Basics: bivalence, excluded middle & noncontradiction

[honestrosewater]

I'm trying to achieve a complete, precise understanding of the differences between these laws. A precise clarification of the following terms as they relate to each other is needed: true, not true, false, negation.

 

[AKG]

LNC: \neg (P \wedge \neg P) : It is not true that some proposition P and it's negation are true.

LEM: P \vee \neg P : A proposition is true or it's negation is true, and the two need not be exclusive.

Principle of Bivalence: Either P is true, or it is false, and the two are exclusive.

Now, it is an entirely different subject to define truth, and different theorists will define it differently. Anyhow, given some definition for "true," then "not true" is, quite basically, what you can infer from the laws stated above. "Negation" and "not true" go hand in hand. The negation of P is equivalent to the proposition that P is not true, if we go with the convention that "P" is equivalent to "P is true" or "it is true that P." So explaining what "not true" is can be done by explaining "negation." What is negation, how does it work? Well, it works according to the 2 laws (not talking about the principle yet) above (according to most logics). How does implication work, i.e. what does "implies" mean? To get the answer, you might look at modus ponens and modus tollens, which essentially define how "implies" works. Similarly, the laws above essentially define how negation works.

The principle of bivalence basically says that \neg P implies P is false, or rather, that if P is not true, then it is false (and vice versa). This is a slightly stronger condition. If we say that "this statement is false" is a proposition (which is arguable), then the LNC simply says that it is not both true and not-true. And in fact, it is easy to say that it is not true. By the principle of bivalence, this would render the propsition false, but it's also not-false. We can easily get away with saying that it is not-true and not-false, but if we add the principle of bivalence, then we get contradictions, if we suppose that it is a reasonable proposition. Also, some logics subdivide "not true" into two categories, including "false" and "possible" and some divide up truth values infinitely, and place them on a scale from 0 to 1, 0 representing falsehood, 1 representing truth, and numbers in the middle representing something between truth and falsehood (fuzzy logic).

If you want to think of them as sets, then the law of excluded middle say that:

P \cup \neg P = U

Where U is the universal set. So these sets together make up the whole unviersal set, but it doesn't say that they don't overlap. The law of non-contradiction says that:

P \cap \neg P = \emptyset

It says that the two sets do not overlap (but doesn't say that they cover the whole universal set). Together, they say that the two partition U. The principle of bivalence says that \neg P is not divided up into further sets, it is only one set, and it can be labeled "false." Some logics say that it can be divided up into two sets, some say that it can be divided up into infinitely many sets.

 

[honestrosewater]

Oh, you rock! I will read it thoroughly tomorrow and reply. Thank you! :!)

 

[honestrosewater]

Thanks again. To be sure I understand fully, I'll rephrase your comments.

LNC: a proposition p cannot be both true and not true (in the same instance).
LEM: for all p, "true" or "not true" are the only options.
POB: There is only one reason for p to be not true: because it is false.

Assuming LNC, LEM & POB, if p="There will be a sea battle tomorrow" is not true, then there will not be sea battle tomorrow. Some logics reject or modify LEM so they can have POB's falsifying power while allowing other options in addition to "true" and "not true". Correct?

BTW I like the set explanations, especially the partitioning concept.

 

[Pavel]

LEM: P \vee \neg P : A proposition is true or it's negation is true, and the two need not be exclusive.


If you want to think of them as sets, then the law of excluded middle say that:

P \cup \neg P = U

Where U is the universal set. So these sets together make up the whole unviersal set, but it doesn't say that they don't overlap. The law of non-contradiction says that:

Can you please give an example where P and \neg P overlap while covering the whole set? I always thought the LNC dictates that P and \neg P must be exclusive in the LEM if you want them to be consistent within the same formal logical system. I'd love to see an example.

Thank you.

 

[AKG]

Thanks again. To be sure I understand fully, I'll rephrase your comments.

LNC: a proposition p cannot be both true and not true (in the same instance).
LEM: for all p, "true" or "not true" are the only options.
POB: There is only one reason for p to be not true: because it is false.

Yes, but be careful to note that (with regards to LEM) this doesn't mean things can't be "false," they can be false, it is simply that "false" would be a "subset" of the "not true" option. Also, the principle of bivalence states that something is either true, or false. The explanation you gave for POB seems okay, but I think it loses some information if you don't assume the other two laws.

