What LEM is not

[Magnus Anderson]

The reason why modern logic rejects your view, is because it is simply unsustainable. No formal system can function in the way that you propose. Such system would be utterly unusable if it tried. That is why a proposition is defined in modern logic as a "formal syntactic object", while every syntactically valid sentence of the designated form is a proposition in the formal language.

I think I have to go back and repeat that the meaning of a word cannot be true or false but merely more or less useful in relation to a goal ( and there are many different goals to choose from. )

In other words, there is no "true" meaning of the word "unicorn". The word can mean anything you want it to mean. The question is merely: it is useful given what you want to achieve? We can talk about the true meaning of a word in a given language, e.g. the true meaning of the word "unicorn" in standard English. That is true. But otherwise, the word has no "true" meaning.

Moreover, you can always coin a neologism instead of taking an existing term and redefining it. The latter is what these systems of logic are doing. They are redefining terms. They are taking established terms such as "proposition" and "truth value" and changing their meaning to suit their purposes. That sort of thing is dangerous because it can, and it obviously does, lead to equivocation. Neologisms make it difficult to equivocate.

And finally, when evaluating the truth value of a statement, the only meanings that matter are the ones assigned to the words of that statement by the author of that statement. You have to understand the language the author is speaking in before you can understand what he is saying ( which is a prerequisite for evaluating the truth value of what he's saying. ) If you don't do that, you will necessarily end up misinterpreting him. And if you misinterpret him, you won't be evaluating the truth value of what he's saying.

If I say "Dogs are cats", you are not free to simply interpret that statement using standard English language. Perhaps that's not the language I am speaking. Perhaps I am speaking a variation of it where the word "dog" means the same thing as the word "cat". It's your duty as a reader to figure that out. Simply using whatever language you wish is not an option.

And in the case of LEM, the term "proposition" means "an idea that a portion of reality is such and such".

[godelian]

In other words, there is no "true" meaning of the word "unicorn". The word can mean anything you want it to mean. The question is merely: it is useful given what you want to achieve? We can talk about the true meaning of a word in a given language, e.g. the true meaning of the word "unicorn" in standard English. That is true. But otherwise, the word has no "true" meaning.

That is true for natural language. In the context of modern logic, however, there are rather precise model-theoretical semantics to a proposition expressed in the formal language of a theory.

Moreover, you can always coin a neologism instead of taking an existing term and redefining it. The latter is what these systems of logic are doing. They are redefining terms. They are taking established terms such as "proposition" and "truth value" and changing their meaning to suit their purposes. That sort of thing is dangerous because it can, and it obviously does, lead to equivocation. Neologisms make it difficult to equivocate.

That is not really what happened. We have come a long way from Aristotle's take on logic. Over the centuries, the ongoing investigation ended up producing increasingly precise results. Modern logic is the result of that centuries-long investigation. You can't just stop reading until halfway the 18th century.

And finally, when evaluating the truth value of a statement, the only meanings that matter are the ones assigned to the words of that statement by the author of that statement. You have to understand the language the author is speaking in before you can understand what he is saying ( which is a prerequisite for evaluating the truth value of what he's saying. ) If you don't do that, you will necessarily end up misinterpreting him. And if you misinterpret him, you won't be evaluating the truth value of what he's saying.

That is true for natural language. That is not true for the formal language of modern logic. That is also not true for programs written in programming languages. Modern logic is no longer done in natural language, exactly for that reason.

If I say "Dogs are cats", you are not free to simply interpret that statement using standard English language.

English is not a suitable language for modern logic. Any serious investigation should start by picking a formal language, such as, for example, the language of first-order predicate logic. We no longer use natural language to pinpoint the precise interpretation of logic sentences.

Perhaps that's not the language I am speaking. Perhaps I am speaking a variation of it where the word "dog" means the same thing as the word "cat". It's your duty as a reader to figure that out. Simply using whatever language you wish is not an option.

Modern logic is not possible in natural language. Software programming is also not possible in natural language. The very first task is to enforce unique readability and therefore unique parse trees for the sentences being expressed. This is not possible in natural language, because natural language is simply not suitable for that purpose.

And in the case of LEM, the term "proposition" means "an idea that a portion of reality is such and such".

The LEM is a law of modern logic.

Older forms of logic are deprecated and are no longer viable for the purpose of serious investigation into the nature of logic. These older forms of logic have been superseded by modern logic. That is why it does not make sense to keep using 18th century definitions.

