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Chapter 19: Living in the Real World

Chapter 19: Living in the Real World. Introductory Remarks (p. 190).

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Chapter 19: Living in the Real World

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  1. Chapter 19:Living in the Real World

  2. Introductory Remarks (p. 190) • The joy and misery of ordinary English is that you can say the same thing in many ways. Arguments aren’t always presented in standard form. Sometimes elements are missing (Chapter 17). Sometimes there are nonstandard quantifiers. Sometimes there appear to be more than three terms, and you have to substitute synonyms or antonyms — the latter often requires recognizing logically equivalent forms of categorical propositions (Chapter 18). This chapter pulls together everything you have done on categorical propositions, while adding a few new elements. We will be reducing arguments to standard form (restating them in standard form).

  3. Nonstandard Quantifiers (pp. 190-192) • If you have a statement with a quantifier other than ‘All’, ‘No’, or ‘Some’, you’ll need to restate it in standard form. • Ways to say All: every, whatever, any, the only, the, and sometimes a or an. • B. Ways to say No: not any, none, not even one • C. Ways to say Some … are … : at least on, a few, more than none, there is a, there exists a, many, several, diverse, numerous, various, (usually) a or an, and sometimes the. • D. Ways to say Some … are not …: not all, not every

  4. Nonstandard Quantifiers (pp. 190-192) • Only and none but • It’s a universal affirmative, but you must convert the terms with respect to the order given. • If the quantifier is ‘only’ and the above does not yield a valid syllogism, but the statement would be true if the terms were in the original order, simply replace ‘only’ with ‘all’ and test the syllogism. • If all the elements are there but the order is wrong, ask yourself what arrangement would yield a true statement. • “No S unless P” means No S are non-P (All S are P).

  5. Nonstandard Quantifiers (pp. 190-192) • Complex quantifiers • “All but S are P” means both All non-S are P and No S are P. • “Almost all (not quite all, only some) S are P” means both Some S are P and Some S are not P. • If a statement with a complex quantifier is a premise, choose whichever statement will yield a valid syllogism. If you check one and it does not yield a valid syllogism, then you should also check the other. • If a statement with a complex quantifier is the conclusion, the syllogism will almost always be invalid (but see sidebar on p. 191). Show that the argument is invalid (choose the statement accordingly, the one that does not follow from the premises). • I. No quantifier: Ask what categorical statement containing the terms given is true.

  6. Examples with Nonstandard Quantifiers Given: Standard form: Each cat is a mammal. All cats are mammals. Every tabby is a cat. All tabbies are cats. Any tabby is a mammal. All tabbies are mammals. Not every dog is a cat. Some dogs are not cats. None but terriers are dogs. All dogs are terriers. There is a terrier that is not a cat Some terriers are not cats. Every cow is a vertebrate. All cows are vertebrates. Only bovines are Jerseys. All Jerseys are cows. Not any Jerseys are invertebrates. All Jerseys are vertebrates.

  7. Reducing the Number of Terms (pp. 192-194) • If you have two synonymous terms, replace one of them with the other. • If you have a pair of antonyms (complementary terms), obvert and convert as needed to reduce the terms to the same form. • Be cautious of prefixes: ‘flammable’ and ‘inflammable’ are synonyms. • Be aware of contexts: they can tell what counts as complementary terms.

  8. Singular Propositions (pp. 194-195) • Singular propositions concern individuals. • They can be treated as if they are universal or particular, but remember: • Treating “Anne is a cheerleader” as a universal, “(All) Anne is a cheerleader” is shorthand for “All things identical with Anne are cheerleaders.” • Treating “Anne is a cheerleader” as a particular, “(Some) Anne is a cheerleader” is shorthand for “Some things identical with Anne are cheerleaders.” • If one singular proposition is a premise and the other is a conclusion, you should treat both as universals orboth as particulars. • Once you have reduced the arguments to standard form, you test them with rules or Venn diagrams.

  9. Example 1 Given: • Only some people seeking a baccalaureate degree are vociferous students, since not every undergraduate is a junior or senior, and none but reticent young scholars are freshmen and sophomores. Reductions: • Some undergraduates (UG) are not juniors or seniors (JS). • All freshmen and sophomores (non-JS) are reticent young scholars (non-vociferous students = non-VS). • Some baccalaureate seekers (=UG) are vociferous students (VS) And • Some UG are not VS. Notice that ‘only some’ is a complex quantifier. • The second premise, as given, is the major premise. To reduce the number of terms, you merely need to contrapose the second premise.

  10. Example 1 • All VS are JS. • Some UG are not JS. • Some UG are VS. And • Some UG are not VS. Since at most one statement follows from a pair of premises (but see the sidebar on p. 191), you must choose the conclusion that does not follow and show that the syllogism is invalid: • All VSD are JSU. • Some UGU are not JSD. • Some UGU are VSU. • The syllogism violates rules 3 and 5, or, if you prefer: • Notice that what does follow is “Some UG are not VS.” Since at most one statement is entailed by the premises of a syllogism, and the conclusion as given is complex, we must show that one of the possible conclusions does not follow.

  11. Example 2 Given: • Since every current U.S. Senator from Virginia is a Republican, we may conclude that some Senators from Virginia are not running for office, since at least one candidate is a Democrat. • The conclusion is, “Some Senators from Virginia are not running for office.” People running for office are candidates. So, the conclusion can be restated as, “Some Senators from Virginia are not candidates,” which matches a term in one of the premises. It’s a political context, so the terms ‘Democrats’ and ‘Republicans’ can be taken as complementary. (This isn’t quite right — there are Libertarians and independents, for example — but it is not unreasonable given the context.) If you want to stick with Republicans, you can contrapose the conclusion and replace ‘Democrats’ with ‘non-Republicans’ (and obvert the result) to form this argument:

  12. Example 2 • All current U.S. Senators from Virginia are Republicans. • Some candidates are not Republicans. • Some candidates are not current U.S. Senators from Virginia. • The argument is valid. • You don’t like to talk about Republicans? The following will work just as well: • No current U.S. Senators from Virginia are Democrats. • Some candidates are Democrats. • Some candidates are not current U.S. Senators from Virginia.

  13. Example 2 • And, of course, you can be really perverse in your wording: • Some Democrats are not non-candidates. • All Democrats are non-current U. S. Senators from Virginia. • Some non-current U.S. Senators from Virginia are not non-candidates. • Your mission, should you decide to accept it, is to show that each of the above formulations is valid, and that considerations of conversion, obversion, and contraposition show that the several formulations of the premises and conclusion are logically equivalent to one another.

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