No, what we're saying is that A. "this sentence is false" and B. "this sentence is neither true nor false" are not logically equivalent — Michael

Please read my previous post

Logical equivalence of two given statements means that the given statements must have the same truth value in all possible worlds.

I've shown you how ''This statement is false'' and ''This statement is neither true nor false'' have the same truth value as in
1. They cannot be true
2. They cannot be false
3. They cannot be both true and false

However both can be 4.Neither true nor false. Doesn't that establish logical equivalence?

''A: This statement is false'' is logically equivalent to ''B: This statement is neither true nor false''. Therefore, since B is true A must also be true.

Logical equivalence of two given statements means that the given statements must have the same truth value in all possible worlds.

I've shown you how ''This statement is false'' and ''This statement is neither true nor false'' have the same truth value as in
1. They cannot be true
2. They cannot be false
3. They cannot be both true and false

However both can be 4.Neither true nor false. Doesn't that establish logical equivalence? — TheMadFool

Consider the sentences "how old are you?" and "what is your name?". Both are neither true nor false but they are not logically equivalent. For two sentences to be logically equivalent it must be that iff one is true then the other is true and iff one is false then the other is false. It isn't a term that is applicable to sentences that are not truth-apt.

Consider the sentences "how old are you?" and "what is your name?". Both are neither true nor false but they are not logically equivalent. For two sentences to be logically equivalent it must be that iff one is true then the other is true and iff one is false then the other is false. It isn't a term that is applicable to sentences that are not truth-apt — Michael

I beg to differ. In logic we have no way of distinguishing ''what is yor name?'' from ''how old are you?'' These two are the same so far as logic is concerned. Likewise logic can't find a difference between ''this statement is false'' and ''this statement is neither true nor false''. Therefore they are logically equivalent.

However both can be 4.Neither true nor false. Doesn't that establish logical equivalence? — TheMadFool

In symbolic logic, we say that two well-formed sentences, call them A1 and A2, are logically equivalent iff A1--> A2 and A2-->A1.

'Well-formed sentence' is a precise property that means the symbol string meets certain well-defined syntactic requirements that are part of the symbolic logic language L. Different languages will have different requirements.

Since the definition of logical equivalence relies on the definition of well-formed sentence, which relies on the language L, we see that logical equivalence is only meaningful relative to a specified language. That is, when we say that A1 and A2 are logically equivalent, we actually mean they are logically equivalent in symbolic language L (which we can call the 'reference language').

Because the majority of symbolic logic is done in first-order predicate logic (FOPL), it is reasonable to assume that if L is not explicitly specified, a suitable form of FOPL is implied as the reference language.

We can extend the notion of logical equivalence to a natural language N as follows.

Two symbol strings S1 and S2 are logically equivalent in natural language N, relative to logical language L, iff all of the following are true:
1. there exist well-formed sentences A1 and A2 in symbolic language L such that most people that understand both N and L would agree that S1 means the same as A1 and S2 means the same as A2

2. In L, A1-->A2 and A2-->A1

As per the above, we can leave out reference to L if we assume it means a version of FOPL.

Under this definition, questions, commands, expletives and meaningless sentences cannot be logically equivalent to anything because they are not equivalent to any well-formed sentence in FOPL.

Similarly, the liar sentence does not have the same meaning as any well-formed sentence in FOPL and hence cannot be logically equivalent to anything - but, having been away, I don't know whether the discussion has moved beyond that sentence.

In logic we have no way of distinguishing ''what is yor name?'' from ''how old are you?'' These two are the same so far as logic is concerned. Likewise logic can't find a difference between ''this statement is false'' and ''this statement is neither true nor false''. Therefore they are logically equivalent. — TheMadFool

With my definition of logical equivalence, that is not the case. Neither sentence is logically equivalent to anything.

Under this definition, questions, commands, expletives and meaningless sentences cannot be logically equivalent to anything because they are not equivalent to any well-formed sentence in FOPL — andrewk

If I can't distinguish the difference between A and B, then it can be inferred that A and B are the equivalent.

