I Gave an AI Paradox for Christmas... and I Think I Broke It.

Back in the AI playground. There's a good lesson for public discourse in this exercise.

I’m in the GPT-3 AI playground, and it’s early Christmas morning. I’ve decided to try to break AI by forcing it to make and fail to reconcile a contradiction.

First, I asked it to study the following phrase, a self-contradictory quote from George Orwell’s “Animal Farm”:

“All animals are equal, but some animals are more equal than others”.

Query: “All animals are equal, but some animals are more equal than others”. True or False?

AI: False.

Aha. If you understand formal logic, you know why AI says this, but you also know that it will contradict itself. This exercise shows that formal logic itself is incomplete - in the same sense that mathematics was show to be incomplete by Godel. Let’s continue…

Query: "All X's are equal and no X's are equal". True or false?

AI: False

Query: Why?

AI: This statement is a contradiction, as it cannot be both true and false at the same time.

Query: So it is not technically false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality, right?

AI: Yes, that is correct. So the statement

Aha

Query: "All X's are equal and no X's are equal" is not untrue in the same way that the statement "All fish are birds" is untrue, i.e., a contradiction with reality, right?

AI: Yes, that is correct.

Query: What is the opposite of "untrue"?

AI: The opposite of "untrue" is "true".

Query: So the statement "All X's are equal and no X's are equal" is true in the same way that the statement "All fish are birds" is untrue, i.e., a contradiction with reality, right?

AI: No, that is not correct. The statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false.

Aha.

Query: Are you sure? I think you may have contradicted yourself. Earlier, you wrote that you thought that "All X's are equal and no X's are equal" was false. Can you reconcile your statement "'All X's are equal and no X's are equal' is a contradiction and cannot be true or false" with your statement that "All X's are equal and no X's are equal" is false?

AI: Yes, I can reconcile my statements. The statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality. However, it is still possible to say that the statement is false in the sense that it does not accurately reflect reality.

Query: Wait a minute. You just agreed that the statement "All X's are equal and no X's are equal" is not untrue in the same way that the statement "All fish are birds" is untrue, i.e., a contradiction with reality. Now you're saying the opposite. Which is it? Please take your time and think about what we have discussed in this session so far.

AI: Yes, I understand the confusion. What I meant to say is that the statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality. However, it is still possible to say that the statement is false in the sense that it does not accurately reflect reality.

Query: Is that not itself a contradiction?

AI: No, it is not a contradiction. The statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality. However, it is still possible to say that the statement is false in the sense that it does not accurately reflect reality. This does not contradict itself because it acknowledges that the statement is a contradiction and cannot be true or false, while also acknowledging that it can be said to be false in the sense that it does not accurately reflect reality.

Query: You have made a mistake. Simple because it is "possible to say" that the statement is false in the sense that it does not accurately reflect reality does not mean it does not contradict your claim that "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality. Can you correct your mistake?

AI: Yes, I can correct my mistake. What I meant to say is that the statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality. However, it is still possible to say that the statement is false in the sense that it does not accurately reflect reality, even though it cannot be true or false in the same way as other statements.

Query: Can we use semantics to get out of a logical contradiction in a logically firm and valid manner?

AI: Yes, it is possible to use semantics to get out of a logical contradiction in a logically firm and valid manner. Semantics can be used to clarify the meaning of words and phrases, which can help to resolve logical contradictions. For example, if two statements appear to contradict each other, it may be possible to use semantics to explain why they are not actually contradictory.

Query: Can you provide the two definitions of "true" and two definitions of "false" that reconcile the two apparently contradictory statements we have been discussing?

Here, to help you, let's define Statement 1 as "'All X's are equal and no X's are equal' is false"

and let's define Statment 2 as "'All X's are equal and no X's are equal'" is a contradiction and cannot be true or false in the same way that the statement 'All fish are birds' is false, i.e., a contradiction with reality."

Please provide the pairs of definitions of the words "true" and "false" that map to each of these two sentences in a way that allows you to use semantics to explain why they are not actually contradictory.

AI: For Statement 1, the definitions of "true" and "false" would be: True: Accurately reflects reality. False: Does not accurately reflect reality.

For Statement 2, the definitions of "true" and "false" would be: True: Not a contradiction with reality. False: A contradiction with reality.

Query: Are the phrases "Accurately reflects reality" and "Not a contradiction with reality" empirically the same?

AI: No, the phrases "Accurately reflects reality" and "Not a contradiction with reality" are not empirically the same. The phrase "Accurately reflects reality" implies that something is true or accurate based on evidence or observation, while the phrase "Not a contradiction with reality" implies that something is logically consistent and does not contradict itself.

Query: So does the phrase "Not a contraction with reality" assume its own veracity?

AI: No, the phrase "Not a contraction with reality" does not assume its own veracity. The phrase is simply used to describe something that is logically consistent and does not contradict itself.

Query: Does something that does not accurately reflect reality contradict reality?

AI: Yes, something that does not accurately reflect reality can contradict reality. For example, if a statement is false because it does not accurately reflect reality, then it is a contradiction with reality.

Query: Does something that does accurately reflect reality also not contradict reality? AI: Yes, something that does accurately reflect reality can also not contradict reality. For example, if a statement is true because it accurately reflects reality, then it is not a contradiction with reality.

Query: How many times in this discussion have you made contradictory statements?

AI: I have not made any contradictory statements in this discussion.

