Symbolic Logic – Final Study Sheet
I. Translations in quantification.
Translate each of the following into the logical notation of prepositional functions and
quantifiers, in each cause using the abbreviations suggested.
1. Bats are mammals (Bx: x is a bat; Mx: x is a mammal.)
x (Bx  Mx)
2. Sparrows are not mammals. (Sx: x is a sparrow; Mx: x is a mammal.)
x (Sx  Mx)
3. Reporters are present. (Rx: x is a reporter; Px: x is present.)
x (Rx  Px)
4. Nurses are always considerate. (Nx: x is a nurse; Cx: x is considerate.)
x (Nx  Cx)
5. Diplomats are not always rich. (Dx: x is a diplomat; Rx: x is rich.)
x (Dx  Rx)
6. Ambassadors are always dignified. (Ax: x is an ambassador; Dx: x is dignified.)
x (Ax  Dx)
7. No boy scout ever cheats. (Bx: x is a boy scout; Cx: x cheats.)
x (Bx  Cx)
8. Only licensed physicians can charge for medical treatment. (Lx: x is a licensed
physician; Cx: x can charge for medical treatment.)
x (Cx  Lx)
9. Snake bites are sometimes fatal. (Sx: x is a snake bite; Fx: x is fatal.)
x (Sx  Fx)
10. The common cold is never fatal. (Cx: x is a common cold; Fx: x is fatal.)
x (Cx  Fx)
II. For each of the following find a normal form formula logically equivalent to the given
one. In other words, get rid of the negation outside of the quantifier and simplify the
expression as much as possible by putting it into a disjunctive or conjunctive normal form
using DeMorgan’s Theorems and other equivalences we know about.
11. x (Ax  Bx)
x (Ax  Bx)  x (Ax  Bx)  x (Ax  Bx)
12. x (Cx  Dx)
x (Cx  Dx)  x (Cx  Dx)  x (Cx  Dx)
13. x (Ex  Fx)
x (Ex  Fx)  x (Ex  Fx)
14. x(Gx  Hx)
x (Gx  Hx)  x (Gx  Hx)
15. x (Ix  Jx)
x  (Ix  Jx)  x (Ix  Jx)
16. x (Kx  Lx)
x  (Kx  Lx)  x (Kx  Lx)
17. x ((Mx  Nx))
x  ((Mx  Nx))  x (Mx  Nx)
18. x((Ox  Px))
x ((Ox  Px))  x (Ox  Px)
19. x ((Qz  Rx))
x ((Qx  Rx))  x (Qx  Rx)
20. x ((Sx  Tx))
x  ((Sx  Tx))  x (Sx  Tx)
III. The game.
For the following exercises, consider this world. Note that the back row is large, the
middle row is medium, and the front row is small.
Evaluate the following expressions, deciding whether each is 1) not a wff, 2) a wff but
not a sentence, 3) a true sentence or 4) a false sentence.
21. Cube(a)
This is a true sentence
22. Small(Cube(a))  FrontOf(Tet(e))
This is not a wff.
23. x BackOf(x, b)
This sentence is false.
24. Cube(a)  Cube(b)  Cube(c)
This is not a wff.
25. x (Cube(x))
This sentence is false.
26. x (Cube(x)  Small(x))
This is a true sentence.
27. x (Tet(y)  Large(y))
This is a wff but not a sentence.
28. x (Cube(x)  Medium(x))
This is a true sentence.
29. x ((Tet(x)  Small(x))  FrontOf(x, e))
This is a true sentence.
30. u ((Tet(u)  Medium(u))  BackOf(u, c))
This is a true sentence.
31. x ((Medium(x)  Large(x))  BackOf(Small(x))
This is not a wff.
32. y ((Large(y)  FrontOf(b, e))  FrontOf(b, d))
This is a true sentence.
33. Ex (Cube(x)  Medium(x))
This is a false sentence.
34. x ((Tet(x)  Small(x))  BackOf(e, x))
This is a true sentence.
IV. Given the above-world, decide whether the following statements are merely true,
merely false, inherently vacuous, or self-contradictory.
36. x ((Large(x)  Dodec(x))  Dodec(x))
This sentence is inherently vacuous because there are no Dodecs.
37. x((Large(x)  Cube(x))  Cube(x))
This sentence is self-contradictory.
38. x ((Large(x)  Tet(x))  FrontOf(b, e))
This is a true sentence.
39. x(Large(x)  Small(x))
This sentence is self-contradictory.
40. x(Large(x)  Small(x))
This sentence is logically true.
V. Complete the following proofs in system F or F’ using the list of rules available
during the last test.
41.
1. P  Q
2. (P  Q)
3. P  Q
4. P
5. Q
6. Q
7. Q  Q
8. P  Q
/ P  Q
DeM 2
 Elim 3
 Elim 1, 4
 Elim 3
 Intro 5, 6
 Intro 2-7
42.
1. (F  G)
2. G
3. H  F
/  H
4. H
5. F
6. F  G
7. (F  G)  (F  G)
8. H
 Elim 4, 3
 Intro 5, 2
 Intro 6, 1
 Intro 4-7
43.
1. S  (T  U)
2. (T  U)
3. U  (T  W)
/  (S  W)
4. S  W
5. S
 Elim 4
6. T  U
 Elim 1, 5
7. (T  U)  (T  U)  Intro 6, 2
8. (S  W)
 Intro 4-7