Assuming LNC, LEM & POB, if p="There will be a sea battle tomorrow" is not true, then there will not be sea battle tomorrow. Some logics reject or modify LEM so they can have POB's falsifying power while allowing other options in addition to "true" and "not true". Correct?

I'm not sure, I don't have any formal education in logic (not until next semester at least), so I can't tell you too much about what the various logics do. Some logics reject LEM to allow for possibilities like "possible" or "indeterminate" that are entirely distinct from "true" and "false." I think intuitionist logics do this, and they, as far as I know, substitute justification for truth, sort of. I think this gets into a more metaphysical/epistemilogical thing, though. Are mathematical propositions true, and our proofs convince us of that, or are they true upon being proven? This depends on which theory of truth you subscribe to, I would think. If mathematical propositions are inherently either true or false (and this has relation to metaphysics, because Platonists will see mathematical propositions as inherently true or false, since mathematical things really exist in a Platonic sense), then it makes no sense to reject LEM. Whether we know if the proposition is true, or even if it is indeterminate such that we can never tell if it is true, does not effect whether it is, "truly" true or not. However, if something is true upon being proven, then "false" can be equivalent to "proven false", true is equivalent to "proven true," possible is equivalent to "not proven to be true or false (yet)," and indeterminate would be equivalent to "can't be proven true nor false."

However, as far as I know, the POB implies LNC and LEM. What does the POB say without LEM? Something can be true, false, or some other option? Well that principle isn't saying anything at all. It is only saying something when it says that a proposition is either true or false, but never both, and never none. In saying that, of course, it implies LEM and LNC. (quickly note that POB without LNC says something can be true or false, and sometimes both, which makes the word "false" pretty meaningless, and again, the POB becomes meaningless). Actually, just disect the word "bivalent." bi:2, valent:values. There are only 2 values, so it certainly implies LEM.

 

[AKG]

Can you please give an example where P and \neg P overlap while covering the whole set? I always thought the LNC dictates that P and \neg P must be exclusive in the LEM if you want them to be consistent within the same formal logical system. I'd love to see an example.

Thank you.

I'm not sure what you're asking for. If we assume LNC, then P and \neg P do not overlap, i.e. they are exclusive. On it's own, I don't see the LEM dictating that the two don't overlap, but I can't think of an example when they do. However, I'm sure one can invent a bizarre logic where the two overlap. Indeed, it all depends on how you want to define \neg anyways. If you want to define it in some bizarre, unnatural sense, then you can certainly have overlap. I believe that, in the most natural sense, LNC and LEM are undeniable, however, some people like different theories of truth in which "possible" is a reasonable value for a proposition, and in such theories, logics will differ from logics that use LNC and LEM.

However, if you have LEM and don't have LNC, then you don't have the problem of the two being consistent (I can't tell if that's what you were asking though, you're question is unclear).

 

[honestrosewater]

Great, I appreciate your patience. POB implies LNC and LEM. That's what I needed. I thought I read somewhere that some logics assume POB but reject LEM, and I couldn't figure out how that was possible. It isn't possible.
Right

 

[Pavel]

I'm not sure what you're asking for. If we assume LNC, then P and \neg P do not overlap, i.e. they are exclusive. On it's own, I don't see the LEM dictating that the two don't overlap, but I can't think of an example when they do. However, I'm sure one can invent a bizarre logic where the two overlap. Indeed, it all depends on how you want to define \neg anyways. If you want to define it in some bizarre, unnatural sense, then you can certainly have overlap. I believe that, in the most natural sense, LNC and LEM are undeniable, however, some people like different theories of truth in which "possible" is a reasonable value for a proposition, and in such theories, logics will differ from logics that use LNC and LEM.

However, if you have LEM and don't have LNC, then you don't have the problem of the two being consistent (I can't tell if that's what you were asking though, you're question is unclear).

Well, I'm not implying any bizarre logic or some interpretation in an unnatural sense. I'm talking about our traditional classical logic system in which LNC, LEM, and POB are axioms, self evident truths that we have to take for granted. You’re right in that they’re undeniable, as in doing so, you would contradict yourself. It also means that they don’t imply each other, they’re fundamental atomic units of our classical logic system. Now, what I was saying was that for these axioms to play well together in one formal consistent system as our logic, the LEM has to be exclusive. The cat is either dead or not dead. Those both states “occupy” the full set and there’s no intersection between the two. If there is, then that entails there are situations where the cat is both dead and alive, which contradicts another axiom - LNC. That’s the way I see it, the LEM is exclusive. If not, I asked you for an example of the intersection where truth and the denial of a proposition are both asserted.

