Is this an example for the Gettier problem in mathematics?

[godelian]

Are the LNC and the LEM one and the same law?

Law of Noncontradiction (LNC): ¬(P∧¬P)
Law of Excluded Middle (LEM): P∨¬P

If you apply the Morgan's Laws, you can see that:

¬(P∧¬P) ⟺ (¬P ∨ P)

Hence, they seemingly express one and the same law ... but they don't.

ChatGPT: Why does the Law Of the Excluded Middle not follow from the law of noncontradiction using the Morgan's law?
(Remark: Aka, what's wrong with the proof?)

Can LEM Follow from LNC via De Morgan's Laws?

To derive P∨¬P (LEM) from ¬(P∧¬P) (LNC), let's consider the logic:

Start with ¬(P∧¬P), the statement of LNC.
Using De Morgan's laws, ¬(P∧¬P)≡¬P∨¬(¬P)
Simplify ¬(¬P) to P, giving:
¬(P∧¬P)≡¬P∨P.
Rearrange the disjunction:
¬P∨P≡P∨¬P.

At this stage, it seems we've derived LEM from LNC. However, this reasoning assumes that the rules of classical logic, such as De Morgan's laws and the double negation elimination (¬(¬P)≡P¬(¬P)≡P), are valid.

Why This Is Circular Reasoning

The validity of De Morgan's laws and double negation elimination depends on classical logic, which assumes the Law of Excluded Middle as a foundational principle. In other words:

To use De Morgan's laws and double negation elimination, you're already working within a framework where LEM is valid.
Therefore, deriving P∨¬P (LEM) from ¬(P∧¬P) (LNC) using De Morgan's laws is circular.

The problem is that the proof for De Morgan's laws uses truth tables. The use of truth tables implies the LEM already. So, it is indeed circular.

Is this a Gettier problem?

So, the expression is indeed technically true:

¬(P∧¬P) ⟺ (¬P ∨ P)

The expression is also indeed technically provable, i.e. perfectly justified.

So, it is a "justified true belief" (JTB) that the LEM necessarily follows from the LNC, but it is actually false.

So, is this a Gettier problem?

[godelian]

ChatGPT: The expression, ¬(P∧¬P) ⟺ (¬P ∨ P), is true and provable. It suggests that the Law Of the Excluded Middle follows from the law of noncontradiction. This is, however, false. Is this an example of a Gettier problem in which a Justified True belief is still somehow false?

Analysis of the Expression

- The equivalence ¬(P∧¬P)  ⟺  (¬P∨P) is a tautology in classical logic.
- In classical logic, left and right sides are logically equivalent.

Is It a Gettier Problem?

No, because:

- The statement ¬(P∧¬P)  ⟺  (¬P∨P) is true within the framework of classical logic, and the belief that they are equivalent is justified by standard logical reasoning.
- There is no "coincidental" or "flawed" justification involved. The logical equivalence is established within the rules of classical logic, and there’s no element of chance or misleading evidence.
- The issue raised—whether the law of the excluded middle "follows" from the law of non-contradiction—is not about a gap between justification and truth. It's more about the philosophical interpretation of logical principles and the framework (classical vs. non-classical logics).

If someone incorrectly interprets this equivalence to mean something philosophically false (e.g., that the law of the excluded middle is a consequence of the law of non-contradiction in all logical systems), that’s a misinterpretation or a category error, not a Gettier-style problem.

The expression ¬(P∧¬P)  ⟹  (¬P∨P) can be read as "The LNC implies the LEM" but that is a misinterpretation:

ChatGPT: The expression ¬(P∧¬P)  ⟹  (¬P∨P) can be read as "The LNC implies the LEM" but that is a misinterpretation.

You're absolutely correct that reading ¬(P∧¬P)  ⟹  (¬P∨P) as "The Law of Non-Contradiction (LNC) implies the Law of the Excluded Middle (LEM)" is a misinterpretation. This stems from conflating logical equivalence within a framework (classical logic) with a metaphysical or foundational relationship between principles.

Why "LNC implies LEM" Is Misleading

Foundational Principles Are Not Deductive:
Foundational principles like the LNC (¬(P∧¬P)) and LEM (P∨¬P) are usually considered axioms or basic truths in classical logic.
Axioms aren't "implied" by one another in the usual sense—they serve as independent starting points for reasoning.

Implication Suggests a Directional Relationship:
A  ⟹  B suggests that B depends on A being true. However, in classical logic, the LNC and LEM are interdependent and cannot be prioritized as one "causing" the other. They coexist as part of the logical system.

Proper Interpretation

Instead of saying "The LNC implies the LEM," a more accurate interpretation is:

"The LNC and the LEM are logically equivalent in classical logic." This equivalence reflects the structure of classical logic, but it does not establish a foundational or causal relationship between the two principles.

As axioms of classical logic, the LNC and LEM are not independent. This situation is accepted but considered somewhat "undesirable":

ChatGPT: Do axioms need to be independent of each other?

No, axioms in a formal system do not necessarily need to be independent of each other, though independence is generally considered desirable.

Trade-offs in Axiom Design
...
In formal systems, balancing independence, simplicity, and utility often involves trade-offs:
...
Classical Logic: In classical logic, axioms like the Law of the Excluded Middle (LEM) and the Law of Non-Contradiction (LNC) may appear interdependent within the system but are treated as fundamental because of their philosophical significance.

Conclusion

While independence is a desirable property in an axiom system, it is not strictly necessary. Practical, philosophical, or pedagogical concerns can justify the inclusion of dependent axioms, provided they enhance the system’s clarity, usability, or expressive power without undermining its logical foundations.