9 All Greek to me. Anyway(Y)

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.

The logical high ground here is yours; I'm just pointing out that the linguistics isn't always so simple. — Srap Tasmaner

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.

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. — 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".

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

Is this a bad example?:s — TheMadFool

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.

I still don't get it. Let me ask you 2 questions:

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.

What would a TRIvalent system look like? — TheMadFool

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".

2. In LEM what is the ''middle'' that is ''excluded''? — 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.

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 — Metaphysician Undercover

Ok. This is understandable.

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. — 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?

Is ''the apple is red'' AND ''the apple is not red'' also excluded [by LEM]? — TheMadFool

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.

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. — 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...

2. If it isn't then it leads us to a contradiction and also, why? — TheMadFool

Well, dialetheists can live well with contradictions, since they also drop ex falso quod libet, so contradictions don't trivialize the system.

No, since disjunction is inclusive. — Nagase

So, if LEM doesn't exclude [P & ~P] what is this ''middle'' that's being ''excluded''?

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. — 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.''???!!!

So, if LEM doesn't exclude [P & ~P] what is this ''middle'' that's being ''excluded''? — TheMadFool

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).

That is to say LEM and LNC are logically equivalent. — 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.

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.''???!!! — 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.

LEM literally says "either p or ~p", not "either p or ~p, but not both". — Nagase

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. That is, the "middle" position, that Socrates is neither mortal nor not-mortal, is excluded by logic, and therefore either the first possibility (Socrates is mortal) or its negation (it is not the case that Socrates is mortal) must be true. — Wikipedia

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.

Is ''the apple is red'' AND ''the apple is not red'' also excluded? — TheMadFool

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.

2. If it isn't then it leads us to a contradiction and also, why? — TheMadFool

I don't see why you say this.

So it is excluded ''that neither Socrates is mortal nor Socrates is not mortal'' — TheMadFool

Yes, that's the LEM, it cannot be the case that both "Socrates is mortal", and "Socrates is not mortal" are false.

That means it is excluded that (P & ~P). That's the LNC: ~(P & ~P). — 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.

What is denied by LEM is that there is a third option, that the apple is neither red nor not red. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

Can you expand on this a bit. Sorry for the trouble.

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. — TheMadFool

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.

Can you expand on this a bit. Sorry for the trouble. — 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.

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. — TheMadFool

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.

We have to agresive here.

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 Laws of Non-Contradiction, Excluded Middle, and Bivalence

The Law of Non-Contradiction (LNC): ~ [X & ~X].

• Nothing can both be and not be.
• A proposition X and its logical negation ~X cannot both be true together.
• A proposition X cannot be both true and false.
• The joint affirmation of contradictories is denied!
• Something cannot both be and not be.

The Law of Excluded Middle (LEM): X V~X.

• Either a proposition X is true or its negation ~X is true.
• It cannot be the case that neither X is true nor ~X is true.
• A proposition X cannot be neither true nor false (i.e., not true).
• A proposition X and its negation ~X cannot both be false together!
• Excluded middle logically excludes the "joint denial of contradictories (X, ~X)," also called "nor" operator, which stands for neither - nor:

The Law of Bivalence (LOB): X xor ~X

• A proposition can only bear/carry one truth value, that truth value being either true or false, not both, and not neither!
• A proposition X and its negation ~X can neither be true together nor false together.
• A proposition X is either true or false; where the "or" operator is to be understood as an exclusive-or [i.e., exclusive disjunction: = ‘xor’], which logically excludes both the “and” and the “nor” operations of contradictories X and ~X:
• The conjunction (the “and” operation) of X and ~X is called the “joint affirmation” of contradictories (X,~X), which yields the both-and-option which states: both X and ~X are true. Therefore, the law of bivalence excludes this option: {i.e., ‘X is true’ and ‘~X is true’}. Therefore, the “joint affirmation” of X and ~X is denied by the law of bivalence.
• The “joint denial” of contradictories X and ~X is the neither-nor-option that says, “neither X is true nor ~X is true”. This joint denial is also excluded by the law of bivalence. This neither-nor option is a result of the "nor" operation of contradictories (X, ~X):
• [X nor ~X] = {‘X is false’, and ‘~X is false’};** i.e., “neither X nor ~X is true”.
• The law of bivalence excludes the options in which a proposition X and its negation ~X are both true together or both false together. The joint affirmation (both-and-option) and the joint denial (neither-nor-option) of contradictories are logically excluded by the law of bivalence.

Comparing & Contrasting:
Non-Contradiction (LNC) vs.
Excluded Middle (LEM) vs.
Bivalence (LOB)!

Four a proposition X, the following options exist:
. X
[ii]. ~X
[iii]. Both X and ~X
[iv]. Neither X nor ~X

Each option can be reformulated as follows:
= 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.

Please Note the Important Differences Between the Laws of Excluded Middle and Bivalence!

• Note that the law of excluded middle (LEM) uses the operator "or" (inclusive disjunction), while the law of bivalence (LBi) uses "xor" (exclusive disjunction).
• Further note that LEM (excluded middle) contains a negation ("not" = "~") in its formula; whereas, LBi (bivalence) does not include a negation in its expression!
• Moreover note that LEM is only expressed in terms of the truth value of 'true' and does not include 'false'! Bivalence, on the other hand, is mathematically expressed in terms of 'true' and 'false' and does not include ("not" = "~") as a connective.
• In LEM, the negation operator ("not" = "~") serves as a logical connective, not as a truth function! Whereas, LBi (bivalence) is a principle about negation ("not" = "~") as a truth function!
• Therefore, LEM is a syntactical principle of logic, while LBi is a semantical principle of logic.

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!

• LEM makes the joint denial (the "neither-nor" option) logically impermissible.
• LNC makes the joint affirmation (the "both-and" option) logically impermissible.
• Therefore, LBi - being the conjunction of LEM and LNC - makes both the joint denial and the joint affirmation impermissible!

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.

This is a paraconsistent system, by the way, and some who defende dialetheism employ it. — Nagase

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.

That sounds like Hegel -- we were talking about logic. — Srap Tasmaner

: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.

I've started warming to the general idea of dialectic — Srap Tasmaner

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.

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. — Tristan L

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.

As I understand it, PB states that there are only two (bivalence) truth values viz. true and false — TheMadFool

Right.

the LEM states that (p v ~p) [...] — TheMadFool

Also right, and since it is a law, it states that this is true.

[...] which simply means that given a proposition, either the proposition itself is true or its negation is true. — 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.

(TRUE(p) v TRUE(~p)) = (p v ~p)

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).

Statements about the future are a different story.

Why should they be? They are propositions like others, and they neatly show the difference between LEM and PB since the former applies to them but the latter doesn't.

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.