What LEM is not

[Magnus Anderson]

The Law of Excluded Middle ( LEM ) is captured by the statement "For every proposition P, P is either true or it is false".

This should not be confused with the statement "For every conceivable thing T, T is either X or it is not X".

The two statements are very similar, so it's very easy to confuse them, but they are nonetheless different in that LEM is about the truth value of propositions whereas the second statement is about the classification of phenomena.

In fact, the second statement is more fundamental than LEM and LEM can actually be deduced from it. But more fundamental than the second statement are definitions from which the second statement can be deduced.

The statement "Any given conceivable thing T is either X or it is not X" means "Any given conceivable thing T can either be represented by the symbol X or it can be represented by the symbol not X". The truth value of that statement can be very easily deduced if one understands language, or more specifically, what "not X" symbol denotes.

Whatever the symbol "X" means, i.e. whatever the set of conceivable phenomena that it can represent consists of, the symbol "not X" is a symbol that can be used to represent ANY conceivable phenomenon that cannot be represented by the symbol "X" ( so as long it meets other criteria ) but NO conceivable phenomenon that can be represented by "X".

As an example, if we say that "truth" is a symbol that can be used to denote any proposition that perfectly corresponds to the aspect of reality it is attempting to represent, and if we say that "falsehood" is synonymous with "not truth", then every proposition will fall into one of these categories. No proposition will remain unclassified and no proposition will end up in some other category.

( Bear in mind that "not truth" is not meant to be taken literally or out of context. It does not mean that the symbol can be used to represent ABSOLUTELY ANYTHING that cannot be represented by the word "truth". For example, we can't use the word "truth" to represent elephants, but that does not mean we can use the word "falsehood" to represent them. The reason we can't do that is because it is implied that both the word "truth" and the word "falsehood" can only be used to represent propositions. A less explicit definition of the word "falsehood" would be "a proposition that is not truth". )

And that is how you end up proving LEM.

But to emphasize the importance and the fundamental nature of definitions, consider the case in which the word "falsehood" was defined slightly differently, as a proposition that does not correspond to the aspect of reality it is attempting to represent AT ALL -- a mismatch in every single way. In that case, the words "truth" and "falsehood" would not cover ALL truth values and some truth values would remain unclassified, making the statement "Every proposition P is either true or false" wrong.

But keep in mind that when evaluating the truth value of other people's statements, such as LEM, what matters is THEIR definitions and not the ones you or someone else freely chose.

In the case of LEM, the word "true" and "false" are very clearly defined as complementary.

So LEM remains true.

And binary classification ( i.e. the process of classifying a set of conceivable phenomena into two sets ) remains a separate thing from it.

[Scott Mayers]

The name, "law of excluded middle" simply means that no THIRD option exists and is also where the term 'contradiction" relates:

Con- (with) -tra- (third) -diction (factor).

A 'contradiction' is interpreted today as something we think is a derogatory thing needing a fix. The term though only DESCRIBES a state of something having more than two states or values. The rule to EXCLUDE a 'middle' refers to the exclusion of contradiction when referencing things that are COMPLEMENTARY evaluated, something that is reduced to binary options, like whether a proposition can be true or false.

However, you can EXTEND logic to include multiple values (allowing 'contradiction') by adding more logical instances in parallel. For instance, electronics can add a third value by using two distinct parallel inputs so that you can get {00, 01, or 10} instead of using a third digit with unique properties, such as {0, 1, 2}.

Quantum computing, for instance, tries overcome the binary limitations in electronics by enabling the use of more than one logical dimension as well. Think of how alternating current has the values {-1, 0, and 1} for forward current, backward current, and no current. The quantum computing tries to use this but because accelerating current causes interfering magnetic lines that induce current in other close circuitry, they found that they can only use this if they use superconducting materials that lack the induction that would cause too many output errors.

[Magnus Anderson]

The name, "law of excluded middle" simply means that no THIRD option exists

If what you're saying is that LEM states that things can only exist in two states, then I disagree.

First, LEM is really only about the truth value of propositions ( it does not apply to everything that can be conceived. )

And second, LEM does not say that for each proposition P the number of different states the truth value of P can be in is 2.

[Skepdick]

The Law of Excluded Middle ( LEM ) is captured by the statement "For every proposition P, P is either true or it is false".

That's one formulation.

The other formulation is "In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true."

And so both of these claims are invocations of LEM.

Either "the sky is green" is true; or the negation of "the sky is green" is true.
Either "the sky is green" is true; or it's false.

The reason these are equivalent is precisely becase in Classical/Boolean logic True and False are complementary.

If not True then False.
If not False then True.

If what you're saying is that LEM states that things can only exist in two states, then I disagree.

