PHILOSOPHY 2340
Symbolic Logic
FIRST EXAMINATION
Part I: Terminology and Basic Ideas
True or False (Circle T if the statement is true and F if it is false; 2 points each):
T
T
T
T
T
F
F
F
F
F
1.
2.
3.
4.
5.
T
F 6.
T
F 7.
T
T
T
F 8.
F 9.
F 10.
In FOL, a name can fail to refer to any object. [False]
In FOL, an object can lack a name. [True]
In FOL, a name can refer to more than one object. [False]
In FOL, an object can have more than one name. [True]
The truth table for a tautology will show that the sentence is true in some rows
and false in other rows. [False]
If two sentences are tautologically equivalent, then each is a tautological
consequence of the other. [True]
If C is a logical consequence of P1, . . ., Pn, then the argument from P1, . . .,
Pn to C is logically valid. [True]
“Tet(a)  Dodec(a)” is an atomic sentence. [False]
“a=b” is an atomic sentence. [True]
The disjunction of a sentence and its negation is a tautology. [True]
Short Answer:
11. Consider the following sentence:
¬Tet(a)  ¬Cube(b)  a ≠ b
Is it a tautology? A logical necessity? Why? (5 points)
Not a tautology (not true in every row of its truth table). It is a logical necessity: to
make it false, we would have to make every disjunct false, so it would have to be the case
that a is a tetrahedron, b is a cube, and a = b. But the same object cannot be both a
tetrahedron and a cube.
12. List the idempotence logical equivalences (5 points).
AA⇔A
AA ⇔A
13. List the distributivity logical equivalences (5 points).
A  (B  C) ⇔ (A  B)  (A  C)
A  (B  C) ⇔ (A  B)  (A  C)
14. How many rows will you need to construct a truth table for the following sentence (2
points)?
((A  B)  (C  D))  (C  E)
There are five atomic sentences (A, B, C, D, and E), so we have 25 = 32 rows.
15. Write out the truth table for: A  B. (3 incredibly easy points)
A
T
T
F
F
B
T
F
T
F
AB
T
T
T
F
16. Write out a joint truth table for the following two sentences. Draw a box around the
column under the major operator of each sentence, that is, the column that gives the
truth values for the sentence as a whole. State, based on the information in the truth
table, (a) whether the first and second sentences are tautologically equivalent, and (b)
whether the second sentence is a tautological consequence of the first sentence. (10
points)
first sentence:
(P  Q)  ¬R
second sentence:
¬(¬P  R)
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
(P  Q)  ¬R
T
T F
T
T T
F
F F
F
T T
F
F F
F
T T
F
F F
F
T T
¬(¬P  R)
T F F
T F F
T F F
T F F
F T T
T T F
F T T
T T F
(a) not tautologically equivalent (because different truth values in row 3)
(b) second is a tautological consequence of the first (because there is no row where the
first sentence is true and the second sentence is false)
III. Translations
Please translate the following English sentences into the language of Tarski’s World. (4
points each.)
17. a is larger than b but smaller than e.
Larger(a,b)  Smaller(a,e)
18. a and e are both back of c, but neither is back of d.
BackOf(a,c)  BackOf(e,c)  ¬(BackOf(a,d)  BackOf(e,d))
19. Either a or b is a small cube.
(Small(a)  Cube(a))  (Small(b)  Cube(b))
20. a is a cube and b is not a small dodecahedron.
Cube(a)  ¬(Small(b)  Dodec(b))
21. b is neither identical with a nor adjacent to a.
b ≠ a  ¬Adjacent(b,a)
22. Either c is back of both a and b, or d is not left of either of them.
(BackOf(c,a)  BackOf(c,b))  ¬(LeftOf(d,a)  LeftOf(d,b))
IV. Proofs. (13 points each.)
Make sure to use line numbers, and indicate in the justification for each line which earlier
line(s) are being used as justification! Also make sure not to skip any steps, however
obvious they may seem; every step must be an exact instance of one of the rules. You
may use only Reiteration and the introduction and elimination rules for =, , , , and .
23. C from the following premises (took me 9 steps).
1. (A  B)  C
2. A
24.  (P  Q) from the following premise (took me 10 steps). Hint: you will probably
need a subproof inside another subproof.
1. P  Q