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Concept
Corresponding conditional
Summary
In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedent is the conjunction of the argument's (or derivation's) premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is valid if and only if the negation of its corresponding conditional is a contradiction. Therefore, the construction of a corresponding conditional provides a useful technique for determining the validity of an argument.
Consider the argument A:
Either it is hot or it is cold
It is not hot
Therefore it is cold
This argument is of the form:
Either P or Q
Not P
Therefore Q
or (using standard symbols of propositional calculus):
P Q
P
Q The corresponding conditional C is: IF ((P or Q) and not P) THEN Q or (using standard symbols): ((P Q) P) Q and the argument A is valid just in case the corresponding conditional C is a logical truth. If C is a logical truth then C entails Falsity (The False). Thus, any argument is valid if and only if the denial of its corresponding conditional leads to a contradiction. If we construct a truth table for C we will find that it comes out T (true) on every row (and of course if we construct a truth table for the negation of C it will come out F (false) in every row. These results confirm the validity of the argument A Some arguments need first-order predicate logic to reveal their forms and they cannot be tested properly by truth tables forms. Consider the argument A1: Some mortals are not Greeks Some Greeks are not men Not every man is a logician Therefore Some mortals are not logicians To test this argument for validity, construct the corresponding conditional C1 (you will need first-order predicate logic), negate it, and see if you can derive a contradiction from it. If you succeed, then the argument is valid. Instead of attempting to derive the conclusion from the premises proceed as follows.
Q The corresponding conditional C is: IF ((P or Q) and not P) THEN Q or (using standard symbols): ((P Q) P) Q and the argument A is valid just in case the corresponding conditional C is a logical truth. If C is a logical truth then C entails Falsity (The False). Thus, any argument is valid if and only if the denial of its corresponding conditional leads to a contradiction. If we construct a truth table for C we will find that it comes out T (true) on every row (and of course if we construct a truth table for the negation of C it will come out F (false) in every row. These results confirm the validity of the argument A Some arguments need first-order predicate logic to reveal their forms and they cannot be tested properly by truth tables forms. Consider the argument A1: Some mortals are not Greeks Some Greeks are not men Not every man is a logician Therefore Some mortals are not logicians To test this argument for validity, construct the corresponding conditional C1 (you will need first-order predicate logic), negate it, and see if you can derive a contradiction from it. If you succeed, then the argument is valid. Instead of attempting to derive the conclusion from the premises proceed as follows.
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