Is it possible for an argument to be valid when the conclusion is false statement? How?

It is possible for an argument (proposition) containing false assumptions to be valid.





In Sentential and in First Order Logic, a proposition may be written in the form a 鈥?gt; b, which means the antecedent a implies the consequent b and is true (valid) whenever a is not true or when b is true, written as 卢a or b. (If this is unfamiliar, consult any elementary text in Mathematical Logic). There are thus 4 possible cases as follows:





聽a聽聽聽聽聽b聽聽聽聽聽a 鈥?gt; b


聽T聽聽聽聽聽T聽聽聽聽聽聽聽聽T


聽F聽聽聽聽聽T聽聽聽聽聽聽聽聽T


聽T聽聽聽聽聽F聽聽聽聽聽聽聽聽F


聽F聽聽聽聽聽F聽聽聽聽聽聽聽聽T





The above is called a Truth Table. In rows 1 and 2, b is true so 卢a or b is true. Therefore a 鈥?gt; b is true





In row 4, b is false but so is a. So 卢a is true, and 卢a or b is true. Hence, a 鈥?gt; b is true.





In row 3, however, b is not true, and a is true, which means that both 卢a and b are false. So 卢a or b is false, and thus, a 鈥?gt; b is false.





Now return to row 4: Both the antecedent (premise or assumption) and the consequent (conclusion) are false, and yet the proposition itself is true. As an example, consider the proposition: ';If wishes were horses then beggars would ride.'; Assuming both the antecedent and the consequent are false, we nonetheless find that the proposition itself is valid.





To understand this concept in terms of Set Theory, first note that we define a proposition as valid or true whenever the consequent follows for every member of the class defined by the antecedent. When the antecedent defines a class that is empty (false antecedent) the consequent vacuously follows for every member of the class so defined. Otherwise, there would necessarily exist a member of the class for which the consequent does not follow; and since the class is empty, no such member exists. Propositions with false antecedents are thus described as being ';vacuously true.';Is it possible for an argument to be valid when the conclusion is false statement? How?
Only if one or more of the assumptions are also false.





Example: I can prove ';if 0=1 then 0=2';. Just multiply both sides of 0=1 by 2 to get 0=2. But the conclusion ';0=2'; need not be true, since the assumption ';0=1'; is not true.