 

[loseyourname]

Well, the only linguistic formulations of the law of excluded middle that I've seen all that say that any proposition P must either be true or false, not that it must be either true, false, or both. Even if you formulate it symbolically, it still says the same thing. There is an interesting implication there in that non-contradiction might be redundant. I had never thought of that.

 

[AKG]

No, the POB implies LNC and LEM. It also implies that there are strictly two values, so \neg P cannot be subdivided into things like "P is possible" and "P is false," it is "atomic" in a sense, and simply stands for "P is false." Again, the LNC states that P and \neg P are exclusive, so the LEM is "exclusive" in a logic which holds to the LNC. However, you could come up with a logic, I assume, where LNC is denied, but LEM is held, in which case P and \neg P could overlap (by denial of LNC), but they would still "cover the whole set" (by LEM). Suppose that "dead" and "alive" are the only two options. According to some QM interpretations, I believe it is possible to say that the cat is both dead and alive. This would be an example of a case where the LNC is denied but LEM is asserted, so there still are two options that "cover the whole set", those options being dead or alive, but those options do overlap, as the cat can be both dead or alive. Actually, there is something called "quantum logic," I don't know much about it, but it might be a logic where LNC is denied but LEM might be held.

I think the decisions to agree with or deny these varies laws and principles depend mostly on definitions, and don't really show fundamental contradictions in the nature of logic. How do we define "not" and negation? How do we define implication? How do we define truth? How do we define a proposition? The way one chooses to answer these questions will essentially determine the axioms of his logic. Since these are questions have arbitrary answers, (we are essentially free to define things as we please, although I would argue that those definitions which are most rigorous yet, at the same time, most natural and common, are best), the differences between various logics is arbitrary and insignifcant. However, I hope to learn a little more about propositional logic and logic in general next semester, maybe I'll have something different to say then.

 

[AKG]

loseyourname, stictly speaking, P \vee \neg P states P OR \neg P, as opposed to P XOR \neg P. That lack of an "X" in the expression of LEM means exactly that the P and it's negation are not necessarily exclusive.

 

[loseyourname]

loseyourname, stictly speaking, P \vee \neg P states P OR \neg P, as opposed to P XOR \neg P. That lack of an "X" in the expression of LEM means exactly that the P and it's negation are not necessarily exclusive.

Yeah, you're right. I forgot that logical disjunction is inclusive.

 

[honestrosewater]

That also confused me at first because in PC, LEM and LNC are equivalent tautologies: \neg ( P \wedge \neg P ) \equiv P \vee \neg P
Of course, this is because PC assumes POB.

 

[Pavel]

I’m not sure I can agree with you guys. I’m not saying you’re wrong, but your explanations are not convincing. First, again, let’s not talk about quantum logic, fuzzy logic, or any bizarre logical system or interpretations of it. The laws at hand are defined within our formal classical logic, the one you use when you apply for a job or prove a theorem in your geometry class. The law of excluded middle is defined so because there’s nothing in the middle, no intersection. AKG, when you take your symbolic logic class, you most likely will be proving the validity of the arguments by contradiction, where you assume the conclusion to be false and show its contradiction with one of the premises. This whole, reductio ad absurdum, principle is based on the law of excluded middle. If there was an intersection, you could not prove the validity of the argument. That is, you rely on the logic that either the conclusion is true OR your premise is false, and there is no place for both. That is an exclusive OR. I’m fairly certain, but to give you the benefit of the doubt, I’d like to see an example of an intersection, within our logical system, which was my original question. Let’s leave quantum logic out of this, we don’t use it unless we deal with questions pertaining quantum mechanics. And by the way, I don’t believe your example of the cat being both dead and alive contradicts the LEM. The quantum uncertainty does not assert that both states are true in the same context. It asserts that the state of the cat is unknown, undeterminable, until you check or measure. Then! Only then, one of the states is true, not both. The LEM would be violated if both states were true after you take the measurement, or “look” so to say. That is not the case. Until then, the cat is both dead and not dead in two separate contexts, or worlds, if you will.

 

[AKG]

This whole, reductio ad absurdum, principle is based on the law of excluded middle.

