Philosophy Now Forum

There are symbols such as words.

Then there are concepts. These are attached to symbols.

And then there are sets of conceivable phenomena that can be represented by some symbols ( if any ) and not by others ( if any. )

These 3 things must not be confused ( but they often are by non-analytical people, even by programmers, since programming is predominantly a synthetic endeavour, explaining their attraction to constructivism. )

A concept C attached to a word W is a set of rules that determine the set of all conceivable phenomena that can be represented by W.

If we take the symbol "X" and say that it is "A shape that is a square and not a square" that description is actually a description of the concept attached to it. It establishes what can be represented by the symbol "X", if anything. And given that the description is oxymoronic, the set of all conceivable phenomena that can be represented by "X" is an empty one. In other words, absolutely nothing can be represented by "X".

To argue that "X is representing the oxymoronic concept C that is attached to it, so you're wrong when you say that nothing can be represented by it" would be merely another instance of equivocation, the equivocated word being "represent".

Skepdick wrote: ↑Tue Oct 15, 2024 9:06 am No, you didn't.

You failed to explain how the truth-value of LEM is decided.

I did. You are free to quote it and ask questions if something is unclear or express what you think is wrong with it.

I am certainly not going to be repeating myself.

Magnus Anderson wrote: ↑Tue Oct 15, 2024 9:10 am

Skepdick wrote: ↑Tue Oct 15, 2024 9:06 am No, you didn't.

You failed to explain how the truth-value of LEM is decided.

I did. You are free to quote it and ask questions if something is unclear or express what you think is wrong with it.

I am certainly not going to be repeating myself.

Sure. It's unclear to me what reality has to do with anything....

Which part of reality does -1-1=-2 correspond to?

You are talking about abstract structures. Such as LEM. What makes your abstract imagination true?

Given the existence of undecidable propositions LEM fails even under correspondence.

Magnus Anderson wrote: ↑Tue Oct 15, 2024 8:45 am More accurately, if something is a proposition, then by definition, it has truth value. Conversely, if it has no truth value, then by definition, it is not a proposition, and at best, it merely looks like one.

A proposition in a mathematical theory is represented by a sentence, i.e. a well-formed formula with no free variables. It is a symbol stream that satisfies the grammar of the theory's language. Only syntactic requirements need to be satisfied.

There are no semantic requirements for a sentence to be part of a theory. In other words, you do not need to evaluate a sentence in order to verify if it is valid sentence in the theory. Inspecting its sequence of symbols is enough. So, if it correctly looks like one, then it is indeed one.

If the truth value of a well-formed sentence turns out to be undecidable then it is still a legitimate member of its theory.

Your approach would be completely unworkable. It would not be enough to verify the formal grammar of a sentence to check if it is valid. You would also have to evaluate or execute it. There is absolutely no system that works like that.

godelian wrote: ↑Tue Oct 15, 2024 9:36 am Your approach would be completely unworkable. It would not be enough to verify the formal grammar of a sentence to check if it is valid. You would also have to evaluate or execute it. There is absolutely no system that works like that.

Humans work like that. Evaluation/execution of language is never suspended.

Skepdick wrote: ↑Tue Oct 15, 2024 9:44 am

godelian wrote: ↑Tue Oct 15, 2024 9:36 am Your approach would be completely unworkable. It would not be enough to verify the formal grammar of a sentence to check if it is valid. You would also have to evaluate or execute it. There is absolutely no system that works like that.

Humans work like that. Evaluation/execution of language is never suspended.

In the context of formal systems, how can the following be implemented: "an undecidable proposition is not a proposition"?

It badly mixes up syntax ("proposition") and semantics ("undecidable"). I find it unreasonable.

godelian wrote: ↑Tue Oct 15, 2024 10:07 am In the context of formal systems, how can the following be implemented: "an undecidable proposition is not a proposition"?

It badly mixes up syntax ("proposition") and semantics ("undecidable"). I find it unreasonable.

It can't be implemented. But you know what it means.

