Logic without the three traditional laws of thought The wikipedia page on the laws of thought mentions the The three traditional laws: The Law of the Excluded Middle, Law of Non-contradiction and The Law of the Identity. These laws are embedded in a vast majority of logic systems, but not all.
In intuitionistic logics, the The Law of the Excluded Middle does not hold. In paraconsistent logics, the Law of Non-contradiction does not hold. As pointed out in this answer, in Schrodinger logics, the The Law of the Identity does not hold.
One interesting thing about these logics is that the removal of the laws has a larger ramification, it affects other inference rules. For example, in intuitionistic logic the law of double negation doesn't hold, and reduction ad absudum may only prove negative statements. Another example, as stated in stanford's entry on paraconsistent logic: "One feature of LP which requires some attention is that in LP modus ponens comes out to be invalid". The removal of these laws makes the system restrictive in more ways than just not being able to use that specific law.
So that raises the quesion, what about a system where none of them hold? Can such a system make any deductions at all?
Note that this question is mostly for exploration/amusement/curiosity. It does not require an example of an actual formal system where these laws don't apply, as a formal system without any of these would hardly be of any use, and therefore, it has probably never been developed. However, it still feels like an interesting question.
Is it a requirement for at least one of these to hold for a system to be logical, or is it possible to build some system, even if very restrictive, in which none of these laws hold, but that is still capable of producing sound arguments?
 A: Lukasiewicz 3-valued logic rejects both the law of noncontradition and the law of the excluded middle as general laws applicable to all propositions, although they do still apply in special cases. As a propositional logic, it does not make specific use of the law of identity.
This does impose certain restrictions on deduction. In particular, the presence of doubtful conditionals imposes limitations on the free use of common methods of deductive inference such as modus ponens and modus tollens. Proof by contradiction becomes much more difficult.  However,these methods do hold in modified or restricted form. With due attention to the necessary restrictions and requirements, it is possible to conduct valid and sound deductive reasoning in this system.
A: There are plenty of inference rules in classical logic which do not rely on any of those three laws. Some examples include:

*

*Modus ponens: $A, A\rightarrow B\vdash B$.


*Left and right conjunction elimination: $A\wedge B\vdash A$ and $A\wedge B\vdash B$.


*Absorption: if $A, A\vdash B$ is a valid sequent, then $A\vdash B$ is a valid sequent. (Actually this is just a special case of absorption, but oh well.)
These are simply unrelated to LEM/noncontradiction (negation doesn't appear) or to identity (they make sense in propositional logic which has no notion of equality in the first place).
So we can drop all three "laws of thought" and still have a nontrivial deductive system. Of course, this raises the question of whether such a system still counts as a "logic," and there is no accepted notion of what exactly a logic is. But I believe the current stance is a very pluralistic one, according to which even such weak systems still count as logics. (FWIW I hold such a pluralistic view.)