In modern logic, a proposition is not some kind of vague "idea". It is a well-formed symbol stream that satisfies the requirements of a particular syntax. There may not even be any "idea" associated with it, if only, because syntactic constructions do not necessarily have any semantics attached to them. Therefore, the term "idea" does not have any legitimate use in a definition that will help defining a formal system. The term "reality" is also not suitable for the construction of a formal system. Modern logic rejects the use of such terminology because it will only lead to problems further down the line. It just does not work like that anymore. We have come a far way from outdated 18th century views on what logic is.

[Magnus Anderson]

That is true for natural language. In the context of modern logic, however, there are rather precise model-theoretical semantics to a proposition expressed in the formal language of a theory.

That is true for natural language. That is not true for the formal language of modern logic. That is also not true for programs written in programming languages. Modern logic is no longer done in natural language, exactly for that reason.

Again, you're missing the point. You keep insisting on misinterpreting what other people are saying. LEM was formulated using a natural language. If you want to evaluate its truth value, you have to first understand the natural language with which it was stated. You focus way too much on subsequent developments, particularly in mathematics, worshipping them as if they make everything that preceded them obsolete and as if they are flawless or at least better in every single regard. The thing is that the people involved in these subsequent developments are not authorities on what LEM is. They did not invent it. That's why you should shift your attention away from modern mathematics towards the original formulation of LEM, understand what it is saying and evaluate its truth value. Don't allow yourself to be confused by subsequent developments ( happens quite enough. )

And remember that just because it's new it's not necessarily incompatible with the old. Fuzzy logic is perfectly compatible with classical. It's certainly not rejecting LEM. You have to misunderstand what LEM is in order to believe otherwise.

That is why it does not make sense to keep using 18th century definitions.

If you're evaluating the truth value of a statement made by someone in 18th century, then you have to understand the 18th century definitions, and specifically, the ones used by the author.

In modern logic, a proposition is not some kind of vague "idea". It is a well-formed symbol stream that satisfies the requirements of a particular syntax. There may not even be any "idea" associated with it, if only, because syntactic constructions do not necessarily have any semantics attached to them. Therefore, the term "idea" does not have any legitimate use in a definition that will help defining a formal system. The term "reality" is also not suitable for the construction of a formal system. Modern logic rejects the use of such terminology because it will only lead to problems further down the line. It just does not work like that anymore. We have come a far way from outdated 18th century views on what logic is.

That's how a number of modern systems of logic define the term "proposition". Outside of these systems, the word "proposition" is used to denote a non-linguistic, non-symbolic, entity. A string of symbols, such as a string of letters, is not a proposition but at best a symbol representing a proposition. There was no need to redefine the term "proposition", they could have simply used the term "sentence" instead. A sentence may or may not represent a proposition, and as such, it may or may not have truth value. As such, for each sentence, you can say that it is either true, false or that it has no truth value because it is not a proposition. If you don't know which one of these three is the case, you can simply say that its status is "unknown". But they did redefine the term anyways and they did it because they did not know anything better at the time. Add to that various pressures ( e.g. time pressure. )

I am not going to address the rest of your quote because even the parts that I did address have very little to do with the actual subject and it's actually a serious distraction.

The subject is LEM, what it is and what it is not, and not modern developments in mathematics.

[godelian]

[LEM was formulated using a natural language.

Arithmetic was originally also formulated using natural language. This has been superseded by the more modern use of decimals, arithmetic operators, and other symbolic language.

You focus way too much on subsequent developments, particularly in mathematics, worshipping them as if they make everything that preceded them obsolete and as if they are flawless or at least better in every single regard. The thing is that the people involved in these subsequent developments are not authorities on what LEM is. They did not invent it.

I also don't think that arithmetic should still be carried out in natural language. Without decimals and operator symbols such as "+ - x /" arithmetic would be much harder to perform correctly. It would also be really hard to build calculators. It would also fail to capture modern notions such as group theory and other algebraic structures. It would essentially be unusable in a modern context.

For all practical purposes, natural language is obsolete for the purpose of arithmetic.

The same holds true for logic. In modern times, the consensus is that it needs to be done in formal symbolic language.

The subject is LEM, what it is and what it is not, and not modern developments in mathematics.

What you are arguing about the LEM is incorrect in terms of modern symbolic logic. Your argument is based on an outdated 18th century definition that has long been superseded. It is not even correct anymore in terms of 19th century boolean algebra. That is how outdated it is.

[Magnus Anderson]

What you are arguing about the LEM is incorrect in terms of modern symbolic logic. Your argument is based on an outdated 18th century definition that has long been superseded. It is not even correct anymore in terms of 19th century boolean algebra. That is how outdated it is.

And you're still getting distracted, forcing me to repeat myself.

