# Law of Excluded Middle:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is. The principle should not be confused with the principle of bivalence, which states that every proposition is either true or false, and has only a semantical formulation.

# Principle of Bivalence:

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic. In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.

The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form "P ∨ ¬P".The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle.

I'm not quite sure I get the difference. It seems that 'excluded middle' is a syntatic problem and 'bivalence' would be a semantic one. Is this correct? Also, it seem that in the realm of bivalence, stating that "P" is false, doesn't necessarily mean "non-P" is true, which would be the case with the principle of the excluded middle. Is this correct?

I don't understand precisely in which situations one or the other principle are at play, it seems that they may appear together, but not necessarily. Can someone give me examples and help me clarify the differences?

• I think that POB allows only two truth values for any proposition but it doesn't exclude the possibility that a proposition and its negation have the same truth value. – Alfred Centauri Jul 10 '12 at 23:24
• After reading the answers here I'd suggest that to avoid a muddle you start again from scratch and read 'Aristotle's Interpretatione: Contradiction and Dialectic' by CWA Whittaker. – user20253 Mar 21 '19 at 11:13
• The proper EXPRESSION of the LEM is that no two propositions can be simultaneously true and false at the SAME LOCATION, TIME and CONTEXT of the language used. If you were SPECIFIC in detail at least one of those qualities will distinguish two similarities of propositions. The other rule expresses that objectively there are only 2 truth values and NO MORE. Ojective knowledge is NOT science. Nor does objective knowledge require sense verification. Objective knowledge exists independently of your awareness by definition. – Logikal Mar 21 '19 at 17:36
• It seems to me the answer is given in the question. Any confusion may arise from the unrigorous expression of the LEM quoted. The LEM does not state "for any proposition, either that proposition is true, or its negation is." It states that where this is the case the LEM holds, and where it is not the case the LEM does not hold. Read literally the quoted definition given is for the principle of bivalence, not the LEM. This allows a clear distinction to be made between the PB and LEM. . . . . . – user20253 Oct 1 '19 at 13:29

OK, I think I've got it now:

• A certain logic validates the Law of the Excluded Middle (LEM) if the following is a theorem in the logic: p v not p

• A certain logic abides by the Principle of Bivalence (PB) if every well formed expression according to the logic has exactly one truth value: true or false

Some semantics may make it the case that LEM is true and PB is not true. Consider the following supervaluationist treatment of vague predicates. A statement such as

Schiphol is bald

will be supertrue (superfalse) iff under all (no) acceptable precisifications of the predicate "bald", the sentence comes out true. A precisification will have the form "... has n hairs", where, e.g., n = 0 is acceptable, but n = 10^6 is not. Sadly, the sentence above is supertrue -- which is the supervaluationist criterion for accepting it as true.

Luckier folk, such as, say, Andy, might come out bald according to some precisifications and not bald according to others. Thus,

Andy is bald

is neither supertrue nor superfalse: it lacks truth value, according to supervaluationism. PB, therefore, is false: that sentence is neither true, nor false. Now, what happens with a sentence of the form [p v not p], such as

Andy is bald or Andy is not bald

Well, such sentences will be true for all precisifications, because either Andy has n hairs or he doesn't, for all n. Therefore, the sentence comes out supertrue -- this is the supervaluationist for accepting it as true. Its negation ("it's not the case that Andy is bald or Andy is not bald"), by the same token, comes out superfalse.

The same will happen with every other vague sentence: the supervaluationist semantics validates LEM. Supervaluationism is a semantics that validates LEM but not PB.

• what do you think about the semantic/syntatic distinction as stated in the wiki articles? – Tames Jul 12 '12 at 17:15
• @Tames you know what, I think I need to revise my answer. I'm not so sure of what I've written there. – Schiphol Jul 12 '12 at 18:51
• I've changed my reply completely. I'm pretty sure this is bad behaviour. Whoever voted me up, please feel free to withdraw your vote! – Schiphol Jul 12 '12 at 19:09
• Hmm... sounds more interesting now! But in the case of "Andy is bald or Andy is not bald", does LEM still hold? Because it seems that negation of it would have exactly the same value, as in "Maybe Andy is bald"(the negation "Maybe Andy is not bald" means the same thing), or not? Can "maybe" and "maybe not" statements be judged as "true"? (it seems they can't be wrong, because they express doubt) – Tames Jul 12 '12 at 20:26
• The "Andy is or is not" sentence is supertrue (that is, true), and its negation is superfalse, no? In every precisification Andy has that number of hairs or he doesn't. I don't think the "maybe" sentence is parallel: this other sentence is not universally true, for example. I have tried to make it clearer in the answer, let me know what you think! – Schiphol Jul 13 '12 at 2:28

This is the initial thread to the discussion:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is. The principle should not be confused with the principle of bivalence, which states that every proposition is either true or false, and has only a semantical formulation.

