It’s really hard to answer these sorts of questions universally because there’s a bunch of ways of setting up things that are strictly speaking different but which yields the same results overall. For instance, some take ¬P to be a primitive notion, whereas I am more used to defining ¬P to mean P⟹⊥. However, pretty much always the inference rules or axioms for taking ¬P to be a primitive are set up in such a way that it is equivalent to P⟹⊥.
If you define it that way, the law of noncontradiction becomes (P∧(P⟹⊥))⟹⊥ is pretty trivial, because it is just a special case of (P∧(P⟹Q))⟹Q, and if you don’t have the rule (P∧(P⟹Q))⟹Q then it seems like your logic must be extremely limited (since it’s like an internalized version of modus ponens, a fundamental rule of reasoning).
I have a bunch of experience dealing with logic that rejects the law of excluded middle, but while there are a bunch of people who also experiment with rejecting the law of noncontradiction, I haven’t seen anything useful come of it, I think because it is quite fundamental to reasoning.
Also, does “A” mean the same thing as “A = True”? Does “~A” mean the same thing as “A = False”?
Statements like A=⊥ are kind of mixing up the semantic (or “meta”) level with the syntactic (or “object”) level.
It’s really hard to answer these sorts of questions universally because there’s a bunch of ways of setting up things that are strictly speaking different but which yields the same results overall. For instance, some take ¬P to be a primitive notion, whereas I am more used to defining ¬P to mean P⟹⊥. However, pretty much always the inference rules or axioms for taking ¬P to be a primitive are set up in such a way that it is equivalent to P⟹⊥.
If you define it that way, the law of noncontradiction becomes (P∧(P⟹⊥))⟹⊥ is pretty trivial, because it is just a special case of (P∧(P⟹Q))⟹Q, and if you don’t have the rule (P∧(P⟹Q))⟹Q then it seems like your logic must be extremely limited (since it’s like an internalized version of modus ponens, a fundamental rule of reasoning).
I have a bunch of experience dealing with logic that rejects the law of excluded middle, but while there are a bunch of people who also experiment with rejecting the law of noncontradiction, I haven’t seen anything useful come of it, I think because it is quite fundamental to reasoning.
Statements like A=⊥ are kind of mixing up the semantic (or “meta”) level with the syntactic (or “object”) level.
G2g