Why don't paradoxes violate the law of non-contradiction?

[Ben_Sinner]

I’ve seen it said that a paradox, for example, the Liar’s Paradox (“This sentence is false”) doesn’t violate the law of non-contradiction even though it may appear it does.

Why don’t paradoxes violate the law of non-contradiction?

I’ve heard it is because they are self-referential. What about the paradoxes that don’t claim to be self-referential?

For example, the Card Paradox

*Front side of card says: The back side of this card is true.

Back side of the card says: The front side of this card is false.*

Each side of the card appears to be true and false at the same time.

 

[Peter_Plato]

I’ve seen it said that a paradox, for example, the Liar’s Paradox (“This sentence is false”) doesn’t violate the law of non-contradiction even though it may appear it does.

Why don’t paradoxes violate the law of non-contradiction?

I’ve heard it is because they are self-referential. What about the paradoxes that don’t claim to be self-referential?

For example, the Card Paradox

*Front side of card says: The back side of this card is true.

Back side of the card says: The front side of this card is false.*

Each side of the card appears to be true and false at the same time.

Try…
maverickphilosopher.typepad.com/maverick_philosopher/2010/05/five-grades-of-self-referential-inconsistency-towards-a-taxonomy.html

 

[Blue_Horizon]

I’ve seen it said that a paradox, for example, the Liar’s Paradox (“This sentence is false”) doesn’t violate the law of non-contradiction even though it may appear it does.

Why don’t paradoxes violate the law of non-contradiction?

I’ve heard it is because they are self-referential. What about the paradoxes that don’t claim to be self-referential?

For example, the Card Paradox

*Front side of card says: The back side of this card is true.

Back side of the card says: The front side of this card is false.*

Each side of the card appears to be true and false at the same time.

I can see the Liar’s paradox is self referential and so is not really a valid proposition in the first place.

I would say the Card Paradox is simply self-contradictory much like what happens when an irresistible force meets an immovable object.

 

[Rhubarb]

The law of non-contradiction states that contradictory statements cannot both be true or false at the same time. So, (p and ~p) cannot share the same truth-value.

This doesn’t seem to be what’s happening with the card paradox. The contradiction arises because of the position of the sentences (with indexicals) of the card, which to me is self-referential. Without the card being arranged in just that way, the sentences wouldn’t be contradictory.

 

[Beryllos]

I’ve seen it said that a paradox, for example, the Liar’s Paradox (“This sentence is false”) doesn’t violate the law of non-contradiction even though it may appear it does.

Why don’t paradoxes violate the law of non-contradiction?

I’ve heard it is because they are self-referential. What about the paradoxes that don’t claim to be self-referential?

For example, the Card Paradox

*Front side of card says: The back side of this card is true.

Back side of the card says: The front side of this card is false.*

Each side of the card appears to be true and false at the same time.

Isn’t the Card paradox also self-referential, referring to itself through a chain of references?

 

[JuanFlorencio]

The law of non-contradiction states that contradictory statements cannot both be true or false at the same time. So, (p and ~p) cannot share the same truth-value.

This doesn’t seem to be what’s happening with the card paradox. The contradiction arises because of the position of the sentences (with indexicals) of the card, which to me is self-referential. Without the card being arranged in just that way, the sentences wouldn’t be contradictory.

I will offer set of propositions equivalent to the “card paradox”:

A: B is false
B: A is true

But proposition A can be written as ~B. So, we have:

~B
B

Then

~B ^ B

Which is a contradiction.

 

[Rhubarb]

I will offer set of propositions equivalent to the “card paradox”:

A: B is false
B: A is true

But proposition A can be written as ~B. So, we have:

~B
B

Then

~B ^ B

Which is a contradiction.

~B isn’t the same as A. ~B is “it is not the case that A is true”, not “B is false.” They are not logically equivalent.

 

[Blue_Horizon]

“NOT (A is true)” is **not **the same as “B is false.”

You have my attention.
Because…?

 

[JuanFlorencio]

~B isn’t the same as A. ~B is “it is not the case that A is true”, not “B is false.” They are not logically equivalent.

Proposition A can have only two logical values: true or false. If proposition A is true, then B is false; therefore, ~B is true. If proposition A is false, then B is true; therefore, ~B is false.

As A and ~B have the same truth values, they are logically equivalent.

 

[Peter_Plato]

I’ve seen it said that a paradox, for example, the Liar’s Paradox (“This sentence is false”) doesn’t violate the law of non-contradiction even though it may appear it does.

