Question

If P, then Q,

where P and Q are both sentences. P is the antecedent of the conditional—the ‘if’ part—and Q is the consequent—the ‘then’ part. The corresponding conditional of an argument contains the conjunction of the premises in the antecedent and the conclusion in the consequent. Thus, for argument (I) we get the following corresponding conditional:

If all women are mortal and Madonna is a woman, then Madonna is mortal.

Likewise, for argument (J), we get the following corresponding conditional.

If all fish are blue and George W. is a fish, then George W. is blue.

An argument is valid if and only if its corresponding conditional is a logical truth. Note that corresponding conditionals are sentences, not arguments. Thus, corresponding conditionals are never valid. Similarly with corresponding theories.