They can be seen as fundamental methods by which we build truth tables. In this sense their truth is something beyond the scope of truth tables.

No. There are scores of deductive logics, many of which violate *any* axiom you can think of.

Logics where the Law of Identity is either not present or is restricted? Check out Non-reflexive logics and quasi-set theory (Newton da Costa).

Logics where the Law of Non-contradiction fails to be strictly true? Check out paraconsistent logic, more specifically, the theory known as Dialetheism (namely, from logician Graham Priest).

Logics where the Law of the Excluded Middle fails to be a tautology? Check out Intuitionistic Logic, the most studied type of paracomplete logic.

There is more in logical heaven than one might assume.

Not really. I mean, one can develop a Paraconsistent Logic (including a Paraconsistent meta theory) which has truth tables and which is dialetheic, meaning it does not treat the Law of Non-contradiction as always true (in fact, it is both true and false in dialetheic paraconsistent semantics).

Logics where the Law of the Excluded Middle fails to be a tautology? Check out Intuitionistic Logic, the most studied type of paracomplete logic. — MindForged

Also any multi-valued logic, e.g. that used by SQL.

Yes and no. SQL's logic is many-valued, but that's not what causes it to violate Excluded Middle. Multiple truth-values causes a violation of the Principles of Bivalence. Lots of Many-valued logcs still validate Excluded Middle, since the LEM isn't about the semantic values relating to propositions. For example, Intuitionistic Logic has only has 2 truth-values, yet it still violates LEM.

SQL's logic violates LEM and Bivalence, as well as the Law of Identity when dealing with the NULL object (NULL = NULL does not return true, it returns the 3rd truth-value "UNKNOWN").

Sorry about being a pedantic ass, lol.

Lots of Many-valued logcs still validate Excluded Middle — MindForged

Yes, my mistake. For a moment I thought of the LEM as stating that a statement is either true or false, when it's actually stating that either a statement or its negation are true.