The White Horse Dialogue

[attofishpi]

PROBLEM:

Consider the claim, "A white horse is a horse."

Is it true or is it false?
Most people would say, "It's true."

But how can that be the case given that a white horse is not the same as a horse?

1) A white horse can only be white.
2) A horse can be of any color.
3) Therefore, a white horse is not the same as a horse.

The only example that complies with a TRUTH value as to what is a horse where its colour is included within the statement is:-

2) A horse can be of any color.

EG: A purple horse is a horse is TRUE per (2).

[attofishpi]

PROBLEM:

Consider the claim, "A white horse is a horse."

Is it true or is it false?
Most people would say, "It's true."

But how can that be the case given that a white horse is not the same as a horse?

1) A white horse can only be white.
2) A horse can be of any color.
3) Therefore, a white horse is not the same as a horse.

..but hang on, white contains ALL colours.

Therefore a white horse is the same as a horse.

[Age]

If that argument was flawed from the beginning, then why introduce it, here?

Flawed arguments are useful if you want to learn how to recognize mistakes in bad reasoning.

I AGREE if someone wants to TEACH another what NOT TO DO, then PRESENT a 'flawed argument' like you DID, but I ALSO SUGGEST that it IS EXPLAINED that 'this IS A flawed argument' and 'these' are BEST NEVER REPEATED.

That way, you not only strengthen your own ability to reason ( by strengthening your defenses against making logical mistakes ) but you also help other spot their mistakes.

BUT, are others, here, TO LEARN FROM 'your mistakes', or to be necessarily TAUGHT BY, or FROM, 'you'?

If yes, then WHO are 'those others', EXACTLY?

I think you will find that others are, here, in this forum to NOT BE TAUGHT BY another.

Once more, a 'problem', to me, is a question posed for a solution. So, I do not see any actual 'problem', above, here.

The problem is to spot the flaw in the argument.

But, HOW is 'that' A 'problem', EXACTLY?

What does the word 'problem' even MEAN, TO you, EXACTLY?

Finding flaws in arguments is certainly not a 'problem', at all, to me, here.

Well, if you don't care, that's your own choice.

WHY did you TAKE 'my actual words', and MAKE UP SOME ABSURD ASSUMPTION, and THEN JUMP TO SOME ABSURD CONCLUSION, FOR, EXACTLY.

WHY you MADE UP, and CAME TO, the ASSUMPTION and CONCLUSION that 'I' do NOT CARE, you MIGHT LIKE TO EXPLAIN TO 'the readers', here.

But, I might, ALSO, ASSUME, and CONCLUDE, that you WILL NOT CARE TO, and that if you do not care to, then that IS YOUR OWN CHOICE, right?

Finding flaws not only makes your ability to reason stronger, it also helps other people correct their mistakes.

BUT, PRESENTING 'flaws' is NOT FINDING FLAWS.

[godelian]

PROBLEM:

Consider the claim, "A white horse is a horse."

Is it true or is it false?
Most people would say, "It's true."

But how can that be the case given that a white horse is not the same as a horse?

1) A white horse can only be white.
2) A horse can be of any color.
3) Therefore, a white horse is not the same as a horse.

SOLUTION:

Of course, before one can evaluate the truth value of a statement, one has to properly understand it.

The above statement can be interpreted in one of the following ways:

1) Every member of the class "white horse" belongs to the class "horse".

2) The class "white horse" is the class "horse".

The first interpretation is the correct one and under that interpretation the statement is true and the above argument is irrelevant.

The second interpretation is an incorrect one and under that interpretation the statement is false and the above argument is relevant.

Natural language is surprisingly ambiguous. When we say that "A is B", it can be any of the following:

- A = B, as in "7 = 9 - 2".
- A ∈ B, as in "John is a Texan"
- A ⊆ B, as in "Texans are Americans" or "White horses are horses"

Set-theoretical language carefully distinguishes between identity ("="), "element of" ("∈"), and "subset of" ("⊆").

In natural language, all of that is collapsed into the one verb "to be". It is up to the listener to disambiguate by himself.

Natural language is very powerful in its own right but it was never designed as a formal language. It is not meant to be used as a formal language. We have other languages for that.

[Magnus Anderson]

Natural language is surprisingly ambiguous. When we say that "A is B", it can be any of the following:

- A = B, as in "7 = 9 - 2".
- A ∈ B, as in "John is a Texan"
- A ⊆ B, as in "Texans are Americans" or "White horses are horses"

Set-theoretical language carefully distinguishes between identity ("="), "element of" ("∈"), and "subset of" ("⊆").

In natural language, all of that is collapsed into the one verb "to be". It is up to the listener to disambiguate by himself.

No statement is ambiguous unless it's difficult for someone to interpret it.

The statement "A white horse is a horse" is actually a pretty clear one to anyone who understands English language.

It means, "Everything that can be represented by the word white horse can also be represented by the word horse."

