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

[JuanFlorencio]

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.

I will take the first of your two compound sentences first:

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

But the back side says: “the front side is false”

So, replacing it in your sentence, we get

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

Which is a contradiction.

Now, I will take your second sentence:

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

But again the back side says: “the front side is false”

So, replacing it in your sentence, we get

2a) The front side is false if and only if the front side is false.

Which is a tautology.

 

[JuanFlorencio]

“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

Originally I had only sentences A and B. With them we could built the contingent argument that you are proposing:

A
B​

A&B

Now, as A is “B is false” I wrote it as ~B. Then, substituting it in the argument above, we have

~B
B​

~B&B

Which is not contingent any more, but a contradiction.

 

[Rhubarb]

Originally I had only sentences A and B. With them we could built the contingent argument that you are proposing:

A
B​

A&B

Now, as A is “B is false” I wrote it as ~B. Then, substituting it in the argument above, we have

~B
B​

~B&B

Which is not contingent any more, but a contradiction.

That’s not how substitution works in logic. Substitution is, say, going from A&B to (P&Q)&(R&S).

 

[JuanFlorencio]

That’s not how substitution works in logic. Substitution is, say, going from A&B to (P&Q)&(R&S).

Granted: I did not use the word “substitution” as it is used when you are formally demonstrating something. I used it according to the daily use. Nevertheless, the word describes what I did: I exchanged one sentence with another equivalent to it. I have demonstrated the equivalence in a previous post using something similar to a truth-table:

"***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.***"

 

[Rhubarb]

Granted: I did not use the word “substitution” as it is used when you are formally demonstrating something. I used it according to the daily use. Nevertheless, the word describes what I did: I exchanged one sentence with another equivalent to it. I have demonstrated the equivalence in a previous post using something similar to a truth-table:

"***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.***"

Right. But to be logically equivalent they have to share truth-values across ALL interpretations. Not just the one we’re working under. For instance, “P” and “(PvQ)&~Q)” That’s what makes a valid pattern of inference. A and ~B are not logically equivalent, they just happen to both have the same truth-value right now. Logical equivalency indicates when sentences express the same proposition; that have the same logical content. P expresses the exact same thing as (PvQ)&~Q. A and ~B does not express the same proposition.

 

[JuanFlorencio]

Right. But to be logically equivalent they have to share truth-values across ALL interpretations. Not just the one we’re working under. For instance, “P” and “(PvQ)&~Q)” That’s what makes a valid pattern of inference. A and ~B are not logically equivalent, they just happen to both have the same truth-value right now. Logical equivalency indicates when sentences express the same proposition; that have the same logical content. P expresses the exact same thing as (PvQ)&~Q. A and ~B does not express the same proposition.

I see.

It is true that when we are demonstrating something, many times we need to use rules of inference, which allow us to go from one statement to another. One example is the De Morgan laws, which you mentioned earlier. When we use those laws we can go from a disjunctive statement to a conjunction, or vice versa, depending on which law we use. The original and the final statements are formally equivalent.

That is one thing.

Another thing is when we are formalizing an argument. Then we have to write our statements using symbology in order to have the formal argument whose correctness needs to be determined. Let’s suppose that one of the statements says: “it is not true that the dog is not heavy”. We can re-state it is as “the dog is heavy”. And then we symbolize this sentence, which is equivalent to the other. There is nothing wrong about that, and it makes our life easier.

I started from an intermediate point, because one of the sentences refers to the other. But it says: “B is false”. Isn’t it obvious to you that this is equivalent to say “~B”, no matter what “B” means?

 

[Peter_Plato]

I will take the first of your two compound sentences first:

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

But the back side says: “the front side is false”

Yes, we know what the back side says, but we don’t know whether it is true or not, which is the point.

The front side is true IFF the back side is true, but there is no way to determine whether the back side actually IS true since it merely refers us back to the front side. It’s actual truth value is indeterminable since it merely passes or defers the determination of its truth value back to the front side. We never know because we never can tell whether the statement on the back side is actually true.

Ergo, the front side is true IFF the back side is determinably true – which it isn’t.

Ditto with the rest of your argument.

So, replacing it in your sentence, we get

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

Which is a contradiction.

No, “the front side is false” only if the back side is true, but that hasn’t been determined by what the front side says. You are assuming that both the front and back sides actually are true, but there is no way to determine that they are. This is big time passing the buck – no determination as to truth value is being made anywhere.