I would think it's based on the principle of bivalence.

I’m fairly certain, but to give you the benefit of the doubt, I’d like to see an example of an intersection, within our logical system, which was my original question.

There can be none, since our logical system asserts the prinicple of bivalence (which implies LNC), or, even if we say that it does not assert POB, it does at least assert LNC, which says there can be no overlap. I don't believe there can be an overlap, but the LEM without the LNC does not say that there can be no overlap. Personally, I think it's unnatural to do without the LNC, I'm just saying that if it were to be done, then we could have overlap.

And by the way, I don’t believe your example of the cat being both dead and alive contradicts the LEM. The quantum uncertainty does not assert that both states are true in the same context.

It was never about contradicting the LEM. It was about contradicting the LNC. I'm not sure that you know what it is you're asking, and if you do, it is not clear to me at all what it is.

The LEM would be violated if both states were true after you take the measurement, or “look” so to say. That is not the case. Until then, the cat is both dead and not dead in two separate contexts, or worlds, if you will.

No, the LEM is not violated, the LNC is violated.

 

[Pavel]

I'm not sure that you know what it is you're asking, and if you do, it is not clear to me at all what it is.No, the LEM is not violated, the LNC is violated.

You're probably right: I'm not sure myself what it is I'm asking. I think I'm trying to define the boolean logic with boolean logic, and that's the problem.

But let me try again, if you will. You said

LEM: P \vee \neg P : A proposition is true or it's negation is true, and the two need not be exclusive.

What exactly did you mean by "the two need not be exclusive". Please be specific with an example of it not needing to be exclusive and then, for contrast, an example of it needing to be exclusive. I think it'll be become clear to me as soon as you do it.

Thanks,

Pavel.

 

[honestrosewater]

Pavel,
By definition, P \vee \neg P means there are three possibilities: P,\ \neg P,\ (P \wedge \neg P). The "or" is defined to mean "or/and", and "or/and" is not exclusive; It does not exclude (P \wedge \neg P) as a possibility.
Don't take my word for it though.

As an example, if P is undecidable in X, then the LNC is an axiom or theorem of X. Right? If so, create a system that includes the LEM but not the LNC and translate "This statement is unprovable" into a proposition in the system. Don't take my word for it though.
____
Do you see what I'm saying? The LEM allows P to be 1) true, 2) false, or 3)true and false.
It's the LNC that excludes #3 as a possibility, thus making a proposition whose truth-value is #3 undecidable. Undecidable propositions are the example you are looking for. I think.

 

[AKG]

What exactly did you mean by "the two need not be exclusive". Please be specific with an example of it not needing to be exclusive and then, for contrast, an example of it needing to be exclusive. I think it'll be become clear to me as soon as you do it.

In the strictly logical sense, "OR" is not exclusive. So, if I ask, what kind of clothing do you want to buy, and you answer, "a shirt OR a pant," then you would find it acceptable if we bought just a shirt, just a pant, or both. "XOR" means "exclusive-OR". If you were to say, "a shirt XOR a pant," then that means you would want just a shirt, just a pant, but not both. I believe, in most situations, we use "or" as logic uses "XOR." When someone asks you what you want to eat, and you say, "italian or chinese" you mean either italian, or chinese, but not both. "XOR" is like saying EITHER option 1 OR option 2, BUT not both.

When I say that LEM says that they need not be exclusive, then that means that the case could be that P, ~P, or both. Now you would ask, "both P and ~P? That's weird." Indeed, it is, and that's what LNC is for. LEM says P, ~P, or both, and LNC says, no, not both. Putting them together, we get either P or ~P, just one or the other, not both. Now, I don't think I can give an example where we have P and ~P. LEM allows it, but LNC does not, so to give an example, I would have to deny LNC. I don't think I could make a rational English sentence using the word "not" if I were to deny LNC. Technically, such an example is possible, but I don't think you would buy it as a reasonable example because it would sound so unnatural.

 

[AKG]

honestrosewater

Are undecidables true and false, or are they just undecidable?

 

[honestrosewater]

Are undecidables true and false, or are they just undecidable?

Undecidable propositions are neither true nor false. I am just going to find a good logic text and start over from the beginning.

