1. For Friday, read chapter 6, section 2. As nongraded HW, do the problems on p. 184- 185. Graded Homework #7 is due on Friday at the beginning of class. For practice with symbolization, try http://www.poweroflogic.com/cgi/menu.cgi http://www.poweroflogic.com/cgi/menu.cgi (9.1, C, D, E and F – see ‘help’ link for symbol use; note about upside-down ‘A’)

2. Follow-up from Monday: There is a trick for symbolizing this sentence with a quantifier as the main connective. Some philosophers are good, and some philosophers are not good. (  x)[(Px & Gx) & (  y)(Py & ~ Gy)] But of the symbolizations most students are likely to consider, this is the best one. (  x)(Px & Gx) & (  x)(Px & ~ Gx)

3. In many cases, use of a quantifier as the main connective is unavoidable, even though the structure of the sentence suggests otherwise. If, every time you pick a value for x, you want to talk about the same individual throughout the entire instance, the initial quantifier should be the m.c. If any witness told the truth, then he or she is honest. (T_: _ told the truth; W_: _ is a witness; H_: _ is honest) (  x)[(Wx & Tx) → Hx]

4. Speakers have their names listed in the program only if they are famous. (S_:_ is a speaker; P_:_s’s name is listed in the program; F_:_ is famous) (  x)[(Sx & Px) → Fx]

5. Some experienced mechanics are well paid only if all the inexperienced ones are lazy. (E_: _ is experienced; M_: _ is a mechanic; W_: _ is well paid; L_: _ is lazy) (  x)[(Ex & Mx) & Wx] → (  x)[(Mx & ~Ex) → Lx ] The subject matter is largely the same on both sides of the m.c. arrow, but you’re not picking individuals, one at a time, and saying that the entire statement is true of the same individual. Compare 13 and 14 on p. 165.

6. Creating Instances (  x)(Px & Gx) To create an instance of a quantified statement, first drop the quantifier Px & Gx Then replace the variables that were bound by that quantifier with an individual constant Pa & Ga You must replace all of those variables with the same constant.

7. Truth-conditions for the Quantifiers A universally quantified formula is true iff all of its instances are true. A universally quantified formula is false iff at least one of its instances its false. An existentially quantified formula is true iff at least one of its instances is true. An existentially quantified formula is false iff all of its instances are false.

8. Stating an Interpretation Think of an interpretation as a specification of a hypothetical situation. To specify an interpretation, one must specify a domain (i.e., say which objects exist in this hypothetical “world”). Greek letters are used to indicate that the objects themselves are being specified, not the names for them. Then, for each of the predicates in the statements of interest specify the objects that satisfy those predicates. Think of these as the objects that have the property expressed by the predicate.

9. Format of Interpretation The fundamental way of stating an interpretation is by using notation from set theory. For example, D = { , ,  } Ext(F) = { , ,  } Ext(G) =  Ext(H) = { ,  } Ext(I) = {  } Ext(J) = { ,  } Unless otherwise specified, assume the natural assignment of individual constants to objects (‘a’ refers to alpha, ‘b’ refers to beta, and so on).

10. Represented Graphically The graph on p. 179 represents the same interpretation, where D = { , ,  } is represented by listing these three objects vertically off to the left; Ext(F) = { , ,  } is represented by put F above a vertical column and putting plus-marks in that column opposite each of the three objects; and so on, for the other four predicates (using minus- marks to show when an object doesn’t satisfy a given predicate). Evaluate individual sentences on p. 179 to see whether they’re true or false on this interpretation. Explain your answer.

11. Truth and Falsity Relative to an Interpretation Evaluate individual sentences on p. 179 to see whether they’re true or false on this interpretation. Explain your answer. The procedure: make all possible instances of the quantified portions of a formula; use the chart to assign truth-values to atomic formulae in those instances; then use the truth-conditions for the operators to work your way up to a single truth-value for the entire statement.