Now, I will take your second sentence:

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

Again, we don’t know that the front side is, indeed, false, so we can’t determine if the backside is true.

But again the back side says: “the front side is false”

So, replacing it in your sentence, we get

2a) The front side is false if and only if the front side is false.

Which is a tautology.

It isn’t a tautology because we could only replace the back side with “the front side is false” if we have some independent way of knowing that the back side is actually true. You ASSUME that it is by your jumping sides, but we don’t know and can’t tell that it is merely by jumping sides. That is why it is a recursive and indeterminable loop rather than a tautology.

 

[JuanFlorencio]

Yes, we know what the back side says, but we don’t know whether it is true or not, which is the point.

The front side is true IFF the back side is true, but there is no way to determine whether the back side actually IS true since it merely refers us back to the front side. It’s actual truth value is indeterminable since it merely passes or defers the determination of its truth value back to the front side. We never know because we never can tell whether the statement on the back side is actually true.

Ergo, the front side is true IFF the back side is determinably true – which it isn’t.

Ditto with the rest of your argument.

No, “the front side is false” only if the back side is true, but that hasn’t been determined by what the front side says. You are assuming that both the front and back sides actually are true, but there is no way to determine that they are. This is big time passing the buck – no determination as to truth value is being made anywhere.

Again, we don’t know that the front side is, indeed, false, so we can’t determine if the backside is true.

It isn’t a tautology because we could only replace the back side with “the front side is false” if we have some independent way of knowing that the back side is actually true. You ASSUME that it is by your jumping sides, but we don’t know and can’t tell that it is merely by jumping sides. That is why it is a recursive loop rather than a tautology.

We don’t care if the statements are true or false. We focus our attention on the correctness of the forms. Once rewritten, your statements have the following forms:

1. P <—> ~P

And

1. Q <—> Q

Which are a contradiction and a tautology, respectively.

 

[Peter_Plato]

If we had a card that had two sides as follows…

Side A (Front): Coloured red with the words “The back side of this card is blue.”
Side B (Back): Coloured blue with the words “The front side of this card is yellow.”

A would be true since it refers to a determinable property about Side B.
B would be false since it also refers to a determinable property about Side A that, as it happens, makes the statement on Side B false.

Now think about the original cards.

Side A: The back side of this card is true.
Side B: The front side of this card is false.

Unlike the coloured card where the truth value of each of the statements can be determined simply by looking at the other side of the card – Side B is blue and therefore Side A is true – the truth value of the statement, “The back side of this card is true,” is NOT determinable by anything on the back side of the card. In fact, the statement on the back merely refers us back to the front leaving the truth value of the statements on both sides undetermined and undeterminable.

We don’t know if the statement, “The back side of the card is true,” is, itself, true or false. Neither do we know if the statement, “The front side of this card is false,” is itself true or false.

This situation is not at all like the coloured cards where the statement, “The back side of this card is blue,” is determinable (and found to be TRUE) by virtue of the colour of the back side of the card. The statement, “The front side of this card is yellow,” is also determinably FALSE by the red front of the card.

 

[JuanFlorencio]

If we had a card that had two sides as follows…

Side A (Front): Coloured red with the words “The back side of this card is blue.”
Side B (Back): Coloured blue with the words “The front side of this card is yellow.”

A would be true since it refers to a determinable property about Side B.
B would be false since it also refers to a determinable property about Side A that, as it happens, makes the statement on Side B false.

Now think about the original cards.

Side A: The back side of this card is true.
Side B: The front side of this card is false.

Unlike the coloured card where the truth value of each of the statements can be determined simply by looking at the other side of the card – Side B is blue and therefore Side A is true – the truth value of the statement, “The back side of this card is true,” is NOT determinable by anything on the back side of the card. In fact, the statement on the back merely refers us back to the front leaving the truth value of the statements on both sides undetermined and undeterminable.

We don’t know if the statement, “The back side of the card is true,” is, itself, true or false. Neither do we know if the statement, “The front side of this card is false,” is itself true or false.

This situation is not at all like the coloured cards where the statement, “The back side of this card is blue,” is determinable (and found to be TRUE) by virtue of the colour of the back side of the card. The statement, “The front side of this card is yellow,” is also determinably FALSE by the red front of the card.

