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3.1: Deductive Logics

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  • In this chapter and the next we will study two deductive logics—two approaches to evaluating deductive arguments. The first, which is the subject of the present chapter, was developed by Aristotle nearly 2,500 years ago, and we’ll refer to it simply as Aristotelian Logic; the second, the subject of the next chapter, has roots nearly as ancient as Aristotle’s but wasn’t fully developed until the 19th century, and is called Sentential logic.

    Again, these are two different approaches to the same problem: evaluating deductive arguments, determining whether they are valid or invalid. Recall, deductive arguments are valid just in case their premises guarantee their conclusions; and validity is determined entirely by the form of the argument. The two logics we study will have different ways of identifying the logical form of arguments, and different methods of testing those forms for validity. These are two of the things a deductive logic must do: specify precise criteria for determining logical form and develop a way of testing it for validity.

    But before a logic can do those two things, there is a preliminary job: it must tame natural language. Real arguments that we care about evaluating are expressed in natural languages like English, Greek, etc. As we saw in our discussion of the logical fallacies in the last chapter, natural languages are unruly: they are filled with ambiguity and vagueness, and exhibit an overall lack of precision that makes it very difficult to conduct the kind of rigorous analysis necessary to determine whether or not an argument is valid. So before making that determination, a logic must do some tidying up; it must remove the imprecision inherent in natural language expressions of arguments and make them suitable for rigorous analysis. There are various approaches to this task. Aristotelian Logic and Sentential logic adopt two different strategies.

    Aristotelian Logic seeks to tame natural language by restricting itself to a well-behaved, precise portion of the language. It only evaluates arguments that are expressed within that precisely delimited subset of the language. Sentential logic achieves precision by eschewing natural language entirely: it constructs its own artificial language, and only evaluates arguments expressed in its terms.

    This strategy may seem overly restrictive: if we limit ourselves to arguments expressed in a limited vocabulary—and especially if we leave behind natural language entirely—aren’t we going to miss lots of (all?) arguments that we care about? The answer is no, these approaches are not nearly as restrictive as they might seem. We can translate back and forth between the special portion of language in Aristotelian Logic and expressions in natural language that are outside its scope. Likewise, we can translate back and forth between the artificial language of Sentential logic and natural language. The process of translating from the unruly bits of natural language into these more precise alternatives is what removes the ambiguity, vagueness, etc. that stands in the way of rigorous analysis and evaluation. So, part of the task of taming natural language is showing how one’s alternative to it is nevertheless related to it—how it picks out the logically important features of natural language arguments while leaving behind their extraneous, recalcitrant bits.

    These, then, are the three tasks that a deductive logic must accomplish:

    1. Tame natural language.
    2. Precisely define logical form.
    3. Develop a way to test logical forms for validity.

    The process for evaluating real arguments expressed in natural language is to render them precise and suitable for evaluation by translating them into the preferred vocabulary developed in step 1, then to identify and evaluate their forms according to the prescriptions of steps 2 and 3.

    We now proceed to discuss Aristotelian Logic, starting with its approach to taming natural language.

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