Skepdick wrote: ↑Tue Oct 15, 2024 9:11 am Sure. It's unclear to me what reality has to do with anything....

Which part of reality does -1-1=-2 correspond to?

Before I respond to this, can you quote the part that you're addressing?

You are talking about abstract structures. Such as LEM. What makes your abstract imagination true?

Given the existence of undecidable propositions LEM fails even under correspondence.

LEM is true by definition. It has to do with the way words are defined, its truth value is derived from these definitions. Every single claim that LEM is false, or that it is false under certain circumstances e.g. in this or that system of logic, is based on a misinterpretation of what LEM really is.

"For every proposition P, either P is true or it is false."

Since "false" is defined as "not true", this amounts to "For every proposition P, either P is true or it is not true."

To deny this would be to say that P is BOTH not true AND not not true. And that is a contradiction.

godelian wrote: ↑Tue Oct 15, 2024 9:36 am A proposition in a mathematical theory is represented by a sentence, i.e. a well-formed formula with no free variables. It is a symbol stream that satisfies the grammar of the theory's language. Only syntactic requirements need to be satisfied.

There are no semantic requirements for a sentence to be part of a theory. In other words, you do not need to evaluate a sentence in order to verify if it is valid sentence in the theory. Inspecting its sequence of symbols is enough. So, if it correctly looks like one, then it is indeed one.

If the truth value of a well-formed sentence turns out to be undecidable then it is still a legitimate member of its theory.

Your approach would be completely unworkable. It would not be enough to verify the formal grammar of a sentence to check if it is valid. You would also have to evaluate or execute it. There is absolutely no system that works like that.

If you want to correctly evaluate the truth value of a statement, you have to understand how the author of that statement defined the words he used in its construction.

In the case of LEM, you have to understand what the author of LEM meant by words such as "proposition", "true" and "false".

You are not free to define these words however you want. You are not free to use other people's definitions ( regardless of how popular or original they are. ) Doing that would lead to you misinterpreting the statement you're addressing and thus failing to evaluate its truth value.

In the case of LEM, the term "proposition" means "an idea that a portion of realiity exists in certain state". A proposition is not a statement, it's not a sentence, it is a non-linguistic entity, an idea that some portion of reality is such and such. As such, it must consist of the following two components: the subject ( the referred portion of reality ) and the predicate ( the description of the referred portion of reality. )

Propositions are commonly expressed as "A is B" where "A" stands for the subject and "B" for the predicate. If at least one of these components is missing, then we do not have a proposition. For example, if the symbol "A" is an oxymoron, then nothing can be represented by it, and so no portion of reality has been referenced. That would be the case of "missing subject". Statements such as "Square-circles are circles" and "The set of all sets that do not contain themselves contains itelsf" would be an example. Similarly, if the symbol "B" is missing altogether, e.g. "A is", then we have the case of "missing predicate".

Of course, people are free to redefine the word "proposition" as it suits them ( though, if you ask me, it's better to use neologisms since neologisms maintain a clear distinction between distinct concepts. ) But none of that is really relevant when it comes to evaluating the truth value of LEM.

I already mentioned Łukasiewicz logic earlier in this thread. That system of logic redefines the term "proposition" so that it can also be used in reference to non-propositions and the term "truth value" so that a value, that is not a truth value, can be assigned to non-propositions. That's all perfectly legitimate -- albeit potentially dangerous since it can inspire equivocation -- and useful in its own way. But it's not an argument against LEM.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:50 am If you want to correctly evaluate the truth value of a statement, you have to understand how the author of that statement defined the words he used in its construction.

This is only possible if the well-formed sentence has unique readability, which allows us to syntactically determine from its parse tree if this sentence is a legitimate proposition. We do not need to know what the symbols refer to.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:50 am In the case of LEM, you have to understand what the author of LEM meant by words such as "proposition", "true" and "false".

The grammar of the formal language in use decides this.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:50 am In the case of LEM, the term "proposition" means "an idea that a portion of realiity exists in certain state".

