Test One, PHL 111Name:Solution Key

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Section One: Blocks worlds.

Tet(a) / a is a tetrahedron / RightOf(a,b) / a is located nearer to the right edge of the grid than b
Cube(a) / a is a cube
Dodec(a) / a is a dodecahedron / FrontOf(a,b) / a is located nearer to the front of the grid than b
SameCol(a,b) / a is in the same column as b
SameRow(a,b) / a is in the same row as b / BackOf(a,b) / a is located nearer to the back of the grid than b
Adjoins(a,b) / a and b are located on adjacent (but not diagonally) squares
LeftOf(a,b) / a is located nearer to the left edge of the grid than b / Between(a,b,c) / a, b and c are in the same row, column, or diagonal, and a is between b and c
leftmost(x) / refers to the leftmost object in x’s row / frontmost(x) / refers to the frontmost object in x’s column
rightmost(x) / refers to the rightmost object in x’s row / rearmost(x) / refers to the rearmost object in x’s column

1. Is this sentence true in the world pictured above: ‘FrontOf(frontmost(a),a)’?

No. (The function frontmost(a) refers to a, and a is not in front of itself.)

2. Why is the pictured world not a counterexample to the following invalid argument:

| LeftOf(c,d)Because in the pictured world, the conclusion is true. (In a counterexample,

| LeftOf(d,a)the conclusion must be false.) (If you’re wondering why the argument is

| Between(d,a,c)invalid, it’s because LeftOf(c,d) and LeftOf(d,a) can both be true without a, c

and d being in the same row. But this was not necessary to work out to

explain why the argument is invalid.

3. Modify the world so that it is a counterexample for the following argument:

| o = leftmost(b)Either o names the same object as b, o has to be in the same row as b,

|LeftOf(b,o)to its left. In any other place, either the premise is false or the conclusion

is true.

Section Two: Short-answer questions. Six points each.

  1. Your textbook used the following argument to illustrate invalidity. Explain as clearly as you can why the argument is invalid. “Lucretius is a man. After all, Lucretius is mortal and all men are mortal.”

Counterexample: Lucretius is a goldfish (conclusion is false), but the premises (‘Lucretius is mortal’ and ‘All men are mortal’) remain true. A counterexample is the best way to explain an arguments invalidity.

  1. Suppose the sentence ‘A  B’ says something that is true in every possible world. Would the sentence ‘A  B’ also then be true in every possible world? Why or why not?

If ‘A  B’ is true in every possible world, since ‘A  B’ is true whenever ‘A  B’ is true (right??), ‘A  B’ must also be true in every world.

  1. Max and Jill each own a white Toyota Yaris with exactly the same equipment. Would that make the following FOL sentence true: ‘car(max) = car(jill)’? Why or why not?

No. Since they’re two different cars, car(max) and car(jill) refer to different (but similar) objects.

  1. Explain why an argument with a contradiction [a statement that is false in all possible worlds] for a premise is always valid.With a contradiction for a premise, there will never be a counterexample-world.
  2. Write a tautology that has a negation as its main connective.There are many; here’s one: (A A).

Section Three: True/False

  1. F Whenever an argument is valid it has a false conclusion if it has a false premise.
  2. F An argument must have a true conclusion if it is valid.
  3. F Every argument with only true premises and a true conclusion is sound.
  4. F Every argument with only true premises and a true conclusion is valid.
  5. T An argument with a contradiction [a statement that is false in all possible worlds] for a premise is always valid.

For the next five questions, refer to the following truth table which shows the truth values of four oddly-named compound sentences. All four are composed of the atomic sentences A, B and C and the Boolean connectives, but all you know about what the sentences say is the information in this truth table.

A / B / C / Sentence Bob / Sentence Sam / Sentence Sue / Sentence Jill / B  (A C)
T / T / T / F / T / T / T / F
T / T / F / T / T / F / T / T
T / F / T / T / T / F / T / F
T / F / F / T / T / F / F / F
F / T / T / F / T / T / F / F
F / T / F / T / T / F / T / F
F / F / T / T / T / F / F / F
F / F / F / F / T / T / F / F
  1. TSentence Bob entails Sentence Sam. (The argument “Bob, therefore Sam” is valid.)
  2. FIn the argument with premises Sam, Sue and the conclusion Jill, the T-F-F line is a counterexample.
  3. F“(Sentence Bob)  (Sentence Jill)” is a tautology.
  4. TSentence Bob is logically equivalent to the negation of Sentence Sue.
  5. FFrom Sentence Jill, it follows validly that B  (A C).

Section Four: Translation exercises: Use the predicates and functions in brackets to translate the following English sentences into FOL.

