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.
In logic, a statement or sentence can be expressed as a conditional statement, often written in the form "If P, then Q." The antecedent, represented by P, is the condition or premise, while the consequent, represented by Q, is the conclusion or result.
To determine the corresponding conditional of an argument, you need to identify the premises and the conclusion of the argument. The premises are usually expressed as independent statements, and the conclusion is the statement that follows from the premises.
For example, let's take argument (I):
Premise 1: All women are mortal.
Premise 2: Madonna is a woman.
Conclusion: Madonna is mortal.
To obtain the corresponding conditional, you combine the premises in the antecedent and the conclusion in the consequent:
Antecedent: All women are mortal and Madonna is a woman.
Consequent: Madonna is mortal.
So the corresponding conditional for argument (I) is:
"If all women are mortal and Madonna is a woman, then Madonna is mortal."
Similarly, applying the same process to argument (J):
Premise 1: All fish are blue.
Premise 2: George W. is a fish.
Conclusion: George W. is blue.
The corresponding conditional would be:
"If all fish are blue and George W. is a fish, then George W. is blue."
It's important to note that the validity of an argument is determined by whether its corresponding conditional is a logical truth. A logical truth is a statement that is always true, regardless of the specific values of the variables involved. Thus, corresponding conditionals themselves are not considered valid arguments, but rather valid statements.
The concept of corresponding theories is not explicitly mentioned in your question, but it likely refers to a collection of corresponding conditionals or conditional statements that collectively represent a theory. In this case, the term "valid" may be used to describe whether the theory as a whole is logically consistent or sound.