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11.3: Logical Forms of Statements and Arguments

  • Page ID
    22022
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    The logical form of an argument is composed from the logical forms of its component statements or sentences. These logical forms are especially helpful for assessing the validity of deductive arguments. For instance, consider the following argument, which is in standard form:

    If all crystals are hard, then diamond crystals are hard.
    Diamond crystals are hard.

    ─────────────────────

    All crystals are hard.

    This is a deductively invalid argument, but it can be difficult to see that this is the case. The difficulty arises from the fact that the conclusion is true and all the argument's premises are true. One way to detect the invalidity is to abstract away from the content of the argument and to focus at a more general level on the logical form of the argument. The argument has this logical form:

    If Cryst, then Diam.
    Diam.

    ───────────

    Cryst.

    This form is an instance of the fallacy of affirming the consequent. The term Cryst abbreviates the clause "All crystals are hard." The term Diam abbreviates the clause "Diamond crystals are hard." It is easier to see that the form is invalid than it is to see that the original argument is invalid. The form is invalid because so many other invalid arguments have the same form. For example, suppose Cryst were instead to abbreviate "You are a Nazi" and Diam were to abbreviate "You breathe air." The resulting argument would have the same form as the one about diamonds:

    If you are a Nazi, then you breathe air.
    You do breathe air.

    ─────────────────────

    You are a Nazi.

    Nobody would accept this Nazi argument. Yet it is just like the argument about diamonds, as far as form is concerned. That is, the two are logically analogous. So, if one is bad, then both are bad. The two arguments are logically analogous because both have the following logical form:

    If P, then Q.
    Q.

    ──────

    P.

    It is really the logical forms of the diamond argument that make it be invalid not that it is about diamonds. If someone were to say of the argument about diamonds, "Hey, I can't tell whether the argument is valid or not; I'm no expert on diamonds," you could point out that the person doesn't have to know anything about diamonds, but just pay attention to the pattern of the reasoning.

    Just as valid patterns are a sign of valid arguments, so invalid arguments have invalid patterns. but every valid argument has an invalid pattern.

    That remark needs to be understood very carefully. Every valid argument with two premises has the invalid logical form of

    P, Q, therefore R.

    To be valid, an argument needs just one of its forms to be valid. To be invalid, an argument needs all of its forms to be invalid. Tricky, no? Let's repeat that:

    Here is an example of the point being made. Is the following argument valid?

    It is raining there only if there are clouds overhead there.
    It is raining there.
    So, there are clouds overhead there

    Here is a logical form of the argument:

    P
    Q
    So, R

    That is an invalid form because not all arguments of that form are valid. But the original argument was valid. That is because it also has a valid form, namely

    Rain only if Clouds.
    Rain.
    So, Clouds

    Because of our understanding of equivalence, we can say it is the same form as

    If Rain, then Clouds.
    Rain.
    so, Clouds.

    This form is called modus ponens.

    All arguments have patterns or logical forms. The first person to notice that arguments can be deductively valid or invalid because of their logical form was the ancient Greek philosopher Aristotle. He described several patterns of good reasoning in his book Organon, in about 350 B.C. As a result, he is called "the father of logic." He started the whole subject with this first and yet deep insight into the nature of argumentation.

    In our example, the terms Rain, Clouds, Diam and Cryst served as logical symbols that abbreviated sentences. We will be introducing more logical symbolism as this chapter progresses. The reason for paying attention to logical symbols is that when arguments get complicated, a look at their symbolic logical form can show the important heart of the argument. The reason for using symbolism is much like that for translating mathematical word problems into mathematical symbols: the translation makes the mathematics within the statements more visible for those who have a feeling for the symbols. The purpose of introducing symbols and logical forms is to aid in evaluating reasoning that is too complicated to handle directly in the original English.

    However, this chapter has not yet spelled out how to determine the appropriate logical form of a sentence. Determining the appropriate logical form of a sentence takes some care because the same sentence can have more than one logical form depending on how one treats it. The argument about clouds was an example. This point will come up again.


    This page titled 11.3: Logical Forms of Statements and Arguments is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Bradley H. Dowden.

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