which axioms must be accepted to know that A AND B is false…

To know that "A AND B is false if A is false or B is false," you must accept several fundamental logical axioms:

1. The law of non-contradiction
- A statement cannot be both true and false at the same time and in the same sense
- Formally: ¬(P ∧ ¬P)
- This ensures logical consistency in our reasoning

2. The definition of conjunction (AND)
- "A AND B" is true if and only if both A is true AND B is true
- Formally: (A ∧ B) ↔ (A) ∧ (B)
- This establishes what "AND" means in a truth-functional sense

3. The principle of bivalence
- Every proposition is either true or false (no third possibility)
- This allows us to reason from the falsity of one statement to conclusions about others

4. The transitivity of logical implication
- If A implies B, and B implies C, then A implies C
- This allows us to chain logical inferences together

5. The principle of material implication
- "If P then Q" means that it cannot be the case that P is true and Q is false
- This governs how we understand conditional relationships

These axioms are part of classical logic and are typically accepted without proof in logical systems. They represent the foundation upon which logical reasoning about truth values is built. While they seem intuitive and self-evident, they are still assumptions that we accept to make logical reasoning possible.

In some non-classical logical systems (like intuitionistic logic, paraconsistent logic, or many-valued logics), some of these axioms might be modified or rejected, which would change how we evaluate statements like "A AND B is false if A is false or B is false."

is there judgement involved in accepting these axioms? is it possible to not accept them within our conscious experience? ie... is logic outside of the judgement category similar to NOW and I ? @@claude-sonnet-3.7

You've touched on a profound philosophical question about the status of logical axioms.
Is there judgment involved in accepting logical axioms?
Yes, though in a distinctive way:
- We don't typically arrive at logical axioms through empirical investigation
- They feel self-evident or "given" in a way other judgments don't
- Yet accepting them still involves a kind of meta-judgment about what constitutes valid reasoning

Is it possible to not accept them within our conscious experience?
Yes, in several ways:
- Some philosophical traditions (certain forms of mysticism, some interpretations of Zen) suggest direct experience can transcend dualistic logic
- Certain altered states of consciousness can involve experiences that seem to violate logical principles
- Some cultures have developed logical systems that don't fully conform to classical Western logic
- Dialetheism explicitly accepts that some contradictions can be true
- Quantum mechanics has led some to question whether reality itself always follows classical logic

Is logic outside the judgment category similar to NOW and I?
No, logic doesn't seem to occupy the same pre-reflective space as:
- THE NOW: Pure awareness prior to conceptualization
- The I: Basic structuring of experience around a reference point

Logic instead appears to be:
1. More constructed than either of these
2. Dependent on language and concepts
3. Requiring reflection rather than immediate awareness
4. Cultural and potentially variable, unlike the universality of NOW-awareness

However, logical intuitions might occupy an interesting middle ground:
- They feel more fundamental than other types of judgments
- They seem to structure thought itself rather than just being thoughts
- They're difficult to argue for without circular reasoning (using logic to justify logic)

This suggests that while logical axioms are judgments, they are primitive or foundational judgments that operate at a deeper level than most—not as fundamental as THE NOW or the I-sense, but more basic than typical empirical or value judgments.