For Friday, read chapter 6, section 2. As For Friday, read chapter 6, section 2. As nongraded HW, do the problems on p. 184-nongraded HW, do the problems on p. 184-185.185.

Graded Homework #7 is due on Friday at the Graded Homework #7 is due on Friday at the beginning of class.beginning of class.

For practice with symbolization, try For practice with symbolization, try http://www.poweroflogic.com/cgi/menu.cgi

(9.1, C, D, E and F – see ‘help’ link for (9.1, C, D, E and F – see ‘help’ link for symbol use; note about upside-down ‘A’)symbol use; note about upside-down ‘A’)

Follow-up from Monday: There is a trick for Follow-up from Monday: There is a trick for symbolizing this sentence with a quantifier as symbolizing this sentence with a quantifier as the main connective.the main connective.

Some philosophers are good, and some Some philosophers are good, and some philosophers are not good.philosophers are not good.

((x)[(Px & Gx) & (x)[(Px & Gx) & (y)(Py & ~ Gy)]y)(Py & ~ Gy)]

But of the symbolizations most students are But of the symbolizations most students are likely to consider, this is the best one.likely to consider, this is the best one.((x)(Px & Gx) & (x)(Px & Gx) & (x)(Px & ~ Gx)x)(Px & ~ Gx)

In many cases, use of a quantifier as the main In many cases, use of a quantifier as the main connective is unavoidable, even though the connective is unavoidable, even though the structure of the sentence suggests otherwise. If, structure of the sentence suggests otherwise. If, every time you pick a value for every time you pick a value for xx, you want to , you want to talk about the same individual throughout the talk about the same individual throughout the entire instance, the initial quantifier should be entire instance, the initial quantifier should be the m.c.the m.c.

If any witness told the truth, then he or she is If any witness told the truth, then he or she is honest. (T_: _ told the truth; W_: _ is a witness; honest. (T_: _ told the truth; W_: _ is a witness; H_: _ is honest) H_: _ is honest)

((x)[(Wx & Tx) → Hx]x)[(Wx & Tx) → Hx]

Speakers have their names listed in the Speakers have their names listed in the program only if they are famous. (S_:program only if they are famous. (S_:__ is a is a speaker; P_:_s’s name is listed in the speaker; P_:_s’s name is listed in the program; F_:_ is famous)program; F_:_ is famous)

((x)[(Sx & Px) → Fx]x)[(Sx & Px) → Fx]

Some experienced mechanics are well paid Some experienced mechanics are well paid only if all the inexperienced ones are lazy. only if all the inexperienced ones are lazy. (E_: _ is experienced; M_: _(E_: _ is experienced; M_: _ is a mechanic; is a mechanic; W_: _ is well paid; L_: _W_: _ is well paid; L_: _ is lazy)is lazy)

((x)[(Ex & Mx) & Wx] → (x)[(Ex & Mx) & Wx] → (x)[(Mx & ~Ex) → x)[(Mx & ~Ex) → Lx ]Lx ]

The subject matter is largely the same on The subject matter is largely the same on both sides of the m.c. arrow, but you’re not both sides of the m.c. arrow, but you’re not picking individuals, one at a time, and saying picking individuals, one at a time, and saying that the entire statement is true of the same that the entire statement is true of the same individual. Compare 13 and 14 on p. 165.individual. Compare 13 and 14 on p. 165.

Creating InstancesCreating Instances

((x)(Px & Gx)x)(Px & Gx)

To create an instance of a quantified To create an instance of a quantified statement, first drop the quantifierstatement, first drop the quantifier

Px & GxPx & Gx

Then replace the variables that were bound Then replace the variables that were bound by that quantifier with an individual constantby that quantifier with an individual constant

Pa & GaPa & Ga

You must replace all of those variables with You must replace all of those variables with the same constant.the same constant.

Truth-conditions for the Truth-conditions for the QuantifiersQuantifiers

A universally quantified formula is true iff all A universally quantified formula is true iff all of its instances are true. of its instances are true.

A universally quantified formula is false iff at A universally quantified formula is false iff at least one of its instances its false.least one of its instances its false.

An existentially quantified formula is true iff at An existentially quantified formula is true iff at least one of its instances is true.least one of its instances is true.

An existentially quantified formula is false iff An existentially quantified formula is false iff all of its instances are false.all of its instances are false.

Stating an InterpretationStating an Interpretation

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

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

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

Truth and Falsity Relative to Truth and Falsity Relative to an Interpretationan Interpretation

Evaluate individual sentences on p. 179 to see Evaluate individual sentences on p. 179 to see whether they’re true or false on this whether they’re true or false on this interpretation. Explain your answer.interpretation. Explain your answer.

The procedure: make all possible instances of The procedure: make all possible instances of the quantified portions of a formula; use the the quantified portions of a formula; use the chart to assign truth-values to atomic formulae chart to assign truth-values to atomic formulae in those instances; then use the truth-conditions in those instances; then use the truth-conditions for the operators to work your way up to a single for the operators to work your way up to a single truth-value for the entire statement.truth-value for the entire statement.