Source: http://en.wikipedia.org/wiki/Law_of_excluded_middle

This sloppy formulation of the law of excluded middle (for propositions) is slightly inaccurate (i.e., is mistaken) - although the cause of the inaccuracy (mistake) is very natural.

The law of excluded middle for propositions should instead read: Given any proposition, either it's true or it is not true. Or, alternatively, [given a two-valued logic where the two values are true and false] Given any proposition, either it's false or it's not false. More abstractly, but more precisely, it can be expressed as follows: Given any proposition, either it has property P or it doesn't have property P.

One law of excluded middle for natural numbers is: Given any natural number, either it is even or it's not even. One law of excluded middle for animals is: Given any animal, either it's a vertebrate or it's not a vertebrate.

Truth is not the point here - nor is falsity. Instead, the point is the logical exclusivity that (necessarily) holds between IS and ISN'T.

At this juncture, it might be helpful to state the law of excluded middle for properties, which is a second-order logical truth: Given any property and given any individual, either the individual has that property or it does not have that property. [Please note that it doesn't matter what the property is or what the individual is.]

The law of excluded middle for properties is a logical truth, not merely a logical law of classical two-valued logic. [It's very important to realize that not every logical law is a logical truth.]

The principle of bivalence - although a law of classical (two-valued) logic - is NOT a logical truth, because it has the same logical form as some (i.e., at least one) falsehood. The principle of bivalence is that Every proposition is either true or false.

This proposition (call it a principle, if you like) has the same logical form as the known falsehood Every number is either odd or prime. In sharp contrast, every proposition that has the same logical form as the proposition that Every proposition is either true or it isn't (i.e., Every proposition is either true or it is not true) is a logical truth.

The distinction at issue here is well known by experts, but it's a rather technical (though quite important) distinction. The author of the Wikipedia article appears to be admirably informed, but not an expert. [The entry for the principle of bivalence (that directly follows the initial thread) is also goofed up in several respects.]

By the way, there are a great many other issues that very frequently cause confusion concerning such topics as this one. In particular, it's necessary to know/learn the difference between a proposition and a sentence. For example, the declarative sentence I am female expresses a truth when my girlfriend utters it, but it expresses a falsehood when I utter it. And yet this is not a good reason for claiming that some proposition is both true and false.

• "Given any proposition, either it's true or it is not true". This is the PB not the LEM. The LEM applies to pairs of statements that are mutually exclusive and exhaust the possibilities. This is Aristotle's definition. If a pair of statements do not meet this requirement the LEM cannot be applied. Nevertheless, one or both may be true or false. I feel your dismissal of the Wiki explantion is a bit hasty. – user20253 Mar 21 '19 at 11:05
• Your answer should be made clear that your view may be pure mathematics or how science interprets the LEM. What you stated does not hold or fly with the PROPER EXPRESSION of LEM as found in Philosophy. You read it literally like a child would read. Propositions are not literally sentences and you should understand what the proposition expresses -- not what it literally states. – Logikal Mar 21 '19 at 17:28

The difference between Excluded Middle and Bivalence:

Excluded Middle says every proposition of the form P v ~P is true

Bivalence says every proposition is true or it is false

and that's all she wrote

(forget all the technical jousting)

• Welcome to Philosophy.SE and thanks for your answer!! It might help if you could explore your point a little bit further? – Joseph Weissman Apr 3 '15 at 13:29
• I think the short answer is actually better! – Squirtle Apr 10 '19 at 3:16
• Whoever said this is not correct. The rules for contradictory pairs of dialectical propositions is that one is true and the other false. Where this is true the LEM will apply. The rule for contradictory pairs is - In order for the LEM to apply to a proposition it must be of the form P v ~P and it must be true. This is not the LEM. Your formulation describes the rule for contradictory pairs, which must be met before the LEM or LNC can be applied, not the LEM. A subtle point but vital. – user20253 Oct 1 '19 at 13:18

It may help to have an example of a logic where the excluded middle doesn't hold. Probably the most well known one is Intuitionistic Logic, also known as Constructive Logic. It was formulated in the early part of the 20C in reaction to certain (mathematical) existence proofs where certain mathematical objects were shown to exist but no construction given, this was traced to use of the excluded middle. The intuitionists insisted on being given a construction.

It is correct here to say that not true=false. But there are other truth values. So the bivalence law doesn't hold.