Propositions in a formal system refer to abstract, Platonic realities only.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:50 am A proposition is not a statement, it's not a sentence, it is a non-linguistic entity, an idea that some portion of reality is such and such. As such, it must consist of the following two components: the subject ( the referred portion of reality ) and the predicate ( the description of the referred portion of reality. )

In first-order logic augmented with a sufficiently large fragment of arithmetic to enable Gödel numbering, the following is indeed a proposition:

T ⊢S ⟺ K(⌜S⌝)

Meaning: Theory T proves that sentence S has property K.

This construct is syntactic only. The meaning of T,S, and K is irrelevant.

The sentence does not necessarily refer to reality, neither to physical reality nor to any abstract, Platonic one. It is completely divorced from any of the abstract, Platonic models of truth M that interpret T. We do not need to know what it means and we do not need to know if it is even true. This symbol stream represents a proposition because it satisfies the syntactic template for that.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:50 am Of course, people are free to redefine the word "proposition" as it suits them ( though, if you ask me, it's better to use neologisms since neologisms maintain a clear distinction between distinct concepts. ) But none of that is really relevant when it comes to evaluating the truth value of LEM.

The most common definition in mathematical logic is that any valid sentence in the language L of theory T is a proposition. I think, however, that it is acceptable to restrict the term proposition to predicate sentences that fit the template " T ⊢S ⟺ K(⌜S⌝) ". However, I do not necessarily appreciate the pressing need to do that. I don't think that it makes much of a difference. I have never run into any literature that insists on this.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:50 am I already mentioned Łukasiewicz logic earlier in this thread. That system of logic redefines the term "proposition" so that it can also be used in reference to non-propositions and the term "truth value" so that a value, that is not a truth value, can be assigned to non-propositions. That's all perfectly legitimate -- albeit potentially dangerous since it can inspire equivocation -- and useful in its own way. But it's not an argument against LEM.

The potential problems with the LEM do not occur just in many-valued logic systems (such as Łukasiewicz). Even in two-valued logic systems, there are lots of sentences that fit the syntactic template " T ⊢S ⟺ K(⌜S⌝) " and that are logically undecidable. As I have argued previously, if the sentence is decidable, then the LEM is perfectly sustainable:

if S is decidable then S is true or S is false.

This does not work for undecidable sentences.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:12 am LEM is true by definition. It has to do with the way words are defined, its truth value is derived from these definitions.

It's your perrogative to define self-contradictory things as "true". And it's my perogative to dismiss them on grounds of self-contradiction.

LEM is not defined true. LEM is simply defined.

If you define LEM as true then you contradict LEM.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:12 am Every single claim that LEM is false, or that it is false under certain circumstances e.g. in this or that system of logic, is based on a misinterpretation of what LEM really is.

That's a contradiction.

According to LEM itself either LEM is true or LEM is false.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:12 am "For every proposition P, either P is true or it is false."

That proposition is either true or it is false

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:12 am Since "false" is defined as "not true", this amounts to "For every proposition P, either P is true or it is not true."

...therefore either LEM is true or it's not true.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 8:12 am To deny this would be to say that P is BOTH not true AND not not true. And that is a contradiction.

I am not denying LEM. I am provisionally accepting it AND .... applying it to itself.

Therefore either LEM is true or it is false.

If you are insisting that LEM is ALWAYS true; then you are contradicting "For every proposition P, either P is true or it is false.".
A statement that is ALWAYS true is never "either true or false".

LEM with unary truth-value contradicts LEM with a binary truth-value.
If LEM asserts that ANY proposition P, P is either true or false; THEN LEM is false. Because LEM itself fails to satisfy "either true or false"
If LEM asserts that for SOME (but not ANY; and not ALL) propositions P, P is either true or false; THEN LEM is true.

If LEM is not true; then LEM is false (by LEM itself).
Therefore LEM is false.

Proof of negation Q.E.D

godelian wrote: ↑Sat Oct 19, 2024 9:29 am This is only possible if the well-formed sentence has unique readability, which allows us to syntactically determine from its parse tree if this sentence is a legitimate proposition. We do not need to know what the symbols refer to.

