18.1. Write out a complete description of a first-order structure M that would represent Mary Ellen's World. This has been done above except for the packaging into a single function.

DM = {b1, b2, b3, b4}CubeM = {b1, b2, b3}LargerM = {b3,b2, b3,b1, b3,b4, b2,b1, b2,b4}=M = {b1,b1, b2,b2, b3,b3, b4,b4}cM = b1

18.2. (Simon says) Open Mary Ellen's World. The structure M that we have used to model this world with respect to the sublanguage involving only Cube, Larger, and c, is also a good model of many other worlds. What follows is a list of proposed changes to the world. Some of them are allowable changes, in that if you make the change, the model M still represents the world with respect to this language. Other changes are not. Make the allowable changes, but not the others.1. Move everything back one row.2. Interchange the position of the tetrahedron and the large cube.3. Make the tetrahedron a dodecahedron.4. Make the large cube a dodecahedron.5. Make the tetrahedron (or what was the tetrahedron, if you have changed it) large.6. Add a cube to the world.7. Add a dodecahedron to the world.Now open Mary Ellen's Sentences. Check to see that all these sentences are true in the world youhave built. If they are not, you have made some unallowable changes. Submit your modiedworld.

18.3. In the text we modeled Mary Ellen's World with respect to one sublanguage of Tarski's World. How would our structure have to be modified if we added the following to the language: Tet, Dodec, Between? That is, describe the first-order structure that would represent Mary Ellen's World, in its original state, for this expanded language. [Hint: One of your extensions will be the empty set.]

18.4 Consider a first-order language with one binary predicate Outgrabe. Suppose for some reason we are interested in first-order structures M for this language which have the particular domain {Alice; Mad Hatter}. List all the sets of ordered pairs that could serve as the extension of the symbol Outgrabe. How many would there be if the domain had three elements?{Alice,Mad Hatter, Mad Hatter, Alice}, {Alice,Mad Hatter}, { Mad Hatter, Alice}

18.5. In Section 14.4 (page 387) we promised to show how to make the semantics of generalized quantifiers rigorous. How could we extend the notion of a first-order structure to accommodate the addition of a generalized quantifier Q? Intuitively, as we have seen, a sentence like Qx (A(x); B(x)) asserts that a certain binary relation Q holds between the set A of things that satisfy A(x) and the set B that satisfies B(x) in M. Thus, the natural way to interpret them is by means of a binary relation on P(DM). What quantifier corresponds to the each of the following binary relations on sets?1. x (A(x)B(x))2. x (A(x)B(x))3. x (A(x)B(x))

18.6.?While we can't say with precision exactly which binary relation a speaker might have in mindwith the use of some quantiers, like many, we can still use this framework to illustrate thenature of the logical properties like conservativity, monotonicity, and so forth discussed inSection 14.5. Each of the following properties of binary relations Q on subsets of D correspondto a property of quantiers. Identify them.1. Q(A;B) if and only if Q(A;A \ B)2. If Q(A;B) and A A0 then Q(A0;B)3. If Q(A;B) and A0 A then Q(A0;B)4. If Q(A;B) and B0 B then Q(A;B0)5. If Q(A;B) and B B0 then Q(A;B0)