Proof by Contradiction with Multiple Axioms Looking at proofs by contradiction and it seems I've run into something that does not sit well with me. I am fine with the law of the excluded middle (thus not an intuitionist) and more fundamentally the Principle of Explosion seems reasonable.
The standard form is:
$(P \wedge \neg Q \implies \bot) \implies (P \implies Q) $
However I've seen a number which claim to be reductio ad absurdem but follow the following format:
$(\neg P \implies \bot) \implies P $
Which seems to be not entirely robust when people use it in a similar way to as follows. Let:
$P = A \wedge B \wedge C$
Then through reductio ad absurdem they find that P is true. Thus any of A, B or C is true.
Think of the infinite primes proof with the original P statement, "If there are infinite primes and cats are plants". I'm concerned about this use. Thanks.
 A: The RAA principle :

$(¬P → \bot) → P$

can be equivalently rewritten as :

$\lnot \lnot P → P$

which is Double Negation.
If you agree with it, you are using classical logic.
Thus, when you apply it to a "complex" statement like : $P := A \land B \land C$, you have simply :

$\lnot \lnot (A \land B \land C) \rightarrow (A \land B \land C)$.

If we consider the original formulation : $(¬P → \bot) → P$, with $P := A \land B \land C$, the denial of $P$ amounts to :

$\lnot (A \land B \land C)$

which, again in classical logic, is :

$\lnot A \lor \lnot B \lor \lnot C$.

Proving that this assumption implies the falsum (i.e. $\bot$) amounts to saying that no one between $\lnot A,\lnot B,\lnot C$ is true, and thus that $A,B,C$ are all true.
Note : also the above use of disjunction is not intuitionistically allowed ...

In your example, you are trying to derive a contradiction (⊥) from the denial of :
Primes are infinite and Cats are plants
which is :
Primes are not infinite or Cats are not plants.
How can we do this ?
By the ∨-elim rule :
if P⊢A and Q⊢A, then P∨Q⊢A.
Assuming a "standard" theory of numbers, according to which we can prove that "Primes are infinite", we can derive ⊥ from the assumption : "Primes are not infinite" (the first disjunct).
But what about the second disjunct : "Cats are not plants" ?
Where is the contradiction ?
A: The statement $P=$"there are infinite primes and cats are plants" cannot be proven by contradiction.
Since $\neg P$ is "There are finite primes or cats are not plants" is a true statement, the statement $\neg P\implies\bot$ is not true, so you cannot conclude that $P$ is true.