First, LEM is really only about the truth value of propositions ( it does not apply to everything that can be conceived. )

The truth-value of a proposition can only exist in two states: True or False.

But, like I explained, that amounts to either negating the truth-value while maintaining the proposition; or negating the proposition while maintaining the truth-value.

Last but not least... either LEM is true or it's false. Either the proposition "LEM Is true" is true; or its negation "LEM is not true" is true.

[Scott Mayers]

The name, "law of excluded middle" simply means that no THIRD option exists

If what you're saying is that LEM states that things can only exist in two states, then I disagree.

First, LEM is really only about the truth value of propositions ( it does not apply to everything that can be conceived. )

And second, LEM does not say that for each proposition P the number of different states the truth value of P can be in is 2.

No, the LEM is a SUFFICIENT law that COVERS all possibilities, including multi-valued options by what I explained. Thus the example I gave was how the binary true-false option is MINIMALLY true and can EXPAND to a more complex logic that has more than one value.

If you start by assuming any number of values, you'd have to accept an infinity of possible unique values and THEN try to reduce it to the Binary where needed. But this makes the logic much harder to develop without assuming MORE than you normally would be STARTING with the simpler binary valued system.

For example, a multi-value system requires not assuming a normal 'complement', needs a 'cycle' rule, and a 'minimum' and 'maximum' rule. These though require assuming the user of the logic to know math BEFORE you may want to use the very logic system to prove math from simple set theories. For example, a 'cycle' is a rule that has to assume EACH finite number of possible values as its own separate system and when defined, would require you interpret the set of values in a circle. For three-values, this is a circle with points, 0, 1, and 2 (nodes). '0' in always defined the minimal value of all systems but the maximum here is '2'. A 'cycle' rule would then make the Maximum value (M=2) be followed by the Minimal value (m=0). The 'rule' would then be...

a "A-Cycle-B" is "(A plus B) modulus (M plus 1), where "M plus 1" is the the actual number of elements in the system. Then a "Complement", "-A" would be "M minus A". But if you are attempting to develop math FROM this, it begs that the one using the logic has to ASSUME math in order to prove it.

Because you'd start with an infinite possible valued systems, EACH finite logic would have to be defined for ALL infinite possibilities, something you cannot actually do. Thus, beginning with a GENERAL system blind to any particular number of values is not possible to prove using simpler laws.

The LEM is just thus the 'safest' means of COVERING all possible logic systems. Then with the binary true-false value system set up, you can CONSTRUCT the more complex logic systems FROM it by using multiple parallel instances to create those systems with greater values, something MUCH easier to do.

[Magnus Anderson]

That's one formulation.

The other formulation is "In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true."

And so both of these claims are invocations of LEM.

Yes. It's the same statement worded differently.

"For every P, either P is true or negation of P is true."

Since "negation of P is true" is another way of saying "P is false", it's basically saying "For every P, either P is true or P is false".

But it's still not the same as "For every conceivable thing T and every conceivable class C, T either belongs to C or it does not."

The truth-value of a proposition can only exist in two states: True or False.

Not quite.

As an example, the truth value of propositions such as "2 + 2 = 4" can be expressed as a number on a scale from 0 to infinity.

The idea is to find the absolute difference between the correct answer and the provided answer.

In the case of "2 + 2 = 4", that would be "| 4 - 4 | = 0". So its truth value is 0, which means, "not at all false".

In the case of "2 + 2 = 3", that would be "| 4 - 3 | = 1". So its truth value is 1, which means, "a bit false".

In the case of "2 + 2 = 1000004", that would be 1,000,000, which means, "extremely false".

LEM does not prohibit this sort of classification.

[Magnus Anderson]

No

No to what? I made 3 claims in that quote.

[Scott Mayers]

No

No to what? I made 3 claims in that quote.

I am asserting that the LEM states that only one of two values exist and labelled as {false, true} or more clearly, {false, non-false} or {non-true, true} . You disagreed by assuming this was strictly limited to propositions but the laws of logic are not simply about propositional logic, but to any logical system. The choice to use 'true' or 'false' is about them being strictly complementary adjectives.

The common wording that uses the propositional statements is fine because usually the next step in defining other logic systems traditionally begins by using propositional statements with the values 'true' or 'false'. The varied meanings of the actual propositions are irrelevant at this stage, but ARE technically built upon the values.

