Liar's paradox...an attempt to solve it.
Everyone knows the liar's paradox. It is simply the impossibility of assigning a truth value (true/false) to the statement: This statement is false. If it is true then it must be false and if it is false it must be true...round and round we go in a circle.
The end result being that the liar statement is neither true nor false. The most common inference drawn from this juncture is that the liar statement is not a statement/proposition at all.
However consider the following reasoning:
[B]This statement is false[/b] is neither true nor false. So there is a logical equivalence between the two statements below:
[B]1) This statement is false[/b]
AND
[B]2) This statement is neither true nor false[/b]
Notice however that statement 2 does have a truth value. It is TRUE - the liar statement IS neither true nor false. But statement 2 is logically equivalent to statement 1 (the liar statement).
Therefore the liar statement This statement is false is TRUE by virtue of the above logical equivalence.
Is the paradox solved?
The end result being that the liar statement is neither true nor false. The most common inference drawn from this juncture is that the liar statement is not a statement/proposition at all.
However consider the following reasoning:
[B]This statement is false[/b] is neither true nor false. So there is a logical equivalence between the two statements below:
[B]1) This statement is false[/b]
AND
[B]2) This statement is neither true nor false[/b]
Notice however that statement 2 does have a truth value. It is TRUE - the liar statement IS neither true nor false. But statement 2 is logically equivalent to statement 1 (the liar statement).
Therefore the liar statement This statement is false is TRUE by virtue of the above logical equivalence.
Is the paradox solved?
Comments (54)
A(the liar statement) =This statement is false
B=This statement is neither true nor false
As per how the paradox is known A cannot be true and cannot be false. In other words A can neither be true nor can A be false. But this is statement B. And statement B is true as shown above. Therefore A (the liar statement) must also be true
Statement 2 doesn't seem to make sense. Try rewriting it in a way that makes it easier to discuss it with logic.
When you say that the statement "this statement is false" is neither true nor false you're not saying that the statement "this statement is false" means "this statement is neither true nor false". You're conflating statement A with a (different) statement about A. A and B are not logically equivalent.
I think a way to approach the problem is to consider what it means for a statement to be false and then rephrase the statement with this in mind. For example, "this statement does not correspond to a fact" (correspondence theory) or "this statement does not cohere with some specified set of sentences" (coherence theory). We then might say "'this statement does not correspond to a fact' does [not] correspond to a fact" or "'this statement does not cohere with some specified set of sentences' does [not] cohere with some specified set of sentences". Are these contradictions?
At any rate, in my opinion, "This statement is false" is equivalent to assigning "F" to "This statement." But "This statement" isn't actually a proposition. So the solution on my view is that "This statement is false" doesn't actually say anything--it functions kind like a context-free pronoun, where we have no idea what the pronoun is referring to, and so it's not true or false.
Now, if someone said, "Everything I say is a lie", I would take him at his word knowing sometimes need v will tell a lie.
There is always going to be a problem when symbols are given real status beyond just being a symbol.
Dude, these statements contradict each other. ~p . (p.~p)
[B]A: This statement is false[/b] is the original Liar statement.
Upon logical analysis we end up in a true-false never-ending loop. The end result being that we cannot assign a truth value to it i.e.
the Liar statement is neither true nor false.
Therefore the Liar statement can be rephrased as
[B]B: This statement is neither true nor false[/b].
But B is true - the Liar statement is neither true nor false. That is to say B is TRUE. Since B is just a different version of A (they're logically equivalent) it follows that A (the original Liar statement) is also TRUE.
Again, no. You're conflating statement A with a (different) statement about A.
Consider the sentence "this sentence has four words". You can't say that because this sentence has five words that it can be rephrased as "this sentence has five words". That'd be a different sentence.
No I'm not conflating the two statements. Please read below:
A: THIS (A) statement is false is
1. Not true
2. Not false
3. Not both true AND false
The only option left is that A is "neither true nor false.
So now we have the new statement:
THIS (A) statement is neither true nor false. This new statement is TRUE for A is neither true nor false. Thus even A (liar statement) is true
You're still conflating. You have one sentence, "this sentence is false", and you have another sentence, "this sentence is neither true nor false". They're not the same sentence.
That the first one is neither true nor false (according to you) is not that it means "this sentence is neither true nor false".
To give another analogy, the sentence "this sentence has five words" is true, but it would be wrong to then say that "this sentence has five words" and "this sentence is true" are logically equivalent.
If the statement is true or false (has a truth value) then it is not true that it's neither true nor false. It might be true if the 2nd sentence would say that "that sentence is neither true nor false" referring to the 1st sentence.
Please follow the line of reasoning...
The Liar statement: A: This statement is false
Available truth value options for the Liar statement are:
1. True
2. False
3. True and false
4. Neither true nor false
Option 1 and 2 are not possible because of the well-known true-false loop.
Option 3 is a contradiction so again, not possible
The only option available is 4, neither true nor false.
Therefore, This statement is false is saying exaclty what B: This statement is neither true nor false.
But B is a TRUE statement. Therefore the liar statement, which is equivalent to B, is also true.
As others have been pointing out, the liar statement (A) is not equivalent to B. At the very least, you have to provide an argument for why the two statements are (supposedly) equivalent.
If B is TRUE, then it is not "neither true nor false".
Let me try to explain it as clearly as possible.
The liar statement: A: This sentence is false
Please note that A is only concerned about the truth value of A, nothing less and nothig more.
Let us now assess what possible options of truth value are there for A:
1. True
2. False
3. Both true and false
4. Neither true nor false
The above 4 choices are jointly exhaustive and mutually exclusive.
Option 1 and 2 are impossible for the reason that a true-false loop results. Option 3 is impossible because its a contradiction. The only option available is 4 which is ''neither true nor false''.