We don’t know either if the following argument is true or false:

P —> Q
~Q
~P

But we know that it is correct. Similarly, we certainly don’t know if any of the simple sentences which constitute your material equivalences are true or false, but we do know that your first material equivalence is a contradiction, and the second one is a tautology.

 

[Rhubarb]

I see.

It is true that when we are demonstrating something, many times we need to use rules of inference, which allow us to go from one statement to another. One example is the De Morgan laws, which you mentioned earlier. When we use those laws we can go from a disjunctive statement to a conjunction, or vice versa, depending on which law we use. The original and the final statements are formally equivalent.

That is one thing.

Another thing is when we are formalizing an argument. Then we have to write our statements using symbology in order to have the formal argument whose correctness needs to be determined. Let’s suppose that one of the statements says: “it is not true that the dog is not heavy”. We can re-state it is as “the dog is heavy”. And then we symbolize this sentence, which is equivalent to the other. There is nothing wrong about that, and it makes our life easier.

I started from an intermediate point, because one of the sentences refers to the other. But it says: “B is false”. Isn’t it obvious to you that this is equivalent to say “~B”, no matter what “B” means?

“It is not true that the dog is heavy” is not the same thing as “it is not THE CASE that the dog is heavy.” The first sentence speaks to the truth-value of “The dog is heavy” where the second statement is a proposition regarding a dog.

 

[JuanFlorencio]

“It is not true that the dog is heavy” is not the same thing as “it is not THE CASE that the dog is heavy.” The first sentence speaks to the truth-value of “The dog is heavy” where the second statement is a proposition regarding a dog.

Granted. So?

 

[MarcoPolo]

To your subject line, a contradiction would be to say that A and not-A are simultaneous, which is impossible. A paradox is something more like object A is at location B and C simultaneously (which is different than saying it is at A and not-A simultaneously).

 

[Rhubarb]

Granted. So?

So, you are trying to say that A is replaceable with ~B. You said “A= B is false” so you can replace A with ~B. That isn’t right, as you granted.

 

[Peter_Plato]

We don’t know either if the following argument is true or false:

P —> Q
~Q
~P

But we know that it is correct. Similarly, we certainly don’t know if any of the simple sentences which constitute your material equivalences are true or false, but we do know that your first material equivalence is a contradiction, and the second one is a tautology.

Well, no actually.

The logic of Modus Tollens assumes the prior truth of P–>Q which is why the logic is expressed as: “If it is true that P –> Q. ~Q. ~P”

Or “if a statement (P–>Q) is true, then so is its contra-positive.”

That “if” is precisely where your argument fails. We don’t know, nor can we be sure that the “if” with regard to the truth of P–>Q is satisfied.

In other words…

We don’t know if it is true that {If the statement on Side A (The back of this card is true) is true then the statement on Side B (The front of this card is false) is true} is true.

The point being that we can’t know that P is, indeed, true so your argument that Modus Tollens applies here is not to be taken for granted because “if it is true that P –> Q” is not known to be true to begin with. We don’t know that It is true that P–>Q even applies here, so we can’t satisfy the “If it is true that P–>Q…”

You said…

We don’t know either if the following argument is true or false:

P —> Q
~Q
~P

No, we “don’t know,” that is why the logical expression “If it is true that P–>Q,” must itself be established.

 

[JuanFlorencio]

Well, no actually.

The logic of Modus Tollens assumes the prior truth of P–>Q which is why the logic is expressed as: “If it is true that P –> Q. ~Q. ~P”

Or “if a statement (P–>Q) is true, then so is its contra-positive.”

That “if” is precisely where your argument fails. We don’t know, nor can we be sure that the “if” with regard to the truth of P–>Q is satisfied.

In other words…

We don’t know if it is true that {If the statement on Side A (The back of this card is true) is true then the statement on Side B (The front of this card is false) is true} is true.

The point being that we can’t know that P is, indeed, true so your argument that Modus Tollens applies here is not to be taken for granted because “if it is true that P –> Q” is not known to be true to begin with. We don’t know that It is true that P–>Q even applies here, so we can’t satisfy the “If it is true that P–>Q…”

You said…

No, we “don’t know,” that is why the logical expression “If it is true that P–>Q,” must itself be established.