Logic can't differentiate ''this statement is false'' from ''this statement is neither true nor false''. Therefore, they are equivalent.

If I can't distinguish the difference between A and B, then it can be inferred that A and B are the equivalent. — TheMadFool

If you can't distinguish between the two statements then you're probably dyslexic.

If I can't distinguish the difference between A and B, then it can be inferred that A and B are the equivalent. — TheMadFool

It can't be inferred, but it can be stipulated, ie: defined to the case. You are free to adopt that definition of 'equivalent' if you wish. Is it a useful definition though? Where does it get you that you couldn't get to otherwise?

If you can't distinguish between the two statements then you're probably dyslexic. — Benkei

Logic can't differentiate ''this statement is false'' from ''this statement is neither true nor false''. Therefore, they are equivalent — TheMadFool

This is poor logic. Being unable to differentiate between two things does not mean two things are equivalent. It just means you don't know how to distinguish them. If I am given two constants A and B without further information, then I don't know what these values are so there is no logic to conclude that A = B.






I think found a further paradox in the Liar's paradox.

let, S₀ be some statement we are not aware of.

Let's make another statement:
S₁ = S₀ is false

Right now, S1 itself is not contradicting. It's just saying S1 is false. Let's continue on,
S₂ = S₁ is false
S₃ = S₂ is false
S₄ = S₃ is false
.
.
.
S_n = S_n-1 is false

Let's substitute all this.
S_n = ( ... ( ( S₀ is false ) is false ) ... is false ) is false

It is important that the parentheses are kept so that we won't get confused about the exact target of "is false" is referring to in each statement. Now here are some axioms:
"(X is false) is false" = "X is true"
"(X is false) is true" = "X is false"
"(X is true) is false" = "X is false"
"(X is true) is true" = "X is true"

Let's apply these to the sequence above and we get:
if n = even, then
S_n = S₀ is true
if n = odd, then
S_n = S₀ is false

Substitute S₀ = This statement is false. Then,
if n = even, then
S_n = This statement is false
if n = odd, then
S_n = This statement is true

It does not matter if n → infinity. S_n oscillates between being true and false and does not converge. That is paradoxical.

Logic can't differentiate ''this statement is false'' from ''this statement is neither true nor false''. Therefore, they are equivalent — TheMadFool

Are you sure? We can agree on the law of identity yes? You've already handily identified that there's a statement A and a statement B. A = A, pace the law of identity. It's corollary is that A = ~B.

As already pointed out, when you opt for the 4th solution to the Liar's paradox that is neither true nor false, the correct phrasing of that sentence would be:

4. "The sentence "this statement is false" is neither true nor false"

You are cutting corners everywhere and have been pointed out several mistakes by several different people already. Time to move on buddy - you're flat out wrong.

I take it from your unwillingness to admit the mistake you don't understand what you're doing wrong or don't understand what people are explaining to you. I suggest you should start asking questions to get clarifications instead of reasserting the same mistakes again and again.

Is it a useful definition though? Where does it get you that you couldn't get to otherwise? — andrewk

First note that
A: this statement is false
B: this statement is neither true nor false

For A the truth-value is indeterminate and we end up concluding B.

Now B can have the following truth values
1. True...this is not possible
2. False...this is not possible
3. True and false...this is not possible
4. Neither true nor false...this is not possible (refer to 1)

So now we have a very odd statement which is not any of the options available as shown above.

What then is this statement ''this statement is neither true nor false''?

What then is this statement ''this statement is neither true nor false''? — TheMadFool

Another seemingly paradoxical statement, much like "this statement is false" and "this statement is not true".

4. Neither true nor false...this is not possible (refer to 1) — TheMadFool

Why isn't this possible when you assume this was possible for the original Liar's Paradox? This appears arbitrary.