  1. Colbert is goofier and funnier than Leno.

[Goofier(x,y) : x is goofier than y][Funnier(x,y) : x is funnier than y] [colbert] [leno]

Goofier(colbert, leno)  Funnier(colbert, leno)

  1. Either Ann and her boyfriend don’t both go to the party, or Sue’s boyfriend doesn’t go.

[Goes(x) : x goes to the party] [boyfriend(x) : the boyfriend of x] [ann] [sue]

(Goes(ann)  Goes(boyfriend(ann))) Goes(boyfriend(sue))

  1. Ann is taller than both her mother and father.

[Taller(x,y) : x is taller than y] [mother(x) : the mother of x] [father(x) : the father of x] [ann]

Taller(ann, mother(ann))  Taller(ann, father(ann))

  1. Max gave Scruffy to Jill and she gave Max her bike.

[Gave(x,y,z) : x gave y to z] [bike(x) : the bike of x] [max] [scruffy] [jill]

Gave(max, scruffy, jill)  Gave(jill, bike(jill), max)

  1. Ann likes her parents, but they don’t like each other.

[Likes(x,y) : x likes y] [father(x) : the father of x] [mother(x) : the mother of x] [ann]

(Likes(ann, father(ann))  Likes(ann, mother(ann)))  (Likes(mother(ann), father(ann)) Likes(father(ann), mother(ann)))

  1. Max and Jill have the same mother but not the same father.

[father(x) : the father of x] [mother(x) : the mother of x] [max] [jill]

mother(max) = mother(jill) (father(max) = father(jill))

alternative: mother(max) = mother(jill) father(max) ≠ father(jill)

  1. Tom’s secretary is taller than her father but not her husband.

[Taller(x,y) : x is taller than y] [secretary(x) : the secretary of x] [father(x) : the father of x] [husband(x) : the husband of x] [tom]

Taller(secretary(tom), father(secretary(tom))) Taller(secretary(tom), husband(secretary(tom)))

  1. Scruffy is mangy and has not been fed.

[Mangy(x) : x is mangy][Fed(x) : x has been fed] [scruffy]

Mangy(scruffy) Fed(scruffy)

  1. The population of Ann’s hometown is larger than the population of Tom’s hometown.

[>] [population(x) : the population of x] [hometown(x) : the hometown of x] [tom] [ann]

population(hometown(ann)) > population(hometown(ann))

  1. Ann is Jill’s aunt. (Which means that Ann is the sister of one of Jill’s parents.)

[father(x) : the father of x] [mother(x) : the mother of x] [sister(x) : the sister of x] [ann] [jill]

ann = sister(father(jill))  ann = sister(mother(jill))

Section Five: Translate the following into FOL and explain in English the meaning of your FOL predicates, functions and names.I also gave full credit for interpretations that differ from those below.

.

  1. Robin’s nemesis is the sheriff of Nottingham.

[nemesis(x) : the nemesis of x] [sheriff(x) : the sheriff of x] [robin] [nottingham]

nemesis(robin) = sheriff(nottingham)

  1. Dave brought his phone but not his wallet from Syracuse to Oswego.

[Brought(w,x,y,z) : w brought x from y to z] [phone(x) : the phone of x] [wallet(x) : the wallet of x] [dave] [syracuse] [oswego]

Brought(dave, phone(dave), syracuse, oswego)  Brought(dave, wallet(dave), syracuse, oswego)

  1. Max is smart, but Jill is smarter.

[Smart(x) : x is smart] [Smarter(x,y) : x is smarter than y] [max] [jill]

Smart(max)  Smarter(jill, max)

Section Six: Truth Table Exercises:

  1. Tell me whether the following arguments are valid using the truth table provided.

|  ((A  B)  C)It’s not valid; the TTF line is a counterexample.

|---

|  (A  (B  C))

A / B / C / A  B / (A  B)  C /  ((A  B)  C) / B  C / A  (B  C) /  (A  (B  C))
T / T / T / T / T / F / T / T / F
T / T / F / T / F / T / T / T / F
T / F / T / T / T / F / T / T / F
T / F / F / T / F / T / F / F / T
F / T / T / T / T / F / T / F / T
F / T / F / T / F / T / T / F / T
F / F / T / F / F / T / T / F / T
F / F / F / F / F / T / F / F / T
  1. | ((A  B)  (C B))Not valid; counterexample on TFF line.

| (B  C)

|---

| (C  (A  B))

A / B / C / A  B / C B / (A  B)  (C B) / B  C / (B  C) / C  (A  B)
T / T / T / T / T / T / T / F / T
T / T / F / T / F / F / F / T / F
T / F / T / T / T / T / F / T / T
T / F / F / T / T / T / F / T / F
F / T / T / T / T / T / T / F / T
F / T / F / T / F / F / F / T / F
F / F / T / F / T / F / F / T / F
F / F / F / F / T / F / F / T / F
  1. Use a truth table to figure out whether this is a tautological equivalence:

(A  (B C))  ((A C) (A B))It looks like it is.

A / B / C / B C / A  (B C) / (A  (B C)) / A C / (A B) / (A  C) (A  B)
T / T / T / F / T / F / F / F / F
T / T / F / T / T / F / F / F / F
T / F / T / F / T / F / F / F / F
T / F / F / F / T / F / F / F / F
F / T / T / F / F / T / T / F / T
F / T / F / T / T / F / F / F / F
F / F / T / F / F / T / T / T / T
F / F / F / F / F / T / F / T / T