WEEK 6B: CHAPTER 10. SYMBOLIC LOGIC

READ COPI/COHEN, Sections 10.1-3, pp. 342-364

10.1 & 2. CONJUNCTION, NEGATION, AND DISJUNCTION

Introductory note. At one level, logic is logic. As noted earlier, however, there are a number of important differences between the syllogistic logic we have studied so far and symbolic logic, which will be our concern for the remander of the logic component of this course. These differences will be indicated as we proceed.

10.1 THE SYMBOLIC LANGUAGE OF MODERN LOGIC

1. [Copi/Cohen, 342] It should be noted that what Copi/Cohen refers to as "linguistic defects" are often subtleties and nuances in certain types of discourse. These effects are hardly "defects;" indeed, they are precisely the effects which certain authors may be trying to achieve. Although it is certainly true, as Copi/Cohen makes clear in this section, that symbolic logic has greatly increased the rigor and precision by which arguments can be logically analyzed, it is also true that certain dimensions of ordinary language tend to be concealed by this quest for rigor and precision. Instances of this concealment will be indicated as they arise.

2. [Copi/Cohen, 343] One primary difference between syllogistic and symbolic logic is the elements which symbolic logic takes as primitives or as basic givens--for the former, it is classes and propositions, for the latter, it is connectives--e.g., "and," "either...or," "if...then," etc.

10.2 THE SYMBOLS FOR CONJUNCTION, NEGATION, AND DISJUNCTION.

CONJUNCTION

3. [Copi/Cohen, 345] Regarding "We know that every statement is either true or false": Keep in mind that Copi/Cohen is referring to statements as they are being treated logically--of course, not every statement is in fact either true or false, e.g., questions, commands, statements about the future, etc.

4. [Copi/Cohen, 345] It is important to realize that the two possibilities for the "truth value" of a proposition--TRUE and FALSE--are equally values. In other words, a false proposition is just as much an instance of a truth value as a true proposition. Thus from the standpoint of logic, a false proposition is just as "valuable" as a true proposition. The important thing is that the value of the proposition be known or at least determinable, regardless whether that value is true or false.

5. [Copi/Cohen, 345] In the logic we are studying, all propositions are truth-functional. Thus you will not have to be concerned about the subtleties of propositions of the sort "Othello believes that Desdemona loves Cassio."

6. [Copi/Cohen, 346] Note the truth table for CONJUNCTION. Since you will be using this and all other truth tables many times for purposes of doing the exercises, it will not be necessary to memorize these tables. You will learn them by frequency of use.

7. [Copi/Cohen, 347] N.B. The short paragraph just before the start of the discussion of NEGATION is very important. From the standpoint of symbolic logic, words such as "but," "yet," and all the others listed by Copi/Cohen are taken to be logical SYNONYMS of "and." For example, the compound proposition "I was a good logic student but now I am even a better one" is equivalent to "I was a good logic student and now I am even a better one." It would be wise to highlight this paragraph or to note it to yourself somehow so that you know which words in English function as logical synonyms for "and," thereby forming conjunctions.

DISJUNCTION

8. [Copi/Cohen, 348] The fact that the Romans had two sets of words for "either...or" suggests that at least in this one respect, Latin is a more "logical" language than English. In other words, the Romans apparently were sufficiently aware of the difference between the inclusive and exclusive senses of "either...or" to coin different words to capture this difference.

9. [Copi/Cohen, 350] The short paragraph at the top of the page, discussing "unless" as a synonym for "either...or" is important. Don't be confused if you see a sentence with "unless" in it; read the sentence to determine whether the "unless" is equivalent to "either...or."

PUNCTUATION

10. [Copi/Cohen, 351-2] The discussion of the variable meaning of "both" depending on its location in the sentence is very important. Review this section with care.

11. [Copi/Cohen, 352] Thus, "Either John or Mary is an excellent logic student but not both of them" is symbolized (J v M) (J M). [Either J or M and not both J and M] The exclusive sense of "either...or" can therefore be represented logically without introducing any new symbols.

12. [Copi/Cohen, 352] Review the example at the bottom of the page and read with care Copi/Cohen's step-by-step account. Note how one should begin from the inside of a complex expression and work outside in order to determine the truth value of that proposition. Also, and this is VERY IMPORTANT: first identify the type of proposition with which you are working. The example on page 352 is a good one in that if you do not look carefully at the proposition, you might think it is, say, a conjunction because of all the dots it contains, i.e., symbols for "and." But closer inspection reveals that the tilde governs EVERYTHING within the brackets--because it is right outside the opening bracket. As a result, the tilde negates EVERYTHING in the brackets (not each part of the EVERYTHING, but that statement as a whole). Thus the logical form of this proposition is a negation.

INSTRUCTIONS: Do Exercise, pp. 353-357: set I--all; set II, 1-10

10.3. CONDITIONAL STATEMENTS AND MATERIAL IMPLICATION

13. [Copi/Cohen, 357] As Copi/Cohen observes, "conditional," "hypothetical," and "implication" are all synonymous and will be used interchangeably. All are names for propositions of the form "if...then...."

14. [Copi/Cohen, 359] If you are a connoisseur of puns, you might note Copi/Cohen's droll use of "acid test" at the bottom of the page.

15. [Copi/Cohen, 360] The truth table on this page lists all the relations regarding material implication which Copi/Cohen has been discussing up to this point. However, the truth table which is essential for understanding the direct logical structure of the conditional proposition is the one on page 361.

16. [Copi/Cohen, 361] The most important element here is the truth table for the conditional at the bottom of the page. You must become familiar with this truth table (as with the truth tables for conjunction, disjunction, and negation).

17. [Copi/Cohen, 363] The discussion on this page is very important. Note that the logical order and the written order of a proposition may not coincide. In every logic course I have ever given--and this often happens on examinations!--someone misreads a proposition such as "He will be acquitted if he has a good lawyer" by reversing antecedent and consequent. The LOGICAL antecedent is "if he has a good lawyer" even though it is written after the LOGICAL consequent. As Copi/Cohen will say on the next page, you must determine the MEANING of the proposition before you are in a position to reproduce its logical structure. Don't blindly assume that the written order of a proposition is identical to its logical order--it may be, but only the meaning of the proposition determines the proposition's logical structure.

A short way to remember the parts of the conditional is as follows: FIRST, a statement with "if and only if" is a bi-conditional or logical equivalence, not a conditional. Where "only if" occurs alone (NOT in "if and only if"), the statement immediately following it is the consequent. Where "if" occurs alone (NOT in "only if" and NOT in "if and only if," the statement immediately following it is the antecedent. FOR LOGICAL ORDER, these are put into the form: IF ANTECEDENT, THEN CONSEQUENT.

18. [Copi/Cohen, 364] You should be aware of the concepts of "necessary" and "sufficient" condition and how they relate to conditionals. For example, in various scientific contexts one frequently sees a proposition described as a "necessary" condition for something and it is important to know logically what such a claim entails. Note, however, that we will not use these concepts in either the exercises or on examinations.

INSTRUCTIONS: Do Exercises, pp. 365-67: set I, all; set II, 1-10