NOTE: The format of the real first midterm will follow this format exactly. However, since I cannot duplicate our normal symbols for conjunction (dot), conditional (horseshoe) or biconditional (triple-bar) in HTML, I have used the following substitutions: conditional ->, biconditional <->, and conjunction &. The real exam will, however, use the normal symbols.
Midterm I
Philosophy 500 -- Introduction to Logic
Write your name here: ______________________________
The exam consists of 20 questions. Each is worth 5 points. Show all work.
Part One. Give a one sentence definition of the following terms.
1. Argument
A set of statements, one of which, the conclusion, is taken to be supported by the others, called premises.
2. Justified Step
A line in a proof that is either a premise or which follows from previous line(s) according to an inference rule or replacement rule.
3. Logically Equivalent
Two statements are logically equivalent if and only if each entails the other, or equivalently, their truth columns are identical.
4. Tautology
A sentence or statement that is always true.
5. Statement form
A formula whose simplest components are statement variables.
Part Two. Short answer.
6. Explain the difference between a statement and statement variable.
A statement is an actual English sentence, symbolized with a capital letter, while a statement variable is a lower case letter that can stand for any statement, simple or complex.
7. Explain the difference between a valid argument and a sound argument.
A sound argument is a valid argument that in fact has all true premises. A valid argument is one such that if the premises are true the conclusion cannot be false, but it may in fact have false premises and even a false conclusion.
Part Three. Truth tables.
In 9 and 10, use truth tables to determine if the given statement has a form that is contingent, a contradiction, or a tautology. You will first need to translate from the English to proper logical symbolism.
9. If I don't bowl a 300 game, then if I don't bowl a 300 game, I will go to the Bowling Hall of Fame.
B = I bowl a 300 game.
H = I will go to the Bowling Hall of Fame.
~B -> (~B -> H)
| B | H | ~B | ~B -> H | ~B -> (~B -> H) |
| T | T | F | T | T |
| T | F | F | T | T |
| F | T | T | T | T |
| F | F | T | F | F |
The statement is contingent, since both Ts and Fs appear in its column of the truth table.
10. If both either my ex-fiancee is cheap or she has no scruples and I got hosed then either my ex-fiancee is cheap or she has no scuples or I got hosed.
X = My ex-fiance is cheap.
S = My ex-fiance has scruples.
H = I got hosed.
((X v ~S) & H) -> ((X v (~S v H))
| X | S | H | ~S | X v ~S | (X v ~S) & H) | ~S v H | X v (~S v H) | ((X v ~S) & H) -> ((X v (~S v H)) |
| T | T | T | F | T | T | T | T | T |
| T | T | F | F | T | F | F | T | T |
| T | F | T | T | T | T | T | T | T |
| T | F | F | T | T | F | T | T | T |
| F | T | T | F | F | F | T | T | T |
| F | T | F | F | F | F | F | F | T |
| F | F | T | T | T | T | T | T | T |
| F | F | F | T | T | F | T | T | T |
The statement has only Ts in its truth column, and hence it is a tautology.
In 11 and 12, use truth tables to determine if the following argument forms are valid. You may use a partial truth table if you like. Explain your reasoning.
11.
p -> q
~p
Therefore: ~q
| p | q | ~p | ~q | p -> q |
| T | T | F | F | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
The argument is invalid, as can be seen from the third row, in which all the premises (bold) are true and the conclusion (italics) is false.
12.
p -> q
~q -> r
Therefore: ~p v r
| p | q | r | ~p | ~q | p -> q | ~q -> r | ~p v r |
| T | T | T | F | F | T | T | T |
| T | T | F | F | F | T | T | F |
| T | F | T | F | T | F | T | T |
| T | F | F | F | T | F | F | F |
| F | T | T | T | F | T | T | T |
| F | T | F | T | F | T | T | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | T | F | T |
The argument is invalid as can be seen from the second row, in which all the premises (in bold) are true and the conclusion (in italic) is false.
In 13 and 14, use truth tables to determine if the given statement forms are consistent, equivalent, and if either logically entails the other. Explain your answer briefly.