 

[Philocrat]

The System of Bivalence is logically and quantitatively misleading...and thank heavens that we have now supassed it with the invention of fuzzy Logic. Because of this, we have been able to invent machines that are functionally more precise. Machines that were originally based on bivalent logic made so many functional errors and inacuracies such that in some life-critcal environments like hospitals, office buildings, building sites, weather predictions centres and so forth, machines used in life-critical situations were cuasing a great deal of problems for their users simply because of the two-logic decision systems in them.

Although, we are still teaching this logic to our youngsters, precuations must be taken as to when bivalent logic is appropriate to use in various parts of our institutions. Bivalent logic used to naively cut the middle of things off, even when the decision components qauntitatively and logically range over opposite extremes. The modern logic now takes care of this blunder...thank God for that!

I am not saying that we should abandon the yes or no logic, but we must tell those we teach it to when and where to use it.

 

[Philocrat]

And most importantly, many ignorant wars have been fought, and are still being fought, many political and economic blunders of epic scale have been made, many intellectual blunders have been made within the academic systems, we have time and time again discriminated against each other, maginalised, classified, and segregated, all on the basis of the 'this or that', 'yes or no', or 'true or false' logic. Often we engineer people to say 'yes' or 'no', even when we are fully aware that there may exist a middle answer or answers. This is just one of countless examples.

It is more than well overdue for us to now sit back and think, look long and hard at the underlying implications of decision systems (be they human or machine) based on the logic of bivalence. We may try and dudge this issue as much as we like, but sooner or later we will have to address it before its underlying implications flush us into oblovion.

 

[Pavel]

Dudes! (or gals) please, let’s leave undecidability or fuzzy stuff out of this. We’re talking basic Logic 101 here. I think I finally figured out where this confusion comes from; we made it way more complicated than it needed to be.

When you say inclusive or exclusive OR, you’re defining a statement with "variables" so to say (wff), a formula if you will, that says one variable stands to the other variable in the following relation: Inclusive OR is (P OR B); XOR is (P OR B) AND NOT (P AND B). When you plug in P and not-P into the formula of the inclusive OR, you get the law of excluded middle. Now, when you said “they need not be exclusive”, you really meant that for any two variables you plug in, they can be both true, that’s how inclusive OR works, doesn’t it? What *I* meant was that when you have the case where those two variables are the same but one is the negation of the other (P and not-P), then those two variables can NOT be true at the same time. So, you still preserve the inclusive OR as a relation of two variables to each other, but the variables will never be true at the same time in this case, because one variable is the negation of the other. It’s that simple! Construct a truth table for the law of excluded middle and see for yourself. Yes, that’s an inclusive OR, but you will NEVER get 1 1 in the “formula” (wff), because the other variable is 0 by virtue of negation. That is, you either get 1 0 , or 0 1, but never 1 1. It’s in this sense that they’re exclusive! And that’s why you can’t come up with an example of when both P and not-P are true, i.e. they both would have “1” values in the truth table. That would throw the whole classical logic with its POB out of the window. So I don’t know if I got confused for no apparent reason, or you made it confusing, doesn’t really matter, but I wouldn’t use “they don’t have to be exclusive” in the context of the Law of Excluded Middle. You can say that when you describe a generic inclusive OR statement.

 

[loseyourname]

Do you see what I'm saying? The LEM allows P to be 1) true, 2) false, or 3)true and false.
It's the LNC that excludes #3 as a possibility, thus making a proposition whose truth-value is #3 undecidable. Undecidable propositions are the example you are looking for. I think.

Actually, LNC makes #3 false. It is a contradiction, and the law specifically states that you cannot have a contradiction.

 

[AKG]

When you say inclusive or exclusive OR, you’re defining a statement with "variables" so to say (wff), a formula if you will, that says one variable stands to the other variable in the following relation: Inclusive OR is (P OR B); XOR is (P OR B) AND NOT (P AND B). When you plug in P and not-P into the formula of the inclusive OR, you get the law of excluded middle. Now, when you said “they need not be exclusive”, you really meant that for any two variables you plug in, they can be both true, that’s how inclusive OR works, doesn’t it? What *I* meant was that when you have the case where those two variables are the same but one is the negation of the other (P and not-P), then those two variables can NOT be true at the same time. So, you still preserve the inclusive OR as a relation of two variables to each other, but the variables will never be true at the same time in this case, because one variable is the negation of the other. It’s that simple!