It isn't correct to say that something can be true and false simultaneously. So the non-contradiction law does hold.

aside: Whereas classical logic is associated with Boolean algebras and standard set theory, intuitionistic logic has an associated Heyting algebra and categorical set theory (topos).

Here is the question about the law of the excluded middle (LEM) and the principle of bivalence (PB):

I don't understand precisely in which situations one or the other principle are at play, it seems that they may appear together, but not necessarily. Can someone give me examples and help me clarify the differences?

Andrea Iacona in her article "Future Contingents" presents a situation showing why one might want to reject one or the other of these.

The reason regards propositions about the future. If I state today, "It will rain tomorrow," then the principle of bivalence claims that that proposition is either true or false today. But if I know today whether it will rain tomorrow for sure, does that not imply that determinism (or fatalism) is also true?

Human free will is what is at stake in this situation. Those who do not want to accept determinism need to create a plausible logical system that rejects either LEM or PB (at least for some class of propositions) or show that together they do not lead to determinism.

There are four possibilities assuming one wants to continue using deductive reasoning with these propositions. Only three of which Iacona considers plausible:

1. Neither Bivalence nor Excluded Middle An example of this is Lukasiewicz' three-valued logic. Some propositions may have an indeterminate truth-value. However, this also requires rejection of LEM since if P is indeterminate, how can one say that what is normally the tautology P v ~P is anything but indeterminate and no longer a tautology? This example still ties LEM and PB together.
2. Excluded Middle without Bivalence This is "the most plausible reading" of Aristotle's position. It is also the position of supervaluationism. Here is an example where a plausible logic has been constructed that accepts LEM but not PB.
3. Both Bivalence and Excluded Middle This is a position that accepts both but tries to argue that determinism is not a consequence of doing so. It "has been defended by Von Wright (1984), Lewis (1986) and Horwich (1987)".
4. Further Considerations This option rejects LEM but not PB. Although this is also an example where these two are separate, Iacona considers this as not plausible:

The debate on future contingents almost never sees the acceptance of bivalence combined with the rejection of excluded middle, because most thinkers take it for granted that bivalence is at least as controversial as excluded middle.

Here is another part to the question:

I'm not quite sure I get the difference. It seems that 'excluded middle' is a syntatic problem and 'bivalence' would be a semantic one. Is this correct? Also, it seem that in the realm of bivalence, stating that "P" is false, doesn't necessarily mean "non-P" is true, which would be the case with the principle of the excluded middle. Is this correct?

In classical modern logic (not Aristotle's ancient term logic), there is no significant difference between LEM and PB as is shown by this truth table: By PB one can construct the first column, but also by PB the other columns will have to take either one of two values: 'T' or 'F'. There is no third truth-value that Lukasiewicz offered. The tautology is generated in the last four columns by truth-functional (semantic) definitions for the logical symbols. These demonstrate LEM's validity in the table.

That means the difference between LEM and PB that would avoid the determinism issue above will not come from the semantics of modern propositional logic. It will also not come from the syntactic proofs since they must be sound based on this semantics. Something like the option presented by supervaluationism will have to be used to modify this logic.

Alternatively, one might claim that the class of propositions about the future would need to be excluded from deductive logical arguments because PB does not apply to them. They may only be permitted in inductive arguments. However, this would just admit that LEM and PB go together. It is not a way to separate them.

Iacona, A. Future Contingents. Retrieved on October 1, 2019 from the Internet Encyclopedia of Philosophy at https://www.iep.utm.edu/fut-cont/

Principle of Excluded Middle: "A proposition p and its negation ~p cannot be false together."

Principle of Non-Contradiction: "A proposition p and its negation ~p cannot be true together.

Principle of Bi-Valence: "A proposition is either true or false."

PEM and PNC forbid a proposition and its negation having the same truth value.

PB forbids a proposition being both true and false or neither true nor false.

• This is not correct. A proposition and its negation may both be false. It would just be the case that the LEM cannot be applied to them. As far as the LEM goes propositions mat be true, false, neither or both. But if (iff) they are to be subject to the LEM then one must be true and the other false. Aristotle is perfectly clear about this. – user20253 Mar 21 '19 at 11:09

I think this is not quite right, or at least doesn't quite bottom out the issues. I am no great expert but as I see it...

The PBV is not (afaik) part of the laws of A's logic.

The LEM would be a stipulation for true contradictory pairs that must be met for the dialectic process to work properly and to decide between contradictory propositions. That is to say, the LEM will hold wherever the proposition to be tested meets A's rule for contradictory pairs (RCP), which is that it must be one of a pair of which one must be true and the other false. This rule would be inviolable.