Why don’t paradoxes violate the law of non-contradiction?

I’ve heard it is because they are self-referential. What about the paradoxes that don’t claim to be self-referential?

For example, the Card Paradox

*Front side of card says: The back side of this card is true.

Back side of the card says: The front side of this card is false.*

Each side of the card appears to be true and false at the same time.

I would think that…

1. The front side is true if and only if the back side is true.
2. The back side is true if and only if the front side is false.

The IFF (if and only if) dependency means that the truth-bearing quality of either side of the card is dependent upon the truth of the other side but, in an absolute sense, indeterminate. This would be a kind of infinite loop in computer programming terms.

To determine whether the truth-bearing statement on the front side (“The back side of this card is true”) is, itself, true, it is necessary that the truth-bearing statement on the back side is true. But it turns out that the truth of the truth-bearing statement found there (“The front side of this card is false.”) is, itself, dependent upon the truth of the front side which turns out to be dependent on the truth of the back side. Which takes us back to the front side to determine what the front side being false entails regarding what was stated there.

This is where a computer would a “hang” in an infinite, unresolvable loop. Not a contradiction per se, but moreso an endless recursion.

 

[Peter_Plato]

Proposition A can have only two logical values: true or false. If proposition A is true, then B is false; therefore, ~B is true. If proposition A is false, then B is true; therefore, ~B is false.

As A and ~B have the same truth values, they are logically equivalent.

The key here is the “if.” Since either side of the card being true is dependent on the other side, there is no way of resolving the actual or absolute truth value of either side. We can’t say A is true absolutely because we can’t say ~B is true absolutely. They are mutually and unresolvably dependent.

If A is true then B is false and if B is true then A is false, but neither “if” is ultimately determinable. Hence, endless loop.

A and ~B do not have the same truth value because the truth (or falsity) of A is dependent upon the truth (or falsity) of ~B which, in turn, depends on the truth (or falsity) of A, etc., to infinity.

Not a contradiction because we can’t determine finally whether A ( ~B) or ~A (B), B (~A) or ~B (A) are actually true.

 

[Rhubarb]

You have my attention.
Because…?

Because the sentences aren’t logically equivalent. A and ~B I mean. It’s easy to find a domain where they have differing truth-values. Logical equivalence requires having the same truth-value across ALL domains.

If A is “B is false” and B is “A is true” ~B is “~(A is true)” which isn’t the same thing as “B is false.”

At least, this is if memory serves. I still think that with the sentences set up the way they are this counts as self-referential. And it makes it super hard for me to translate into logic.

 

[Rhubarb]

Proposition A can have only two logical values: true or false. If proposition A is true, then B is false; therefore, ~B is true. If proposition A is false, then B is true; therefore, ~B is false.

As A and ~B have the same truth values, they are logically equivalent.

What I said above. (I think) A single domain that returns different truth-values is enough to show quantifictional non-equivalence.

 

[JuanFlorencio]

What I said above. (I think) A single domain that returns different truth-values is enough to show quantifictional non-equivalence.

I am sorry, your response does not respond really. You need to show the error with your single domain that returns different truth values. Please go ahead.

Peter Plato has proposed a different version of the paradox. I will analyse it after Mass.

 

[inocente]

For example, the Card Paradox

*Front side of card says: The back side of this card is true.

Back side of the card says: The front side of this card is false.*

Each side of the card appears to be true and false at the same time.

In 1870, Jill from Jerez bought some cards and happened to write the first statement on one of them.

In 1943, knowing nothing of Jill, Jack from Jodhpur bought some cards and happened to write the second statement on one of them.

In 1965, Jim from Jakarta happened to see one of the cards in an antiques store and bought it. In 2001, his son happened to buy the other card in Jacksonville.

In 2010 when Jim passed away, his son inherited the first card and filed it away with the second, after accidentally getting some honey on them, which glued them together.

But this series of accidents doesn’t make the statements on the cards refer to each other, or imho even much of a paradox, since they’re still completely independent just as they always were.

 

[Abba]

Try…
maverickphilosopher.typepad.com/maverick_philosopher/2010/05/five-grades-of-self-referential-inconsistency-towards-a-taxonomy.html

Peter Plato,

Is the Liar’s Paradox really a paradox as it only appears to be a true statement that annuls itself?

All Cretans are liars.

Well, as Saint Paul said: “This testimony is true.”

It seems just that the quantity and time has been left out and this makes it appear as a paradox. They are all liars (we all are because we have all lied at some time or another) but we are not always lying.

