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Math Help - Can a valid argument have a false conclusion?

  1. #1
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    Can a valid argument have a false conclusion?

    Can't understand the answer to this question. Please help? Question Suppose the conclusion of an argument is a contradiction. What can you conclude about the validity of the argument? Answer If the conclusion is a contradiction, then every row of its truth table contains a \false" value for the conclusion. Thus, in this case, a bad row is a row that contains a \true" value for all of the premises. A set of premises that permits such a row in its truth table is called a \consistent" set of premises. The argument is invalid if the premises are consistent, and it is valid if the premises are inconsistent.
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    Re: Can a valid argument have a false conclusion?

    You can't really conclude anything about the argument itself. The argument starts with one or more premises and proceeds logically (or not) from there. A perfectly logical argument can lead to a contradiction if the original premise(s) are incorrect. Indeed this is a very commonly used strategy for proving things.

    Your last line sums it up nicely, though I'd use the word correct or true rather than consistent. The premise(s) can be consistent and still be incorrect.
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    Re: Can a valid argument have a false conclusion?

    Here’s the question in its entirety.

    Question:
    Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument? What if the conclusion is a contradiction? What if one of the premises is either a tautology or a contradiction?

    Answer:
    The rule for a valid argument is: every row of the truth table that contains a “true" value for all of the premises also contains a “true" value for the conclusion. Let's call a row of the truth table that contains a “true" value for all of the premises and a “false" value for the conclusion a \bad" row. An argument is valid exactly when there is no bad row in its truth table.

    If the conclusion of an argument is a tautology, then every row of its truth table contains a “true" value for the conclusion. Therefore, the truth table contains no bad row, so the “argument is valid.

    If the conclusion is a contradiction, then every row of its truth table contains a “false" value for the conclusion. Thus, in this case, a bad row is a row that contains a “true" value for all of the premises. A set of premises that permits such a row in its truth table is called a “consistent" set of premises. The argument is invalid if the premises are consistent, and it is valid if the premises are inconsistent.

    If one of the premises is a tautology, then every row of its truth table contains a “true" value for that premise. If that is the only premise, a bad row is a row that contains a “false" value for the conclusion, so the argument is valid if the conclusion is a tautology, and it is invalid if the conclusion is not a tautology. If there are other premises, then a bad row is a row that contains a “true" value for each of the other premises, and a “false" value for the conclusion. In that case, the argument is valid exactly when the argument without this premise is valid.





    Now with respect to the conclusion being a contradiction, is it not that an argument is “valid if the premises cannot all be true without the conclusion being true as well.”

    Therefore when plotted on a truth table there must be no row/case present where all the premises are all found to be true but the conclusion false in that same row for an argument to be deemed valid. However the answer seems to be claiming that since the conclusion is a contradiction a row that consists of anything other than consistent values of “true” would deem the argument valid.

    Please, did I accurately represent what was being said as to the validity of the argument when the conclusion is contradictory and if so may I ask that you give an example of such a case as I am new to this and I’m being challenged to wrap my head around some of the principles of this logic. However I do enjoy it.

    Thanks
    Last edited by Gayelle; March 23rd 2014 at 01:46 AM.
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    Re: Can a valid argument have a false conclusion?

    Sorry just read the rule on posting logic questions. I'm presently attempting Sentential Logic.
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    Re: Can a valid argument have a false conclusion?

    Quote Originally Posted by romsek View Post
    Your last line sums it up nicely, though I'd use the word correct or true rather than consistent. The premise(s) can be consistent and still be incorrect.
    "Consistent" is a technical term meaning that all formulas in the set are true under some assignment of truth values to variables. Other ways to say this is that all formulas are true in some row of a truth table, or in some interpretation. The word "correct" is not used in mathematical logic (at least, in this topic), and "true" is relevant to an interpretation: a formula may be true in some interpretation and false in another.

    Quote Originally Posted by Gayelle View Post
    Now with respect to the conclusion being a contradiction, is it not that an argument is “valid if the premises cannot all be true without the conclusion being true as well.”
    That's correct.

    Quote Originally Posted by Gayelle View Post
    Therefore when plotted on a truth table there must be no row/case present where all the premises are all found to be true but the conclusion false in that same row for an argument to be deemed valid. However the answer seems to be claiming that since the conclusion is a contradiction a row that consists of anything other than consistent values of “true” would deem the argument valid.
    Again, "consistent" is a technical term whose meaning in logic is different from its common English meaning. It can only be applied to a set of formulas. With regard to your remark, if the conclusion is a contradiction, then a single row that has anything other than all "true" for premises would not make this argument valid, but if all rows have this property, then the argument is indeed valid.

    Let's think about interpretations (i.e., rows of a truth table) as possible worlds. They give all possible meanings to propositional variables, i.e., basic facts of life. Now, premises are, in general, compound propositions, which may be true or false in any paticular world. If a set of premises is true in at least some world, they are called "consistent" (one could also call such set "possible" or "realizable"). For example, the premises "All inhabitants (of the world) are alive" and "All inhabitants are at least 6 feet tall" are presumable consistent, while "All inhabitants are alive" and "All inhabitants are dead" are inconsistent.

    An argument is valid if each world that makes all premises true also makes the conclusion true. Suppose the conclusion is a contradiction, i.e., it is false in all worlds. What would it take for the argument to be valid? If the set of the premises is consistent, the argument cannot be valid. Indeed, in the world that makes all premises true, the conslusion is still false, which contradicts validity. If, on the other hand, the set of premises is inconsistent, then in each world at least some of the premises and the conclusion are false, which is allowed by the definition of a valid argument.

    If x and y are propositional variables, p1, p2 and p3 are premises are c is a conclsion with the following truth table
    Code:
    x y p1 p2 p3 c
    --------------
    0 0 0  1  0  0
    0 1 1  1  0  0
    1 0 0  0  1  0
    1 1 0  0  0  0
    Then the argument that concludes c from p1, p2, p3 is valid because it never happens that all premises are true: the set of premises is inconsistent. The following argument
    Code:
    x y p1 p2 p3 c
    --------------
    0 0 0  1  0  0
    0 1 1  1  1  0
    1 0 0  0  1  0
    1 1 0  0  0  0
    is not valid because in the world where x = 0 and y = 1, all premises are true, so the conclusion should be true as well.

    Quote Originally Posted by Gayelle View Post
    Sorry just read the rule on posting logic questions. I'm presently attempting Sentential Logic.
    You are currently dealing with the semantics of sentential, or proposition, logic. Semantics is about truth/falsity. The forum rule is about various syntactic ways of constructing arguments that do not rely on the concept of truth, but only use the syntactic structure of premises. There are several such formalisms, hence the rule.
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    Re: Can a valid argument have a false conclusion?

    Thank you!
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