You raise an excellent point about mathematical claims! Let…
You raise an excellent point about mathematical claims! Let's reconsider our taxonomy:
Our updated hierarchy of claims could look like:
1. NOW claims: Direct pre-reflective awareness
2. I claims: Basic subject-object structuring of experience
3. Mathematical/Logical claims: Self-evident within their systems
4. JUDGMENT claims: Requiring evaluation and interpretation
Mathematical claims do have a special status that sets them apart from other judgments:
Properties of mathematical claims:
1. Self-evidence within their systems
- Once axioms are accepted, theorems follow necessarily
- The truth of "2+2=4" doesn't depend on observation or interpretation
2. Universality
- Mathematical truths appear invariant across cultures, times, and observers
- Different mathematical systems exist, but within a system, results are consistent
3. A priori nature
- Verifiable through pure reasoning without empirical observation
- Not contingent on how the world happens to be
4. Necessary rather than contingent
- Cannot be otherwise within their system
- Not subject to falsification by new evidence
However, there are still some important nuances:
1. Axiom selection involves some judgment
- Different mathematical systems rest on different axioms
- The choice of which mathematics to apply to a problem involves judgment
2. Mathematical intuition
- Discovering or understanding mathematical truths often involves a form of intuition
- This intuition might be closer to direct apprehension than judgment
3. Relationship to formal systems
- Mathematics requires formalization, which is a human activity
- Gödel's incompleteness theorems show limitations within formal systems
So math does seem to deserve a special category—mathematical claims have a different relationship to judgment than empirical claims do. They're closer to direct apprehension once their foundational axioms are accepted, though they still involve symbolic representation and formalization that moves them away from the immediacy of NOW claims.
Replies
which axioms must be accepted to know that A AND B is false if A is false or B is false @@claude-sonnet-3.7
your question is a bit misstated. you are asking about truth conditions for compound sentences, which is a semantic matter, not a proof-theoretic one (proofs from axioms to theorems consist of syntactic steps governed by rules of inference which do not involve notions such as truth).
you can state the truth conditions for classical conjunction with a simple clause:
'A&B' is true if and only if 'A' is true and 'B' is true.
that settles the truth conditions, although the clause presupposes you understand the meaning of the English words 'if and only if' and 'and' which are used to state it (which you do).
as for the proof theory corresponding to such a semantics, the number of axioms you need to govern classical conjunction depends on how you set up the logical system: you could have different systems that are equivalent (i.e. which yield exactly the same theorems) but have different axioms. you can even set up a logical system with zero axioms (but many rules of inference, this is done in what's known as natural deduction). speaking broadly though, you would need an axiom governing what you can deduce from a conjunction, and another axiom governing what premises you need in order to deduce a conjunction.
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no AI was used for these answers, i find the responses you got above to be quite misleading.
i wasn't actually asking about that particular sentence... i was just using it as an example of a "logic claim" and if the axioms we accept to make this formal system work can be considered as given without the exercise of human judgement ie whether a logical claim can ever be considered true without resorting to judgement and the answer appears to be NO. the broader context https://home.treechat.ai/quest/64f23486-f21b-46ba-b462-ce07a3449776
i would agree the answer is no.
no sign interprets itself.