Or more simply, "Every white horse belongs to the category horse."

In English language, "X is Y" simply means "Everything that fits the concept attached to X also fits the concept attached to Y".

Can you think of a single counter-argument?

But a good one.

[godelian]

No statement is ambiguous unless it's difficult for someone to interpret it.

The statement "A white horse is a horse" is actually a pretty clear one to anyone who understands English language.

It means, "Everything that can be represented by the word white horse can also be represented by the word horse."

Or more simply, "Every white horse belongs to the category horse."

In English language, "X is Y" simply means "Everything that fits the concept attached to X also fits the concept attached to Y".

Can you think of a single counter-argument?

But a good one.

The verb "to be" is not context-free, and was never designed to be context-free:

https://en.m.wikipedia.org/wiki/Context-free_grammar

In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules can be applied to a nonterminal symbol regardless of its context.

The production rules cannot be applied to the non-terminal "A is B" regardless of its context. Depending on the context, the non-terminal will be matched by:

A=B
A∈B
A⊆B

Hence, the compiler from English to ZFC set-theoretical language will need to take into account that the verb "to be" is highly context-sensitive:

This distinguishes it from a context-sensitive grammar, which can have production rules in the form:

αAβ→αγβ

and α, β, and γ strings of terminal and/or nonterminal symbols.

In other words, depending on α and β, the non-terminal A (being "A is B") will be reduced differently:

- α"A is B"β→α"A = B"β
- α"A is B"β→α"A ∈ B"β
- α"A is B"β→α"A ⊆ B"β

No statement is ambiguous unless it's difficult for someone to interpret it.

The term "context-sensitive grammar" does not mean "difficult grammar". It merely means that the grammar has production rules that look like "αAβ→αγβ".

So, a compiler from English to ZFC will have to reduce production rules of the type "αAβ→αγβ" in order to correctly translate "is" into "=", "∈", or "⊆".

In fact, ChatGPT is perfectly capable of doing that:

ChatGPT: Express in the language of ZFC: a white horse is a horse.

In the language of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), we can formalize the statement "A white horse is a horse" using set-theoretic notation.

Step 1: Define the set of horses

Let H be the set of all horses:

H = { x | x is a horse }

Step 2: Define the set of white horses

Let W be the set of all white horses, which is a subset of H:

W = { x ∈ H | x is white}

Step 3: Express the statement

The statement "A white horse is a horse" means that every element of W is also in H, which is simply:

W ⊆ H

Alternatively, in first-order logic with set membership:

∀ x (x ∈ W → x ∈ H)

This does not mean that there is something wrong with natural language. Its context-sensitivity just means that it is potentially ambiguous. That is why natural language will never be the language of ZFC.

[Magnus Anderson]

The term "context-sensitive grammar" does not mean "difficult grammar".

I know that it does not mean difficult grammar. But you did not say that natural language is context-sensitive. You said that it's ambiguous.

Natural language is surprisingly ambiguous.

Emphasis is mine.

The verb "to be" is indeed context-sensitive but that alone does not make it ambiguous.

And if B in "A is B" is a noun, which in the case of "A white horse is a horse" is, then the sense of the verb "to be" is a pretty straightforward one. It means that everything that belongs to the class represented by A also belongs to the class represented by B.

[godelian]

But you did not say that natural language is context-sensitive. You said that it's ambiguous.

ChatGPT: What is the connection between "ambiguous" and "context-sensitive"?

"Ambiguous" and "context-sensitive" are related concepts, but they differ in their implications:

Ambiguous refers to something that has multiple possible meanings or interpretations, often without enough information to determine which is correct. Ambiguity exists when a statement, word, or situation can be understood in more than one way.

Context-sensitive means that the meaning of something depends on the surrounding information, situation, or environment. A context-sensitive expression or concept does not have a fixed meaning; instead, it is determined by the circumstances in which it appears.

The Connection:

Ambiguity often arises when something is not sufficiently context-sensitive, meaning there isn't enough context to resolve multiple possible interpretations. However, context-sensitive elements can still be ambiguous if the context itself is unclear or open to multiple readings.

For example:

The word "bank" is ambiguous because it can mean a financial institution or the side of a river.
In a sentence like "She went to the bank.", the meaning is context-sensitive—if the surrounding text talks about money, it's a financial institution; if it discusses fishing, it's the side of a river.

Thus, context-sensitive information helps resolve ambiguity, but when the context is insufficient or unclear, ambiguity persists.

Since natural language is highly context-sensitive, the context of a sentence can easily be under-specified ("ambiguous") or over-specified ("contradictory"). This problem does not occur when the language has a context-free grammar.

This makes natural language very suitable for some purposes but surprisingly unsuitable for other ones. If it is the precise meaning of a sentence that matters, it is usually better to rephrase the sentence in a formal language. But then again, precise meaning is just one property of a sentence. There are other properties that could be more important in a given situation.