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Chapter 6.6

Chapter 6.6. Sorites, Negation, and Term Complements. ‘ Sorites ’ defined. Sorites (fr. Gk. soros “ heap ” ): a chain of linked enthymemes with implicit subconclusions:

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Chapter 6.6

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  1. Chapter 6.6 Sorites, Negation, and Term Complements

  2. ‘Sorites’ defined • Sorites (fr. Gk. soros“heap”): a chain of linked enthymemes with implicit subconclusions: • “A series of propositions, in which the predicate of each is the subject of the next, the conclusion being formed of the first subject and the last predicate’” (OED) • Actually, that’s the definition of a hypothetical syllogism, but you get the idea. • Goal is to determine the validity of each link, and hence the validity of the chain.

  3. Procedure for sorites Simplifying and evaluating a sorites is usually not an intuitive process, but must be done more mechanically. • If the argument is in everyday language, translate the everyday language into categorical propositions (A, E, I, O) with like, paired terms. • This may actually be the most difficult step. • Find the conclusion. • The predicate term of the conclusion will direct you to the sorites’ initial proposition, which also contains the predicate term. • The second proposition (etc.) will consist of the one that contains the other term (non-conclusion-predicate term) appearing in the first (etc.) proposition. • Repeat the process for each of the successive propositions. • Treat each successive pair of propositions as an enthymeme and deduce their conclusion (which you will write to the side). • Adopt the sub-conclusion as a new premise to be combined with the following premise into the two premises of a new syllogism. • Repeat steps 3 - 5 until you have accounted for all premises and deduced the ultimate conclusion. • Test each successive syllogism for validity using one of our accepted methods. • Venn diagrams • 5 canons • 15 valid forms (my hit song or Dayton’s stupid diagram) If, at any point, while applying your own logic correctly, the process cannot be continued, then the argument is apparently faulty, and the entire sorites should be deemed invalid. Furthermore, if one way of completing a sorites fails, all will fail.

  4. In re: evaluating individual syllogisms within a sorites— Initially, when identifying a sorites, we must stack up the premises in standard form for a sorites (i.e., the predicate of the conclusion must appear in the first premise of the standard form sorites). But when it comes actually to evaluating the sorites by inspecting the constituent syllogisms step-by-step, the principle of charity allows us to take the premises in each individual syllogism in whatever order is necessary to make the argument valid. For example a Barbara-4 (AAA-4) is valid if the premises (not the terms) can be reversed to make a standard form AAA-1. You must assure, however, that, after the switch, you are dealing with a standard form AAA-1: For example, in any chain of syllogisms in a sorites, if you have (1) Asm Amp Asp the premises can be re-ordered/reversed to become the valid (2) Amp Asm Asp where the validity of (2) is obvious (Barbara) and the validity of (1) is equally demonstrable both by Venn diagram and canons. The operation, however, doesn’t work on an argument that looks like this before the switch: (3) Apm Ams Asp and this after the switch: (4) Ams Apm Asp Although, on first blush, this looks like a valid Barbara, it isn’t in standard form, and its invalidity is demonstrable both by Venn diagram and canons (yielding the fallacy of illicit process/minor, which we learn about in 6.7). In (3), compared to (1), it isn’t the premises that are out of order, but the terms/classes themselves (producing what Carter calls the syllogistic fallacy of false converse, which we will also learn about in 6.7, viz., a fallacious syllogism wherein ‘converting’ any problematic A or O proposition—which, you will recall, isn’t allowed—will nonetheless yield an otherwise valid syllogism). In any case, in any standard form sorites, you may reorder premises (whether A, E, I, O) of any pairings to produce valid syllogisms. If you can’t do this, however, the syllogism is invalid, and so, accordingly, is the sorites.

  5. Negation & term complements • Negation can take different forms. • Analytic: not, no, none, nothing, etc. • Synthetic: un-, in-, im-, dis-, etc. • Negation can indicate different things. • How classes relate (E or O statements) • The “presence” of complements • A syntactic convolution: “none but . . .,” etc. • A contradiction: “It is false that . . . ,” etc.

  6. Dealing with negation & term complements Nixon’s Law: • If there are any simple operations that can be performed as a result of double negation, do so. • Consider whether and how the negation affects the “quality” (positivity v. negativity) of the proposition. • If there appears to be a contradiction, transform the proposition as demonstrated in the Square of Opposition. • If there still appears to be a reference to a complementary class, indicate it via bars (recalling that, in this case as well, “two ‘wrongs’ do make a ‘right’”). If two wrongs don’t make a right, try three. 5. Apply as appropriate the operations of categorical equivalence: obversion AND conversion OR contraposition.

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