But technically, the laws of logic DO only imply that given ANYTHING, call it X, then a non-X exists and it refers to the class of some or all that is not X. X and non-X are "exclusive complements" but in greater valued systems, the complements are not 'negations' and not necessarily exclusive. For instance, in a system that may use, {0, 1/2, 1}, for instance, the middle value exists and its complement is itself. That is, Comp-1/2 == 1/2 [because 1/2 plus 1/2 = 1, as 0 plus 1 = 1 where 1 is the maximum value]

The term, 'truth' [and its relatives, true, not-true, false, not-false] are themselves undefined but simply refer to AGREEMENT between two or more people speaking about the realities in question, whether they be propositions or literal realities. So you are at best pointing out that the things that these values are tied to are infinitely variable. To properly express this might be to first let some symbol, like P, represent what is being evaluated as 'true' or 'false'. To generally state, then you might pick 0 = false and 1 = not-false (or true) and so we can state that:

(0 & P) or (1 & P) = 1 ....means for some P, either P = 0 or P = 1

Because the 0, or the 1, are strict complements, the law of EXCLUDING a middle is more complete here to say,

(0 & P) xor (1 & P) = 1 [the law of excluding the inclusive meaning of 'or']

Technically this is not necessary to state EXCEPT if you use binary complementary values. But I explained why it is too hard to define the general case of multiple values.

The 'P' here can stand for "proposition" if you like. Then your only point is that 'P' itself is irrelevant to specific meaning and has an infinite possible propositions you can replace this. To the actual laws, P does not have to be a proposition but CAN be the very values themselves, such as 0 or not-0 [not-1 or 1,... 0 or 1]

So for P = 0
(0 & 0) or (1 & 0) = 0 or 0 = 0


if P = 1
(0 & 1) or (1 & 1) = 0 or 1 = 1

[Magnus Anderson]

No, the LEM is a SUFFICIENT law that COVERS all possibilities, including multi-valued options by what I explained. Thus the example I gave was how the binary true-false option is MINIMALLY true and can EXPAND to a more complex logic that has more than one value.

If you start by assuming any number of values, you'd have to accept an infinity of possible unique values and THEN try to reduce it to the Binary where needed. But this makes the logic much harder to develop without assuming MORE than you normally would be STARTING with the simpler binary valued system.

You're not assuming the number of values. You're merely choosing into how many different classes you want to classify things.

The number of classes you're going to choose is not a matter of veracity but a matter of utility. There is no "true" number of classes you should choose, just a more or less useful number given a situation. And you're certainly not bound to one classification system. You can use different ones in different situations. In some situations, you may want to use a classification with higher arity; in other situations, you may want to use a classification with lower arity.

LEM is really only about binary classification of truth values. It is unconcerned with any other system of classification. It says nothing about them -- and most importantly, it does not prohibit them.

Unfortunately, there is a lot of confusion surrounding what LEM really is.

As an example, it is said that Łukasiewicz logic, which is a three-valued system of logic, rejects LEM.

But not only does it not reject LEM, it does not even use more than 2 classes for classification of truth values.

Where it differs is that it changes what kind of value is assigned to propositions. In classical logic, each proposition is assigned a truth value. In Lukasiewicz logic, each proposition is assigned either a truth value -- true or false -- or it is assigned a value that is "unknown".

This third value indicates that the truth value of the proposition is either unknown or that it is absent ( meaning that the proposition is not really a proposition, given that every proposition, by definition, has truth value. ) As such, it is not exactly a truth value. A truth value is a value that describes the degree to which the predicate of the proposition corresponds to the subject. This third value means "True, false or inapplicable because it's not really a proposition". The "inapplicable" part is what makes it a non-truth-value.

Even if the third value were a truth value, e.g. "Either true or false", even then, despite the introduction of the third class, the truth value would still be binary.

But really, the third value in Lukasiewicz logic is a truth value only in name. They call every value assigned to propositions "truth value".

Another example would be fuzzy logic, an infinitely many-valued logic, where truth is expressed as a real number ranging from 0 ( meaning "completely false" ) to 1 ( meaning "completely true", or more simply, what is normally meant by "true". ) Anything in between is partial truth.

A common mistake is to think that, just because it works with an infinite number of truth values, that it rejects LEM. But it does not. In fuzzy logic, "true" is represented as "1" and "false" is represented by a number less than 1. In fuzzy logic, all propositions have a truth value that is either 1 or less than 1. That's LEM. And it is obviously true.

Zero LEMs and PBs have been rejected.

[Skepdick]

Since "negation of P is true" is another way of saying "P is false"

Well, duh! That's how tautologies work. What makes these equivalent statements is precisely the "law" of excluded middle.

There are systems of logic in which the negation of P is true is NOT another way of saying " P is false".
There are systems of logic in which the negation of true is NOT false; and the negation of false is NOT true.

e.g there are systems of logic in which the negation of LEM Is true.

And there are systems of logic in which some instances of LEM are true and some instances of LEM are false.

, it's basically saying "For every P, either P is true or P is false".

Precisely that's what "bivalent" means. Bi (two) valent (values).

And then there is non-bivalent/non-Boolean/non-dual logics.

