1. For Wednesday, read chapter 8, section 2 (pp. 257-261)
As nongraded exercises, do problem sets I and II on pp. 261 and 262).
Graded HW #10 is due on Friday at the beginning of class.

2. Evaluating a Formula Relative to an Interpretation
Basic idea: A description of a possible world or situation is given; this is called an interpretation. Then, you’re asked to decide whether a given formula is true or false relative to that interpretation.
“Three people in this room were born in Montana” is not true or false in general. It is true or false only relative to a given situation. (Which room? Which people are in it? Which of them were born in MT?)

3. Elements of an Interpretation
Statement of domain, that is, of all of the objects contained in the interpretation’s toy universe.
Assignment of objects to names, e.g., Ref(a) = object alpha. This is normally left explicit, but must be included if any object has two or more names.
Assignment of extensions to predicates (using ordered n-tuples for multiplace predicates)
Assignment of truth-values to statement letters, if there are any

4. Example
What do we need to specify in an interpretation in order to evaluate ‘($x)[(Gx & Mx) & ("y)Wxy]’?
D = {, }
Ext(G) = {, }
Ext(M) = {}
Ext(W) = {<, >, <, >}
Let’s keep it simple and assume that neither object has more than one name.

5. True or False? ($x)[(Gx & Mx) & ("y)Wxy] D = {, } Ext(G) = {, }
Ext(M) = {}
Ext(W) = {<, >, <, >}
False. Only one object in the domain is both G and M; that’s alpha. But it’s not true of alpha that it W’s everything in the domain (it W’s itself, but not beta).

6. 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.

7. Creating Instances ($x)[(Gx & Mx) & ("y)Wxy]
To create an instance of a quantified statement, first drop the quantifier and outer parens
(Gx & Mx) & ("y)Wxy
then replace the variables that were bound by the dropped quantifier with a single individual constant
(Ga & Ma) & ("y)Way
Then evaluate that instance using the information from the interpretation.

8. In this case, any instance of the entire quantified statement itself has a quantified statement as a subformula.
‘(Ga & Ma) & ("y)Way’ has ‘("y)Way’ as a subformula, which has to be evaluated completely and independently for each of the instances of the existentially quantified formula.
It has two instances ‘Waa’, which is true, and ‘Wab’ which is false. The latter makes the universal false, making this entire instance of the existentially quantified formula false.

9. We don’t need to do much formal work in order to show that the other possible instance of the existential – ‘(Gb & Mb) & ("y)Wby’ – is false.
Mb is false, because beta is not in the extension of M.
That renders ‘Gb & Mb’ false, which makes the entire conjunction false.
Thus, the entire statement in question is false relative to the stipulated interpretation. It’s an existential statement and both of the possible instances of it are false.

10. Problems on p. 256