valued logic which isn't truth-functional.
The truth value of statements about the furure is controversial or so I heard.
You may also want to check out what I said earlier and tell me what you think.
I thought that's precisely why we should avoid them - why add fuel to fire and make the matter more complicated?. Let's consider the LEM and the PB with respect to statements about the present first shall we. Please comment on whether I've understood PB and LEM as it concerns either propositions that are timeless or propositions about the present. Thank you for your concern.