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11.4.2: Arguments, Logical Consequences and Counterexamples

  • Page ID
    36231
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    Let's turn from the semantics of sentences to the semantics of arguments. An argument in Sentential Logic is said to be valid or logically correct just in case the conclusion must be true when the premises are true. When this is the case with some argument, we say its conclusion is a logical consequence of its premises. Here is a valid argument:

    A → D
    A
    ---------
    D

    Here is a recipe or mechanical method for deciding whether an argument is valid. Check the truth table for all the sentences in the argument (namely all its premises plus its conclusion). If in all situations (that is, all row, or all assignments of truth-values to the capital letters) that make the premises be true, the conclusion is also made true, then the declare the argument to be valid. Otherwise, declare it to be invalid. Let’s apply the method to this truth table for the above argument:

    Screen Shot 2019-12-25 at 5.15.27 PM.png

    Any argument written in formal Sentential Logic can be checked for validity by this method of building and inspecting truth tables. An invalid argument will have a row that shows how the premises could all be true while the conclusion is false.

    Exercise \(\PageIndex{1}\)

    Use the truth table method to test whether this argument is valid.

    ~A
    C → A
    ---------
    ~C

    Answer

    Yes, it is valid. It has the form of modus tollens. There is no way for the premises to be true while the conclusion is false without violating the truth tables.

    In a valid argument, there is no row of the truth table in which the premises get Ts while the conclusion gets an F. That is, valid arguments have no counterexamples. Notice that valid arguments can have false premises.

    Exercise \(\PageIndex{1}\)

    Does this argument have a counterexample?

    A → B
    B
    --------
    A

    Answer

    Yes. This argument has the invalid form called affirming the consequent. In a situation where A is false and B is true, we have a counterexample because then the argument has true premises and a false conclusion. For example, here is an argument that commits the fallacy of affirming the consequent: If that boy standing over there is your grandfather, then that boy is a male. That boy is a male. So, that boy standing over there is your grandfather. This has true premises and a false conclusion in any situation where there’s a boy standing over there. Since there’s a situation, even if it’s not a situation in real life, that would make the premises be true and the conclusion false, the argument is invalid.


    This page titled 11.4.2: Arguments, Logical Consequences and Counterexamples is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Bradley H. Dowden.

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