If I say that "Dogs are cats" and you want to evaluate the truth value of that statement, you have to understand the language that I am using. It does not matter how useful that language is compared to some other language for achieving certain ends. You are not speaking, you're not trying to program a machine, you are trying to evaluate the truth value of what someone else is saying.

When they said, "For every proposition P, P is either true or false", what did they mean by "proposition", what did they mean by "true" and what did they mean by "false"? How these terms are defined in other languages is irrelevant. This statement isn't expressed in these other languages.

And note that all we're trying to do here in this thread is understand what LEM is, what LEM is not and whether or not it is true. We are not trying to do anything more than that e.g. we're not trying to build a system of logic.

[godelian]

If I say that "Dogs are cats" and you want to evaluate the truth value of that statement, you have to understand the language that I am using.

This proposition is not expressed in a formal language with a proper symbolic grammar in which its context, theory T, is also expressed. In absence of system-wide premises, i.e. explicitly-stated axioms, this sentence is insufficiently specified.

Leibniz already figured out in the 17th century that the problem was caused by the use of informal language:

https://en.wikipedia.org/wiki/Entscheidungsproblem

The origin of the Entscheidungsproblem goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements.[3] He realized that the first step would have to be a clean formal language, and much of his subsequent work was directed toward that goal.

You are not speaking, you're not trying to program a machine, you are trying to evaluate the truth value of what someone else is saying.

This cannot be done without first solving the problem caused by the ambiguity of natural language. That is a 17th century conclusion.

When they said, "For every proposition P, P is either true or false", what did they mean by "proposition", what did they mean by "true" and what did they mean by "false"? How these terms are defined in other languages is irrelevant. This statement isn't expressed in these other languages.

The LEM is a theorem at the level of a particular theory T that may or may not be provable from T. Examples for T, from which the LEM is provable:

https://en.wikipedia.org/wiki/Decidability_(logic)

Some decidable theories include (Monk 1976, p. 234):[2]

The set of first-order logical validities in the signature with only equality, established by Leopold Löwenheim in 1915.
The set of first-order logical validities in a signature with equality and one unary function, established by Ehrenfeucht in 1959.
The first-order theory of the natural numbers in the signature with equality and addition, also called Presburger arithmetic. The completeness was established by Mojżesz Presburger in 1929.
The first-order theory of the natural numbers in the signature with equality and multiplication, also called Skolem arithmetic.
The first-order theory of Boolean algebras, established by Alfred Tarski in 1940 (found in 1940 but announced in 1949).
The first-order theory of algebraically closed fields of a given characteristic, established by Tarski in 1949.
The first-order theory of real-closed ordered fields, established by Tarski in 1949 (see also Tarski's exponential function problem).
The first-order theory of Euclidean geometry, established by Tarski in 1949.
The first-order theory of Abelian groups, established by Szmielew in 1955.
The first-order theory of hyperbolic geometry, established by Schwabhäuser in 1959.
Specific decidable sublanguages of set theory investigated in the 1980s through today.(Cantone et al., 2001)
The monadic second-order theory of trees (see S2S).

Methods used to establish decidability include quantifier elimination, model completeness, and the Łoś-Vaught test.

A theory T is decidable when it is complete:

https://en.wikipedia.org/wiki/Complete_theory

In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence φ , the theory T contains the sentence or its negation but not both (that is, either T ⊢ φ or T ⊢ ¬ φ).

Proving that a theory T is complete is equivalent to proving that theory T proves the LEM.

And note that all we're trying to do here in this thread is understand what LEM is, what LEM is not and whether or not it is true. We are not trying to do anything more than that e.g. we're not trying to build a system of logic.

The LEM is either a theorem in a particular theory T or it is not. It depends on the theory at hand. Complete theories prove the LEM:

https://en.wikipedia.org/wiki/Complete_theory

Some examples of complete theories are:

Presburger arithmetic
Tarski's axioms for Euclidean geometry
The theory of dense linear orders without endpoints
The theory of algebraically closed fields of a given characteristic
The theory of real closed fields
Every uncountably categorical countable theory
Every countably categorical countable theory
A group of three elements
True arithmetic or any other elementary diagram

The LEM is provable from theory T if there do not exist undecidable propositions in T, meaning that T is complete.

There is nothing ambiguous about the LEM.

[Magnus Anderson]

This proposition is not expressed in a formal language with a proper symbolic grammar in which its context, theory T, is also expressed. In absence of system-wide premises, i.e. explicitly-stated axioms, this sentence is insufficiently specified.