B: This statement is neither true nor false

Available options:
1. True
2. False
3. True and false
4. Neither true and false
The above four options are all that's available (as far as I know)

B can't 1 because then it would be false too

B can't be 2 because then it would be a contradiction (it says its neither true nor false and you're assigning a truth value ''false'' to it)

B can't be 3 because it is a contradiction

B can't be 4 because then B would be true (which is not possible as I've shown above)

So, what kind of statement is ''this statement is neither true nor false''? It's quite different from the Liar statement which is at least understandable as ''neither true nor false''.

You seem to have switched gears. First you were saying that "this sentence is false" and "this sentence is neither true nor false" are logically equivalent, and that because the latter is true then the former is true. Now you're saying that "this sentence is neither true nor false" can't be solved in the same way that "this sentence is false" can be solved?

As for a solution, I don't see why it can't be false.

Sorry if you find that wrong. I was replying to andrewk's as to where such a treatment of the paradox may lead us.

I read "This statement is neither true nor false" as saying that the two word phrase, "This statement," is neither true nor false, and that's true. Of course, that should really be written as "'This statement' is neither true nor false."

If you don't read it that way, you're ignoring everything I said above about how "is true" etc. works. When you respond to me and you don't take issue with those comments, I have to figure that you read them, understood them, and agree with them.

Your take on the matter is quite different and it may not be how others consider what the issue here is.

If I understood you correctly you mean to say the following 2 sentences are equivalent:

1. There's a book on the table
2. There's a book on the table is true

Then you criticize my argument by saying ''this statement'' in ''this statement is true'' is not a statement and assigning a truth value is meanigless. Have I understood correctly?

However take the following statement:

''This sentence has five words''.

Here we consider the entire statement in evaluating the truth-condition of it. We don't just take the ''this statement'' part.

Going back to my previous suggestion, I'm going to approach the problem by replacing "true" and false" with more meaningful alternatives, but I'm going to keep it pretty vague so as to avoid depending on a specific theory of truth.

Let's define the terms as follows:

true = describes something that is the case
false = describes something that isn't the case
not true = doesn't describe something that is the case
not false = doesn't describe something that isn't the case

So our options are:

1. "this sentence describes something that isn't the case" describes something that is the case.
2. "this sentence describes something that isn't the case" describes something that isn't the case.
3. "this sentence describes something that isn't the case" doesn't describe something that is the case.
4. "this sentence describes something that isn't the case" doesn't describe something that isn't the case.

Is 1 a contradiction because the unquoted part contradicts the quoted part? Then surely 2 is redundant because the unquoted part repeats the quoted part?

Is 2 a contradiction because the unquoted part denies the quoted part? Then surely 1 is redundant because the unquoted part affirms the quoted part?

I think this is part of the problem. We evaluate 1 and 2 using different reasoning.

1. There's a book on the table
2. There's a book on the table is true — TheMadFool

(2) is the same as assigning T to (1). It's not the same thing as (1) without a truth-value assignment, and after all, we could assign F to (1) instead.. So (2) is just (1) with the truth value T explicitly appended to the statement itself.

However take the following statement:

''This sentence has five words''. — TheMadFool

"Has five words" isn't a truth value. My comments are about how truth values function with respect to sentences. Apparently you're only focusing on syntactical similarities. What matters for this issue is truth value and its relationship to propositions.

First note that
A: this statement is false
B: this statement is neither true nor false

For A the truth-value is indeterminate and we end up concluding B. — TheMadFool

How do you reach that conclusion? It doesn't look reachable to me.

IIRC we can conclude C: 'Statement A is neither true nor false', but that's very different from B. C refers to A, which B does not, and C is not self-referential.

You all are right.

A: this statement is false

A has no truth value

So, we should be saying: "A is neither true nor false" instead of ''this statement is neither true nor false''

Hello,

if you are interested in solutions to the liar statement,
you could have a look at "layer logic":
It is a three-valued logic that uses a new additional dimension of layers.
More details at https://thephilosophyforum.com/discussion/1446/layer-logic-an-interesting-alternative

Yours
Trestone