I did’t say that the form modus tollens applies here. I could have used any other form as an example to call your attention to the fact that logics does not have to do with the truth or falsehood of arguments, but with their correctness or incorrectness. It is the role of other disciplines, like chemistry, biology, physics, etcetera, to determine if a given statement is true or false. Logics is just a formal discipline, but that is enough.

 

[JuanFlorencio]

So, you are trying to say that A is replaceable with ~B. You said “A= B is false” so you can replace A with ~B. That isn’t right, as you granted.

No, it was not what I granted to you. So far you have not offered any proof that “B is false” and ~B are not equivalent. While Plato tends to think that Logic necessarily deals with the truth and falsehood of propositions, you seem to believe that the word “true” should not appear in any proposition, which is unfounded.

 

[Rhubarb]

No, it was not what I granted to you. So far you have not offered any proof that “B is false” and ~B are not equivalent. While Plato tends to think that Logic necessarily deals with the truth and falsehood of propositions, you seem to believe that the word “true” should not appear in any proposition, which is unfounded.

I said "it is not the case that p"and “p is false” are not the same. You said granted.

I don’t need to prove that those statements are not equivalent. There is a very specific criteria that equivalence is judged. I’ve provided an interpretation where A and ~B are not equivalent. Logical properties, like quantifactional truth, quantifactional falsity, validity, equivalence, etc. are all judged WITHOUT content. Otherwise A and B could be logically equivalent because A=Water is wet and B=Tubas are instruments. You need to judge these things across ALL interpretations. That’s part of the rules. I don’t have the expertise to demonstrate all the axioms and corollaries that make up first-order logic to demonstrate this fact. I do, however, trust the textbooks.

 

[JuanFlorencio]

I said "it is not the case that p"and “p is false” are not the same. You said granted.

I don’t need to prove that those statements are not equivalent. There is a very specific criteria that equivalence is judged. I’ve provided an interpretation where A and ~B are not equivalent. Logical properties, like quantifactional truth, quantifactional falsity, validity, equivalence, etc. are all judged WITHOUT content. Otherwise A and B could be logically equivalent because A=Water is wet and B=Tubas are instruments. You need to judge these things across ALL interpretations. That’s part of the rules. I don’t have the expertise to demonstrate all the axioms and corollaries that make up first-order logic to demonstrate this fact. I do, however, trust the textbooks.

Excuse me, axioms cannot be demonstrated, and I am not asking you to demonstrate any.

You are using the term “interpretation” in the wrong way. I advice you to check your textbooks again. Take your time.

When you say that you cannot demonstrate … but that you trust the textbooks, you are saying that you accept things by authority, without comprehending what you accept. Not so good, specially when it concerns logic.

I am not a textbook, so I have nothing else to say.

 

[Rhubarb]

Excuse me, axioms cannot be demonstrated, and I am not asking you to demonstrate any.

You are using the term “interpretation” in the wrong way. I advice you to check your textbooks again. Take your time.

When you say that you cannot demonstrate … but that you trust the textbooks, you are saying that you accept things by authority, without comprehending what you accept. Not so good, specially when it concerns logic.

I am not a textbook, so I have nothing else to say.

You’re asking me to prove a stipulated rule of first-order logic. There might be a proof out there in some meta-logic text. I don’t know. When the rules say that conjunction returns certain truth values on a truth table, we all roll with it. This is similar.

The term ‘interpretation’ in first-order logic means the specifics of an argument or statement. An interpretation is an assignment of meaning to the symbols of a formal language, such as logic. Once you start ‘filling in the variables’ you are starting the work of giving an interpretation.I will now give one interpretation for modus ponens:

A
A->B​

B

A=There is an apple on the table
B=There is something on the table

Equivalence is a property of sentences in logic. You test quantifactional equivalence PRIOR to giving an interpretation. Or alternatively, after stepping back from your interpretation to work with the logical structure alone. I’m not saying that you cannot talk about truth and falsity in your sentences. (Although I don’t think we can. I can’t say for sure so I haven’t mentioned that) What I’m saying is that “B is false” is not the same thing as “it is not the case that B” You have already granted this.

And yes, when a logician with 20+ years under their belt, functioning as a professor of logic, gives me a definition, I’ll accept it on their authority. I’ve tried explaining several times why I agree with their definition. Two sentences under a single interpretation might happen to share a truth-value. That doesn’t make the sentences logically equivalent, because another interpretation can be found that returns different truth-values for the same sentences. I’ve demonstrated this already. This should have been taught in predicate logic.