None of this would imply anything for the world itself, about which statements may take on various truth-values, even be half-true and half-false.

Thus, say, when Heraclitus states, 'We are and are not' this would violate the PBV but not the LEM. It would not violate the LEM because Heraclitus is not suggesting that either half of his statement is true or false but, rather, that the truth lies elsewhere. His statement does not meet the requirement of the RCP so the LNC/LEM would not be relevant.

This is how it seems to me for now. This would be important because it allows us to use A's logic as the basis for a logic of contradictory complementarity and thus reconcile this logic with the world-view of Heraclitus and his like. If we see the LEM and rule for contradictory pairs as more than a formal device then we will be limiting our world-view.

It is called Excluded Middle because there is nothing in-between those two values: F, and V. In Fuzzy Logic, for instance, there is something in-between: T would be 1, F would be 0, and there is an infinity of values in-between 0, and 1 (0.1, 0.11, 0.23, and so on). Excluding the middle means taking away whatever could be a moderate position, so it is always Yes or No to any question you may have; never a 'more or less' or a 'so so': are you black? Yes. Are you happy? No. If someone asked you, are you rich, and you answered, so so, they would say: No! That is not an ACCEPTABLE answer, mate. In life, it is either an ABSOLUTE YES or an ABSOLUTE NO... That is the World of Classical Logic, or the World of the EXCLUDED MIDDLE... Bi-valence means two values, so that it could refer to any two arbitrary values. If we say Principle of Bivalence in Classical Logic, then it refers to False, and True or 0, and 1. You could not have the middle, and still have three values, so say 0 0.5 1, but no 0.3 or 0.6. Yet, in Classical Logic, you only have two, and that is why we say that, in that world, bivalence is a principle. Notice that it is Law of the EM, but 'principle' of bivalence. That probably means we are surer about not having anything in the middle than we would be of having only two values to hang on to... I have just read about the Principle of Non-contradiction, which is different from the Ex-Falso, which is a consequence of the laws, and principles of CL. In my interpretation, bivalence does not imply exclusion of paraconsistency, so that we could still have 2 values at the same time, or 'the door is open' is true, and 'the door is open' is false at the same point in time, and all else (Ceteris Paribus). That is why we should need a principle to say 'no contradictions accepted', or Principle of Non-contradiction. With this one, we would exclude paraconsistency, so that if 'the door is open' is true, 'the door is open' cannot be false in Ceteris Paribus Worlds: it is either one or, without concomitance, the other.

I think the simplest way to answer this is just to consider a logic with three (or more) (exclusive) truth values. Let's say:

TRUE FALSE UNDEFINED

Obviously, bivalence fails for this logic, since we have more than two truth values. Nevertheless, excluded middle holds. Proof: Every proposition is either true, false or undefined. But if a proposition is false or undefined it is thereby not true. So everything is either true or not true.

Another way to put it: If you think that everything is either true or not true, but you think there are multiple ways to not be true, then you have excluded middle without bivalence.

``````        **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:**

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").
``````

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, but they both cannot be true.

• 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 equivalently stated as follows: A proposition X cannot be neither true nor not true (i.e., false).
• The neither-nor operation of the two following contradictories: [X nor ~X]: that is, joint denial of both X and its negation ~X.
• 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)

LEM states exactly one of X and ~X is true, and the other false, and vice versa, and moreover includes the option where both are true (contradiction), but excludes the option where both are false (joint denial).

The law of bivalence (henceforth, LOB) states that X is either true or false.

• 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 can be more clearly expressed as: 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!

• Please try to base your answers on references validating your assertions. For example, LEM states that "X or ~X" is true, ie. assigns the value "true" to the formula, while LOB asserts that "X" has either the value true or the value false. More crucially, in supervaluationist logics both LEM and LNC hold (are considered as/assigned the value "true"), while LOB does not. You basically brushed over the fact that one states something about the truth value of one particular formula, while the other one states something about possible truth values of propositions as such. – Philip Klöcking Jul 6 at 6:08
• In other words: LEM is about how certain operators manipulate truth values and thus defines what may count as a well-formed formula in a given logic, while LOB is about the possible truth-values in that logic. You make it seem as if they both express truth values of formulae. This happens if you simply express both sentences in classical logic and manipulate according to its rules. – Philip Klöcking Jul 6 at 6:12
• @PhilipKlöcking, I agree. LEM is a syntactical principle (form) which is about negation ("not") as a logical connective, while LOB is a semantical principle (content) and is about negation as a truth-function which outputs truth values. Is all else all right? How did you find my answer? Did I explain well enough? – Karlen Karapetyan Jul 7 at 0:22