| Assignment Two | |
|
Due: 4pm 23/05/2008
|
|
|
1.
[8 marks] |
|
|
Show the following formulas to be S5QT-Valid. a.
(x)(Fx É
[]Gx)
b.
<>($x)Fx
º
($x)<>Fx
|
|
| 2.
[8 marks] |
|
|
Translate the following arguments into S5QT= and test for validity. Create your own dictionary (ignoring tense changes and other grammatical irrelevancies.)
a.
There are more
than seven planets. What's more, any number greater than seven is necessarily
greater than seven. So the number of planets is necessarily greater than seven.
b. Suppose we accept that anything is necessarily identical to itself. Furthermore, if any two things are identical then if one of those things is necessarily identical to itself then those two things are necessarily identical. Why then, it follows that if any two things are identical then they are necessarily identical. Obviously.
|
|
| 3.
[8 marks] |
|
|
Test
the following formulas for validity in S5QT. If it is invalid provide a
counterexample with the calculations to show it is so. a.
[]($x)Fx
É
($x)[]Fx
b. <>(<>($x)Fx & (x)[]~Fx)
|
|
| 4.
[8 marks] |
|
|
Test
the following formulas for validity both in S5QT= and in the free logic FQT=. If
any is invalid in either case provide a counterexample with the calculations to
show it is so. a.
Fa
É
($x)Fx
b.
(x)((x = a) É
Gx) É
($y)Gy
|
|
| 5.
[8 marks] |
|
|
Give formal proofs for the following theorems in T. Use either Natural Deduction or the the Axiomatic method. For either method use the appropriate rules given in the textbook. a. []~q É []~(p & q)
b. [](p É q) É ([](q É r) É [](p É r))
|
|