And it does not have to be expressed in a formal language. You just have to make an effort to understand its expression in a natural language. Certainly, whoever was the first to use it, meant something by it. And I think it's pretty clear what was meant by it. It really isn't that difficult t figure out.

Moreover, if you can't understand what is being said, you can't criticize what is being said. In other words, if you don't know what LEM is, you can't talk about whether it's true or false.

[godelian]

And it does not have to be expressed in a formal language. You just have to make an effort to understand its expression in a natural language. Certainly, whoever was the first to use it, meant something by it. And I think it's pretty clear what was meant by it. It really isn't that difficult t figure out.

Gottfried Wilhelm Leibniz already pointed out in the 17th century that your take on the matter is simply unsustainable.

Moreover, if you can't understand what is being said, you can't criticize what is being said. In other words, if you don't know what LEM is, you can't talk about whether it's true or false.

It is trivially simple what the LEM is:

For every proposition S in L the language of theory T, T proves S or T proves the negation of S.

There are theories in which LEM is provable, meaning that these theories are complete. I have already mentioned examples of theories in which LEM is provable in a previous answer.

[Magnus Anderson]

It is trivially simple what the LEM is:

So why are you mentioning Leibniz?

For every proposition S in L the language of theory T, T proves S or T proves the negation of S.

That's not exactly it.

LEM is simply stated as:

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

Propositions are understood to be language-independent ( not to be confused with statements which are representations of propositions ) and the claim is about the truth value of propositions and not about what can be proven.

[godelian]

For every proposition S in L the language of theory T, T proves S or T proves the negation of S.

That's not exactly it.

LEM is simply stated as:

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

You won't know if P is true or P is false unless you make use of soundness theorem: If S is provable then S is true in all its interpretations.

Furthermore, without L (formal language) and T (system-wide premises), a proposition S cannot be specified unambiguously.

Pure reason is blind. We can get access to truth only by means of proof.

Propositions are understood to be language-independent ( not to be confused with statements which are representations of propositions ) and the claim is about the truth value of propositions and not about what can be proven.

Experience only yields probabilistic truth. It never yields logical truth. Only pure reason can achieve this, but it is blind, meaning that you can only see truth because you can prove it.

You seem to confuse physical truth, which is always probabilistic, with logical truth. Furthermore, systemless reasoning does not work. There is always a system-wide theoretical context as well as a language in which it is expressed.

[Magnus Anderson]

Furthermore, without L (formal language) and T (system-wide premises), a proposition S cannot be specified unambiguously.

That's a position of yours that you have stated more than once by now. And I am telling you that it's horribly mistaken.

What you're doing here, without realizing it, is that you're contradicting yourself by implicitly stating that you don't what LEM is and that you know what LEM is.

LEM was originally expressed using a natural language. Thus, in order to understand what LEM is, you have to understand its expression in a natural language. You refuse to do that. I ask you tell us what they meant by "proposition", "true" and "false" and you avoid doing that.

You say it's impossible to understand that expression because it's ambiguous. That means that you can't and that you don't know what LEM is.

Yet, at the same time, you proceed to express what LEM is using a formal language.

That is a contradiction.

You can't translate a sentence to a different language if you don't understand what that sentence means. You can only mistranslate it. And that's precisely what's happening.

Again, there is absolutely nothing ambiguous about the statement that is "For every proposition P, P is either true or it is false." You may find it difficult to understand due to being accustomed to formal languages but that does not mean it can't be understood. It certainly can. And it isn't really that difficult at all.

You're seriously getting distracted by being overly-focused on modern developments in logic.

[godelian]

Furthermore, without L (formal language) and T (system-wide premises), a proposition S cannot be specified unambiguously.

That's a position of yours that you have stated more than once by now. And I am telling you that it's horribly mistaken.

It is the same position as Gottfried Leibniz took on the matter, and over the centuries it has turned to be incredibly accurate.

LEM was originally expressed using a natural language.

And so was arithmetic. Arithmetic is entirely symbolic now, and we are not going back.

I ask you tell us what they meant by "proposition", "true" and "false" and you avoid doing that.

I have already elaborated at length what exactly these terms mean in modern logic.

Again, there is absolutely nothing ambiguous about the statement that is "For every proposition P, P is either true or it is false."

It is ambiguous because in some theoretical context T, the LEM is provable while in other theoretical contexts, it is not. I have already given examples of both.

[godelian]

You're seriously getting distracted by being overly-focused on modern developments in logic.

