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Math Help - Intro to Logic - Question

  1. #1
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    Intro to Logic - Question

    "D. Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.

    4. An invalid argument, the conclusion of which is a tautology."

    My answer:

    -Possible.
    -Example:
    Rocks are dense.
    Rocks are heavy.
    Therefore, rocks are either rough, or they're not.

    Correct answer: Not possible.

    Where did I go wrong?

    There was another question I wasn't too sure of, however the textbook didn't give an answer for this one. The question is:

    "B. For each of the following: Is it a tautology, a contradiction, or a contingent sentence?

    6. If anyone has ever crossed the Rubicon, it was Caesar."

    My answer: Contradiction.

    Is this correct?
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  2. #2
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    Your example isn't a counterexample, because an invalid argument is one in which, in some model, the premisses are true, and yet the conclusion is false. Your conclusion is true. Indeed, if the conclusion is a tautology, then by definition it's always true, and there's no possibility of the argument being invalid.

    For the second problem, I would say the answer is a contingent sentence. It's certainly not a tautology or a contradiction. It's a contingent sentence, because its truth depends on whether anyone has ever crossed the Rubicon.
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  3. #3
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    "Your example isn't a counterexample, because an invalid argument is one in which, in some model, the premisses are true, and yet the conclusion is false. Your conclusion is true. Indeed, if the conclusion is a tautology, then by definition it's always true, and there's no possibility of the argument being invalid.

    For the second problem, I would say the answer is a contingent sentence. It's certainly not a tautology or a contradiction. It's a contingent sentence, because its truth depends on whether anyone has ever crossed the Rubicon."

    That's exactly what I needed. Thanks very much.
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  4. #4
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    You're welcome. Have a good one!
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  5. #5
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    Indeed, if the conclusion is a tautology, then by definition it's always true, and there's no possibility of the argument being invalid.
    I don't think so. This depends on the definition of an argument and when an argument is invalid. In the terminology of mathematical logic, I would say an argument is a (possibly less formal) derivation. In turn, a (valid) derivation is a structure (a tree, an ordered sequence, or such) built of formulas according to the rules of inference. The three most popular derivation styles in mathematical logic are Natural Deduction (trees of formulas), Sequent Calculus (trees of sequents, i.e., pairs of multisets of formulas) and Hilbert-style derivations (sequences of formulas). A structure like this is an invalid argument if it is not a valid argument, typically when some part is not built according to a correct inference rule.

    Thus, there may be a derivation that has an error in the middle but ends with a tautology. Formally, a sequence consisting of one formula A -> A (for some formula A) is an invalid derivation in the Hilbert system because A -> A is not an axiom. It is, of course, derivable, but the derivation consists of five or six formulas (which is why Hilbert system is rarely used in practice).

    I agree that the statement about the Rubicon is contingent.
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  6. #6
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    Reply to emakarov:

    I doubt that in an Introduction to Logic class (which the OP is in, at least judging by the title of this thread), they would define a valid argument the way you have, though I would agree with you if the class were a more advanced symbolic logic class. Generally, in an intro class, the definition of a valid argument is one in which it is impossible for the premisses to be all true, and yet have the conclusion false.
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