So if I say "Mxcvn is prlsz", you do not have to understand the meaning that I assign to the symbols "mxcvn" and "prlsz" in order to evaluate the truth value of that statement?

Skepdick wrote: ↑Sat Oct 19, 2024 6:10 pm A statement that is ALWAYS true is never "either true or false".

Every statement that is "true" is also "either true or false".

The same applies to every statement that is "false".

One simply has to understand what the symbol "either true or false" means. Again, it's entirely a language issue.

The symbol "either true or false" is a symbol that is defined in such a way that it can only be used to represent truth values that can either be represented by the word "true" or by the word "false".

Only truth values that are neither true nor false can be said to not be "either true or false". But no such truth values exist.

In conclusion, there is absolutely no contradiction whatsoever.

On the other hand, the term "neither true nor false" is an oxymoron since it translates to "not true and not not true", an instance of "X and not X".

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm Every statement that is "true" is also "either true or false".

Every statement that is "true" is also "either true or an anal wart"

Every statement that is "true" is ONLY true. Any OR is supefluous.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm The same applies to every statement that is "false".

Every statement that is "false" is ONLY false. Any OR is superfluous.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm One simply has to understand what the symbol "either true or false" means. Again, it's entirely a language issue.

So understand it in a manner that doesn't amount to "Every apple is either an apple or a chicken."

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm The symbol "either true or false" is a symbol that is defined in such a way that it can only be used to represent truth values that can either be represented by the word "true" or by the word "false".

1. You don't even know what a symbol is.
2. Understand it in a manner that doesn't amount to "Every apple is either an apple or an elephant."

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm Only truth values that are neither true nor false can be said to not be "either true or false". But no such truth values exist.

You are equivocating. You are calling "false" a truth-value. A proposition that has a type Boolean has two possible values. True. False.
Those aren't "truth values" - they are just values.

It's true that 1+1=3 is false.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm In conclusion, there is absolutely no contradiction whatsoever.

I can only explain it to you. I can't understand it for you.

Magnus Anderson wrote: ↑Sat Oct 19, 2024 7:17 pm On the other hand, the term "neither true nor false" is an oxymoron since it translates to "not true and not not true", an instance of "X and not X".

That's incorrect. Your substitution of "false" as "not true" presupposes LEM.

Not true doesn't mean false AND not false doesn't mean true.
True means "truth has been established"
NOT true means "Truth has not been established"
False means "Falsehood has been established".
NOT false means "falsehood has not been established".

If truth is not established AND falsehood is not established it trivially follows that NEITHER True NOR False holds.

On top of making too many mistakes when it comes to very basic things, you also happen to be quite arrogant. That makes it rather difficult to have a fruitful conversation with you. And that is why I asked you to leave this thread. Yet, here you are, complaining about other people not thinking like you, presupposing that you're the one who knows it the best.

At this point, you're doing nothing but repeating yourself, i.e. restating your disagreements, without adding anything of substance.

Skepdick wrote: ↑Sat Oct 19, 2024 7:43 pm Every statement that is "true" is ONLY true. Any OR is supefluous.

That's like saying that, if "x" is "2", then "x" is ONLY "2" and nothing else.

In reality, if "x" is "2", it is also "a number", "a positive number", "a whole number", "an even number", "a prime number", "a number less than 10" and so on.

Note the term "even number" is synonymous with "either 2 or 4 or 8 and so on". In fact, all other terms have this "either/or" form.

It's exactly the same thing as Skepdick being a being, a living being, a human being, a male, a forum member and so on.

You really are trying too hard.

Skepdick wrote: ↑Sat Oct 19, 2024 7:43 pm Your substitution of "false" as "not true" presupposes LEM.

Nah. It's a definition. It presupposes nothing.

Skepdick wrote: ↑Sat Oct 19, 2024 7:43 pm Not true doesn't mean false AND not false doesn't mean true.
True means "truth has been established"
NOT true means "Truth has not been established"
False means "Falsehood has been established".
NOT false means "falsehood has not been established".

These are some circular definitions.