The history of modern logic ended up raising important constructivist concerns with regards to the LEM. In his notes, Robert Harper points out (below) that -- without any mention of a specific or particular theoretical context T -- the LEM is fundamentally undecidable:

https://www.cs.cmu.edu/~rwh/courses/clo ... ts/lem.pdf

15-399 Supplementary Notes:
The “Law” of the Excluded Middle
Robert Harper
February 1, 2005

The Law of the Excluded Middle (LEM) is the proposition A ∨ ¬A. From a classical point of view, LEM is a tautology — it is always true, regardless of whether A is true or false. Constructively, LEM is far from true . . . but neither is it false! One important consequence is that we are free to assume (generally, or specific instances of) the LEM without fear of degenerating into inconsistency.

Neither Provable nor Refutable

From a constructive point of view, the judgement A ∨ ¬A true asserts that the problem expressed by the proposition A is decidable in that we either know a proof of A or we know a proof of ¬A (i.e., we can refute A). On this interpretation, it is obvious that not every proposition is decidable. Indeed, an open problem is precisely a proposition for which we have neither a proof nor a refutation.

Since there are (and always will be) open problems, we cannot expect LEM to be true, constructively speaking.

Since some problems are, in fact, decidable, we cannot expect LEM to be refuted either. That is, we cannot expect ¬(A ∨ ¬A) true to be evident in
general. In fact we can show something even stronger: constructively logic positively denies the falsehood (refutability) of (every instance of) LEM.

Decidability and Undecidability

By the proof given earlier, for no proposition A do we have ¬(A ∨ ¬A) true. Does this mean that no proposition is undecidable? Doesn’t this contradict
well-known results such as the undecidability of the halting problem?

The Halting Problem states that for an arbitrary Turing machine M , either M halts or M diverges. Classically this is a triviality, because it is essentially an instance of LEM. But constructively we can neither prove nor refute it, because there are some machines for which we can prove that they either halt or diverge, and there are some machines for which we have no proof either way. Thus, in logical terms we cannot, in a constructive setting, prove that for arbitrary M , either M ↓ or M ↑. This inability is precisely what is meant by the undecidability of the halting problem!

There are theories, i.e. axiomatic systems, that are provably decidable (and thus for which the LEM is true) and systems that are provably undecidable (and thus for which the LEM is false). The general case, however, is that the LEM is undecidable.

[Magnus Anderson]

I have already elaborated at length what exactly these terms mean in modern logic.

We're running in circles.

I have already told you that it does not matter what these terms mean in modern logic. That's because we're dealing with a statement that came before modern logic, a statement that was expressed using a natural language. If you want to understand what that statement means, which is precisely the subject of this thread, you have to do so by understanding how the author of that statement defined his terms. What these terms mean in other languages, e.g. in modern logic, is irrelevant. Yet, you keep insisting on using the definitions of modern logic to understand that statement. By doing that, you're actively misinterpreting the original statement.

It is the same position as Gottfried Leibniz took on the matter, and over the centuries it has turned to be incredibly accurate.

I advise you to stop throwing references and appealing to authorities and start presenting actual logical arguments in favor of your claims.

And so was arithmetic. Arithmetic is entirely symbolic now, and we are not going back.

And you're still missing the point.

[Skepdick]

I have already elaborated at length what exactly these terms mean in modern logic.

We're running in circles.

I have already told you that it does not matter what these terms mean in modern logic. That's because we're dealing with a statement that came before modern logic, a statement that was expressed using a natural language. If you want to understand what that statement means, which is precisely the subject of this thread, you have to do so by understanding how the author of that statement defined his terms. What these terms mean in other languages, e.g. in modern logic, is irrelevant. Yet, you keep insisting on using the definitions of modern logic to understand that statement. By doing that, you're actively misinterpreting the original statement.

It is the same position as Gottfried Leibniz took on the matter, and over the centuries it has turned to be incredibly accurate.

I advise you to stop throwing references and appealing to authorities and start presenting actual logical arguments in favor of your claims.

And so was arithmetic. Arithmetic is entirely symbolic now, and we are not going back.

And you're still missing the point.

Even when using natural language you are still wrong.

According to your (incorrect) interpretation you are EITHER Magnus Anderson or Skepdick.
According to my (correct) interpretation you are JUST Magnus Anderson.

Either expresses a range of 2 values. One of {Skepdick, Magnus Anderson}
Just expresses a range of 1 value. One of {Magnus Anderson}

either
/ˈʌɪðə,ˈiːðə/
conjunction · adverb
1.
used before the first of two (or occasionally more) given alternatives (the other being introduced by ‘or’).

Oh... and we happen to understand this in formal logic too.

Either(Magnus Andersen,Skepdick)
Just(Magnus Andersen)

Monads. Thanks Leibniz!

https://en.wikipedia.org/wiki/Monad_(fu ... ogramming)