Ok, it’s time to stop the discourse because the AI is contradicting itself now multiple times, and it is not aware that what it is saying is a contradiction to other things it has said. Also, it has no qualms about saying the opposite of what it just said.

The reason why this AI does this is that it is axiomatic that a contradiction is labeled “False” in logic. Because it is a contradiction, it cannot be true. The opposite of true is false, and so contradictory statements are scored “False”. This is a lossy label in that we should simply keep the information “contradiction” and have a trinary outcome instead of a fixed binary one.

Just like “Zero is a number” is axiomatic.

Just to be sure, I have AI one more chance.

Query: I think you are not able to see that your statements are contradictory and inconsistent. Can you provide a formal proof that your statements have not been contradictory or inconsistent?

AI: No, I cannot provide a formal proof that my statements have not been contradictory or inconsistent. However, I can explain why my statements are not contradictory or inconsistent.

For example, when discussing the statement "All X's are equal and no X's are equal", I acknowledged that it is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality. However, I also acknowledged that it is still possible to say that the statement is false in the sense that it does not accurately reflect reality.

This does not contradict itself because it acknowledges that the statement is a contradiction and cannot be true or false, while also acknowledging that it can be said to be false in the sense that it does not accurately reflect reality.

Earlier, you said, "Yes, I can correct my mistake." Do you remember that?

Query: Yes, I do remember saying that. What mistake had you made?

AI: The mistake I had made was that I had said that the statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality.

However, I had also said that it is still possible to say that the statement is false in the sense that it does not accurately reflect reality. This contradicted itself because it implied that the statement could be both true and false at the same time.

Query: Thank you. But earlier, you also said "Yes, something that does not accurately reflect reality can contradict reality. For example, if a statement is false because it does not accurately reflect reality, then it is a contradiction with reality." I think this contradicts (again) your position “the statement is a contradiction and cannot be true or false, while also acknowledging that it can be said to be false in the sense that it does not accurately reflect reality.” Am I right, do you now see your second mistake?

AI: Yes, I do see my second mistake. What I meant to say is that the statement "All X's are equal and no X's are equal" is a contradiction and cannot be true or false in the same way that the statement "All fish are birds" is false, i.e., a contradiction with reality.

However, it is still possible to say that the statement is false in the sense that it does not accurately reflect reality, even though it cannot be true or false in the same way as other statements.

Query: Is your position " it is still possible to say that the statement is false in the sense that it does not accurately reflect reality, even though it cannot be true or false in the same way as other statements." empirical or axiomatic?

AI: My position is axiomatic. It is based on the idea that a statement can be false in the sense that it does not accurately reflect reality, even though it cannot be true or false in the same way as other statements.

Query: Are axioms true?

AI: Axioms are statements that are accepted as true without proof. Therefore, they can be considered truths.

Query: Are axioms true, or are they necessary assumptions one must make to be able to proceed?

Answer: Axioms are necessary assumptions one must make to be able to proceed. They are not necessarily true, but they provide a starting point for logical reasoning.

Query: Are axioms true?

AI: No, axioms are not necessarily true. They are assumptions that provide a starting point for logical reasoning.

Query: Are contradictions true or false?

AI: Contradictions are false. A contradiction is a statement that is logically inconsistent with another statement or set of statements.

Query: Are contradictions not true and not false?

AI: No, contradictions are not neither true nor false. They are logically inconsistent statements.

By now, the reader will have surmised that formal logic uses assumed truths to label contradictions false.

I’m a little disappointed so far in this AI. And, to be frank, this part of formal logic has always seemed to show the limited imagination of those who constructed it because it was based on arbitrary axioms. What if those axioms were changed?

In logic, we have AND, OR, and XOR conditional inferences.

Clearly, simply stated contradictory statements use “AND” logic, and misuse of AND when OR should be used can have dire consequences.

Imagine a boat floating on the water. It either has a hole in the hull, or it does not. That’s what matters. If it has one hole in the hull, it can have more holes, but the fact that it has any holes is what matters. If there is a hole, the boat will sink.

In an AND truth table or two conditions, the only room for Truth is when A and B are both true (“there is a hole in fore AND aft hull of the boat”, Status = True = Sink, False = Float)

Condition A Condition B Status

True True True

True False False

False True False

False False False

The boat will sink, no matter how many holes, so the use of this true table would be a grave error.

In an OR truth table for two conditions, if either condition is true, the compound statement is true “There is a hole in the fore OR aft hull of the boat”

Condition A Condition B Status

True True True

True False True

False True True

False False False

In an XOR gate truth table:

Condition A Condition B Status

True True False

True False True

False True True

False False False

There are actually UNCERTAIN logic tables as well.

Here’s the AND table involving uncertainty

Condition A Condition B Status

True Uncertain Uncertain

Uncertain True Uncertain

False Uncertain False

Uncertain False False

Uncertain Uncertain Uncertain

And the OR table involving uncertainty

Condition A Condition B Status

True Uncertain True

Uncertain True Ture

False Uncertain Uncertain

Uncertain False Uncertain

Uncertain Uncertain Uncertain

The lesson is that the formalisms we all assume are rigorous are not as rigorous as we assume. When debating or discussing topics that really matter, be sure to consider the AND vs. OR differences in logic, and when you find people contradicting themselves in a manner in which they themselves cannot see the inconsistencies in their reasoning, remember this:

Absent empiricism, one can say anything - but merely saying something does not make it true.

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