You're saying that the variables will never be true at the same time, because one variable is the negation of the other. You're implicitly assuming that if one variable is the negation of the other, then both cannot be true. What exactly is this assumption? LNC, precisely. So, LEM on it's own still allows for the three possibilities, and when you assume LNC (which you did impicitly) the third option vanishes. In other words, if I ask you "why does the third option disappear?" you'll say, "because of the way 'negation' is in there." "But what's so special about negation? What is it about negation that makes that third option disappear?" To this, you would have to answer that it is precisely what LNC says that makes 'negation' work in such a way to have the third option disappear.

Now, to be honest, there is some circularity here, I think. Let's replace negation with "gorblab". If we have two rules, LEM' and LNC' as follows, let's see what we get. LEM' states P OR gorblab-P, so we have three options:

1. P
2. gorblab-P
3. P AND gorblab-P

LNC' gives us gorblab(P AND gorblab-P). In other words, gorblab(option 3). If we assume LEM' and LNC', then we have 3 options:

option 1 AND gorblab(option 3)
option 2 AND gorblab(option 3)
option 3 AND gorblab(option 3)

Does "option 3 AND gorblab(option 3)" give us a reaosn to reject option 3? I don't think so. Therefore, I think there's more to negation than LNC and LEM can tell us. If "option 3 AND gorblab(option 3)" is inherently contradictory (note that it's in the form Q and gorblab-Q, where Q is option 3), then we don't even need LNC, it's superfluous. I think this is why you went straight to saying that, by virtue of the second variable being the negation of the first, option three was not allowed. You based that argument off an intuitive notion you had of negation. Now, you could have used LNC to get rid of option 3, but really, we just showed that LNC doesn't do that on it's own, and it only does if we carry that intuitive notion of negation in the first place, i.e. LNC doesn't replace an intuitive notion in a more formal manner because it still needs that intuitive notion to make sense. I believe, however, that the set-interpretation might be closer to a formalization of LNC (That P and ~P are disjoint sets) than ~(P and ~P). Of course, what does it mean for two sets to be disjoint? If x is in a set P, then the negation of "x is in ~P" is true. So, again, we come down to a question of what it means to have a negation. At least we can visualize what it means for sets to be disjoint, even if we can't express it without some circularity.

Indeed, words cannot all be defined completely in terms of words. If so, they'd all be meaningless. You can't build a tower, one brick on top of the next, if you don't have a foundational brick firmly rooted in the ground to start with. Ultimately, there are some words which we will find very difficult to express the meaning of. Try defining "reality" "existence" "actual", etc., without using those terms. I think defining "negation" would be a similar task. We all know, for the most part, what each of us means when we try to talk about these things, although we can't explain them in simpler words. For this reason, I can see why there might be confusion about how "not" should be used, and thus why some people will agree with LNC, and others won't, while both will claim to speak the same language, and claim to be referring to the same concept "negation."

Construct a truth table for the law of excluded middle and see for yourself. Yes, that’s an inclusive OR, but you will NEVER get 1 1 in the “formula” (wff), because the other variable is 0 by virtue of negation.

Not purely by virtue of negation, but by virtue of LNC, and what it says about negation. If you're talking about truth tables, you're already assuming a bivalent logic (in a fuzzy logic, I suppose you'd have more than 1's and 0's, you could have 1/e or 0.298374444444...) and so you're already assuming LNC and LEM. Or maybe you're right. But then what is it about "negation" that makes ~P = 0 if P = 1? Is it that it is fundamentally true that P \neq ~P? Well, what does that mean? It means P is not equal to not-P, or ~(P = ~P). We're essentially back where we've started. If you are going to say anything like "by virtue of negation," you will be necessarily assuming something about what "negation" means. You'd be assuming something it appears I would agree with, but it's something fundamentally inexpressible, and so it's hard to tell another that he's wrong if he treats negation and the implications of negation differently, unless you can claim to have made up the English language, or language in general.

 

[Philocrat]

There is nothing logically and quantitatively wrong in naturally declaring 'This book is both red and not red'. The confusion exists only in ill-devised logical procedures in logical languages. In NL (Natural Language), this would be construed and understood as 'This book is partially red'. In TL (Transitional Logic), LEM is completely written off because anything that ranges over, TL has the full capacity to pick every fact in transit from possibility to necessity. Nothing is ever left unattended in the pathway from point A to B. LEM, if it is still relevant in some branches of logic, needs revision in the context and total respect of how things are done and understood in NL, otherwise its whole enterprise should be condemned to the academic museum.