:blush:

 

[Peter_Plato]

Peter Plato,

Is the Liar’s Paradox really a paradox as it only appears to be a true statement that annuls itself?

All Cretans are liars.

Well, as Saint Paul said: “This testimony is true.”

It seems just that the quantity and time has been left out and this makes it appear as a paradox. They are all liars (we all are because we have all lied at some time or another) but we are not always lying.

:blush:

A paradox only appears to be self-contradictory or self-refuting and may, in fact, turn out to be true in some cases.

The liar’s paradox is only self-refuting if a particular definition of “liar” is applied and even then it is questionable.

For example, if “liar” is taken to mean “someone who never intentionally tells the truth,” that definition far exceeds the common understanding of “liar” as “someone who cannot be trusted to always tell the truth.” Big difference.

Even if – in the liar’s paradox – the more stringent definition of “liar” is used, there is room to permit errors on the part of the liar. Suppose, the liar intends to deceive but is mistaken and ends up, through that error, telling the truth, it is still true to say, “He never intentionally told the truth. His intention was to lie but he was mistaken.”

So, paradoxes are not necessarily self-refuting, they just appear to be because of the laxness of the terms used in the statements being made, in this case the word “liar.”

In order for the liar’s paradox to be self-refuting (maybe,) the rather onerous definition of “liar” would have to be operative, that is: “someone who always and intentionally makes statements that are untrue AND is never mistaken about them.” Even then, it might be possible to come up with an exception that acts to nullify the self-refuting character of the paradox.

 

[Rhubarb]

I am sorry, your response does not respond really. You need to show the error with your single domain that returns different truth values. Please go ahead.

Peter Plato has proposed a different version of the paradox. I will analyse it after Mass.

I misspoke. I meant “interpretation”, not “domain”. The domain is part of the interpretation.

So in the given interpretation, A and ~B return the same truth-values. Consider the following interpretation:

A: 1 is an odd number
B: 2 is an even number

Under this interpretation, A is true and ~B is false. If A and ~B were logically equivalent that move would be a valid pattern of inference. That’s why moves like deMorgan’s laws work - they hold in every interpretation.

 

[JuanFlorencio]

I misspoke. I meant “interpretation”, not “domain”. The domain is part of the interpretation.

So in the given interpretation, A and ~B return the same truth-values. Consider the following interpretation:

A: 1 is an odd number
B: 2 is an even number

Under this interpretation, A is true and ~B is false. If A and ~B were logically equivalent that move would be a valid pattern of inference. That’s why moves like deMorgan’s laws work - they hold in every interpretation.

I don’t think what you say is an interpretation of the formal argument

B
~B

Then

B^~B

Actually, what you are proposing is not an argument, but two independent sentences written one after the other.

Before arriving to the formal argument above (which is contradictory under any interpretation) I started with the following sentences:

A: B is false
B: A is true

And looking at proposition A exclusively (“B is false”), I wrote it as ~B. Then, I made abstraction of the content of B, established both sentences, and concluded with the conjunction B^~B. Where is the error?

 

[Rhubarb]

I don’t think what you say is an interpretation of the formal argument

B
~B

Then

B^~B

Actually, what you are proposing is not an argument, but two independent sentences written one after the other.

Before arriving to the formal argument above (which is contradictory under any interpretation) I started with the following sentences:

A: B is false
B: A is true

And looking at proposition A exclusively (“B is false”), I wrote it as ~B. Then, I made abstraction of the content of B, established both sentences, and concluded with the conjunction B^~B. Where is the error?

An interpretation is filling in the variables, when you assign propositions to your p’s and q’s. It’s also when you flesh out what the domain of discourse is. You can analyze arguments and statements with content, or without content. Without content is when you flesh out truth tables and such. For example:

p
p->q​

q

This can be applied to many interpretation: p=Pope Francis is from Argentina and q=An Argentinean is Pope gives us a different argument than, say, p=Rhubarb likes cheese and q=Rhubarb will eat a quesadilla. Both are different interpretations to the given logic.

“A: B is false” and “B: A is true” doesn’t express an argument either. There’s no conclusion. They are two propositions just like the ones I provided. You don’t need an argument anyway - you can test or logical equivalence between two sentences not part of an argument. That’s what deMorgan’s Laws are.

B&~B is contradictory under any interpretation, yes. But you don’t have B&~B. You have A&B with referential content for this particular interpretation. Even if you have A&~B, you don’t have B&~B. What you have is, in logic:

A
B​

A&B