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Problem of the axiom of regularity in the logic of the reductio ad absurdum?
[this is question 11 in “Principles of Mathematical Logic” by A. Tarski, 1983, page 170].
So, according to Tarski in the deductive calculus, given $P$ and $Q$, the $P\Rightarrow Q$ would be equivalent to the introduction of a new constant, $D$.
Now, $P\Rightarrow Q$ would be equivalent to
$$\exists d\forall a (a\Rightarrow D) \wedge (D\Rightarrow Q)$$
$$\exists d\forall a ((a\Rightarrow d) \wedge (d\Rightarrow Q)$$
and this will be true iff
$$\exists d\forall a
eg (a\Rightarrow D)$$
$$\exists d\forall a
eg (d\Rightarrow Q)$$
Now, we take the $D$ into consideration in $a\Rightarrow D$, so we cannot use $D$ as a value in the $\Rightarrow$-elimination rule. Hence, we cannot “extract” $Q$ by a $\Rightarrow$-elimination rule, and so $Q$ cannot be derived from $P$ by the deduction calculus.
My questions are:
First of all, I’m not sure about the argument given by the author.
The difficulty of deriving $Q$ by the $\Rightarrow$-elimination rule is a problem for us, but the author claims that it is a problem for the deductive calculus. Why?
Thanks in advance for any help.