For every P. some Ps are true. some Ps are false and some Ps are neither true nor false.

But it's still not the same as "For every conceivable thing T and every conceivable class C, T either belongs to C or it does not."

It's exactly the same thing. For every conceivable Proposition P about every coiceivable thing T, P about T belongs to Bool or it doesn't.

The proposition is either Boolean or it's not.

The truth-value of a proposition can only exist in two states: True or False.

Not quite.
[/quote]
Yes, quite. What's the 3rd option?

As an example, the truth value of propositions such as "2 + 2 = 4" can be expressed as a number on a scale from 0 to infinity.

You are welcome to express it that way. In Boolean logic with Excluded middle the value 0 maps to False; the values on the interval (0, ∞) map to True

The idea is to find the absolute difference between the correct answer and the provided answer.

No idea what that means.

In the case of "2 + 2 = 4", that would be "| 4 - 4 | = 0". So its truth value is 0, which means, "not at all false".

In the case of "2 + 2 = 3", that would be "| 4 - 3 | = 1". So its truth value is 1, which means, "a bit false".

In the case of "2 + 2 = 1000004", that would be 1,000,000, which means, "extremely false".

Why have you constructed such a metric? Why aren't you operating on a scale of most true to least true with NO falsehood?

LEM does not prohibit this sort of classification.

It does in the Real numbers.

If there exists a continuous map from R -> {True, False}; or if you prefer, R -> {Least True, Most True}.
For convenience - lets just call it R -> {0,1} where 0 and 1 get to signify whatever meaning you choose them to signify.


There exists no such map because if it did it would break apart the reals into two. And the reals are said to be continuous.

[Magnus Anderson]

Well, duh! That's how tautologies work. What makes these equivalent statements is precisely the "law" of excluded middle.

There are systems of logic in which the negation of P is true is NOT another way of saying " P is false".
There are systems of logic in which the negation of true is NOT false; and the negation of false is NOT true.

e.g there are systems of logic in which the negation of LEM Is true.

And there are systems of logic in which some instances of LEM are true and some instances of LEM are false.

Other systems of logic are not free to assign their own meaning to the words employed by LEM.

In the context of LEM, "negation of P" means precisely what the author of LEM decided that it means. What it means in other systems of logic is irrelevant. They are not authorities on the matter.

It's akin to an Englishman saying "Pie is a type of food" and another objecting by saying that "That's not true. In Spain, it's a body part."

It's a logical fallacy called "equivocation".

Another example would be people arguing against the obviously true claim "Square-circles don't exist" by referencing taxicab geometry.

Equivocation through and through.

[Skepdick]

Other systems of logic are not free to assign their own meaning to the words employed by LEM.

Yes, but they are free to apply LEM to itself.

Either it's true that "For every P, P is either true or false" or it's NOT true that "For every P, P is either true or false"

In the context of LEM, "negation of P" means precisely what the author of LEM decided that it means.

Sure. Let P:= LEM

"negation of P" (where P = LEM) means precisely what the author of LEM decided that it means.

What it means in other systems of logic is irrelevant. They are not authorities on the matter.

Sure. So what does the negation of P (where P = LEM) mean in Classical logic?

It's akin to an Englishman saying "Pie is a type of food" and another objecting by saying that "That's not true. In Spain, it's a body part."

It's nothing like that.

It's a logical fallacy called "equivocation".

It's not.

Another example would be people arguing against the obviously true claim "Square-circles don't exist" by referencing taxicab geometry.

Equivocation through and through.

I have a much better example...

Imagine that somebody proposes that every proposition is either true or false. Is that proposition true or false?

[Magnus Anderson]

It's exactly the same thing. For every conceivable Proposition P about every coiceivable thing T, P about T belongs to Bool or it doesn't.

It's not the same thing because LEM is really only talking about propositions. It's a far more specific claim.

The other claim is talking about EVERY conceivable thing, whether that thing is a proposition or not.

"For every conceivable thing T and for every conceivable set of things S, T either belongs to S or it does not."

That's a far more general claim than LEM.

[Magnus Anderson]

It's nothing like that.

It's exactly like that. And yes, it is an equivocation.

[Skepdick]

It's not the same thing because LEM is really only talking about propositions. It's a far more specific claim.

I am being about as specific as it gets.

LEM itself is a proposition.
It proposes that EVERY proposition is either true or false without explicitly specifying whether LEM itself is true or false.

The other claim is talking about EVERY conceivable thing, whether that thing is a proposition or not.

No, we are talking about propositons ONLY. Such as the proposition known as LEM.

"For every conceivable thing T and for every conceivable set of things S, T either belongs to S or it does not."

That's a far more general claim than LEM.

Sure - you can generalize LEM. ANd with it - you can generalize about the lack of explicit value.