13. p -> ~q , ~p v q
| p | q | ~p | ~q | p -> ~q | ~p v q |
| T | T | F | F | F | T |
| T | F | F | T | T | F |
| F | T | T | F | T | T |
| F | F | T | T | T | T |
The first statement (in italic) does not entail the second (in bold), as can be seen from the second row in which the first is true but the second false. The second does not entail the first, as can be seen from the first row, in which the second true but the first false. They are not equivalent, as can be seen from the first row in which they have different truth values. They are consistent, as can be seen from the last two rows in which both are true.
14. p & ~r , (q -> ~p) v r
| p | q | r | ~p | ~r | p & ~r | q -> ~p | (q -> ~p) v r |
| T | T | T | F | F | F | F | T |
| T | T | F | F | T | T | F | F |
| T | F | T | F | F | F | T | T |
| T | F | F | F | T | T | T | T |
| F | T | T | T | F | F | T | T |
| F | T | F | T | T | F | T | T |
| F | F | T | T | F | F | T | T |
| F | F | F | T | T | F | T | T |
The first statement (italic) does not entail the second (bold), as can be seen from the second row in which the first is true, but the second false. The second does not entail the first as can be seen from the first row, in which teh second is true and the first false. They are not equivalent as can be seen from the first row in which they have different truth values. They are consistent as can be seen from the fourth row in which they are both true.
Part Four. Proofs.
Construct proofs for the following arguments. In 15 and 16 you may use all rules of inference and all replacement rules, but not conditional or indirect proof. In 17 and 18 you may also use conditional and indirect proof. In 19 and 20 you MUST use either condiitonal or indirect proof, or both. You must justify every step.
15.
1. (A v B) -> ~C
2. C v D
3. A \ D
4. A v B [3 add]
5. ~C [1, 4 MP]
6. D [2, 5 DS]
16.
1. [(J -> J)] & K
2. J v ~J
3. ~K \ ~(J & ~J)
4. K [1 simp]
5. K v ~(J & ~J) [4 add]
6. ~(J & ~J) [3, 5 DS]
17.
1. B & D
2. (G -> B) -> (B -> C)
3. (B -> C) -> (D -> E)
4. B -> (G -> B) \ E v G
5. B [1 simp]
6. D [1 simp]
7. G -> B [4, 5 MP]
8. B -> C [2, 7 MP]
9. D -> E [3, 8 MP]
10. E [6, 9 MP]
11. E v G [10 add]
18.
1. H <-> J
2. ~H \ ~J
3. (H -> J) & (J -> H) [1 BE]
4. J -> H [3 simp]
5. |-> J [AIP]
6. | H [4, 5 MP]
7. |- H & ~H [2, 6 conj]
8. ~J [5-7 IP]
19.
1. (P & S) -> (T v W)
2. ~T <-> ~(M & O)
3. ~(W v (~S v M))
4. ~A -> P \ A
5. ~W & ~(~S v M) [3 DeM]
6. ~(~S v M) [5 simp]
7. S & ~M [6 DeM, DN]
8. ~M [7 simp]
9. ~M v ~O [8 add]
10. ~(M & O) [9 DeM]
11. (~T -> ~(M & O)) & ( ~(M & O) -> ~T) [2 BE]
12. ~(M & O) -> ~T [11 simp]
13. ~T [10, 12 MP]
14. ~W [5 simp]
15. ~T & ~W [13, 14 conj]
16. ~(T v W) [15 DeM]
17. ~(P & S) [1, 16 MT]
18. ~P v ~S [17 DeM]
19. |-> ~A [AIP]
20. | P [4, 19 MP]
21. | ~S [18, 20 DS, DN]
22. | S [7 simp]
23. |- S & ~S [21, 22 conj]
24. A [19-23 IP, DN]
20.
1. A -> C \ (A & B) -> C
2. |-> A & B [ACP]
3. | A [2 simp]
4. |- C [1, 3 MP]
5. (A & C) -> C [2-4 CP]