Note again A is only about the truth value of A, nothing less nothing more.
Therefore it is acceptable to replace ''false'' in A with ''neither true nor false'' since the former equates with the latter as I've shown above.
Therefore A can be replaced with B: This statement is neither true nor false. This however is a TRUE statment about A. Therefore, the original Liar statement, A, must also be true by virtue of it being equivalent to B.
Repeating the exact same argument is not an explanation, and it is certainly not any more convincing.
Quoting TheMadFool
This is not a statement about A at all; it is a statement about B. As soon as you change "false" to "neither true nor false," you have a different statement, and the two are not equivalent.
I think you're right. B cannot be true in a direct way. However...
It can be
1. True
2. False
3. Both true and false
4. Neither true nor false
B can be 2(false). So again, can I make a claim that I've opened a new option [2. False] for the Liar statement which was written off in the original paradox?
Again, nothing about B has any bearing whatsoever on what you can claim about A. They are two different statements.
No they're not, but what's your argument or demonstration of that?
Normally, only 1 and 2 are considered truth values. 3 is impossible under classical logic as it violates the law of non-contradiction. 4 is the absence of a truth value.
Below is a new look at the Liar's paradox.
The Liar statement A: This statement is false.
Options available for the truth value of A
1. True
2. False
3. True AND false
4. Neither true nor false
1 and 2 result in the true-false loop and 3 is a contradiction. The last option available is 4 which is neither true nor false.
Note A only concerns itself with the truth value of A. Nothing more nothig less. Therefore we can rephrase A as
B: This statement is neither true nor false.
I'm not making any illegitmate claims about A. It's only about the truth value of A. So, A is neither true nor false.
Let us now check the possible truth values of B.
It is:
1. True
2. False
3. True AND false
4. Neither true nor false
It cannot be 1 in a direct manner as it leads to a true-false loop. It can't be 2 because that leads to the original Liar statement. It cannot be 3 as its a contradiction. The last option is 4 (neither true nor false). Notice that this is what B states. So it must be that A (the Liar statement) is true since B is nothing more than a rephrased version of A.
No, it isn't. B is a different sentence.
You can't go from "'this sentence is false' is neither true nor false" to "'this sentence is false' means 'this sentence is neither true nor false'" (which is what you're doing).
You can't just rephrase A like this, just as I can't rephrase "this sentence has five words" to "this sentence is true".
Let's write this a bit less perfunctory.
Options are:
1. "The sentence "this statement is false" is true"
2. "The sentence "this statement is false" is false"
3. "The sentence "this statement is false" is true and false"
4. "The sentence "this statement is false" is neither true nor false"
Going with option 4, we see that statement B is a statement about A, as such it cannot refer to itself by using the pronoun "this". It should be "that statement is neither true nor false". If it refers to itself again, we get the same problem I pointed out earlier:
Quoting Benkei
I don't agree with that part.
"Is false" "Is true" etc, in my opinion, are equivalent to "spelling out" that we're assigning T or F to some proposition. For example, if we assign T to "The cat is on the mat," it's the same as saying "The cat is on the mat is true" (or vice versa). So "This statement is true" should be the same as assigning T to "This statement."
The subject matter should be whatever the proposition is about, whatever we're assigning true or false to That would putatively be "This statement," but "this statement" isn't a proposition. and unfortunately it functions like a context-barren, free-floating pronoun. We can't include "is true" as part of the proposition, because that's the truth-value assignment (that we're just spelling out and supposedly appending to a proposition), not part of the proposition itself.
With "This statement is neither true nor false," you're not spelling out the truth-value assignment and appending it to a (supposed) proposition.
1. Not true
2. Not false
3. Not true and false
4. Not neither true nor false
It appears to me that the above 4 options are jointly exhaustive and mutually exclusive. I'm at a loss to find out what sort of truth value B has.
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. They're different sentences.
''This statement is false'' is a claim about the truth value of itself. Analysis results in a truth value, if one may call it that, of neither true nor false.
That is to say: ''This statement is neither true nor false'' is a logically acceptable rephrased version of ''This statement is false''. Note, like A, B makes a claim only about truth value of given statement.
Please read my previous post
[B]Logical equivalence[/b] 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.
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.
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.
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.
Quoting TheMadFool
With my definition of logical equivalence, that is not the case. Neither sentence is logically equivalent to anything.
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 you can't distinguish between the two statements then you're probably dyslexic.
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?
Quoting 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, S[sub]0[/sub] be some statement we are not aware of.
Let's make another statement:
S[sub]1[/sub] = S[sub]0[/sub] is false
Right now, S1 itself is not contradicting. It's just saying S1 is false. Let's continue on,
S[sub]2[/sub] = S[sub]1[/sub] is false
S[sub]3[/sub] = S[sub]2[/sub] is false
S[sub]4[/sub] = S[sub]3[/sub] is false
.
.
.
S[sub]n[/sub] = S[sub]n-1[/sub] is false
Let's substitute all this.
S[sub]n[/sub] = ( ... ( ( S[sub]0[/sub] 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[sub]n[/sub] = S[sub]0[/sub] is true
if n = odd, then
S[sub]n[/sub] = S[sub]0[/sub] is false
Substitute S[sub]0[/sub] = This statement is false. Then,
if n = even, then
S[sub]n[/sub] = This statement is false
if n = odd, then
S[sub]n[/sub] = This statement is true
It does not matter if n ? infinity. S[sub]n[/sub] oscillates between being true and false and does not converge. That is paradoxical.
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.
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''?
Another seemingly paradoxical statement, much like "this statement is false" and "this statement is not true".
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''.
As for a solution, I don't see why it can't be false.
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.
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.
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.
(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.
Quoting 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.
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.
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''
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