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Graduate Logic Examination
Spring 2011
You will have two hours for the examination. Your score must be an 80% or above in order to pass the test.
True or false? Worth two points each.
(P ( ( Q ( R) ( (Q ( ( P) ( ( S is in disjunctive normal form.
(x P(x,x) logically implies (x(yP(x,y)
P ((Q ( ( Q) is a logical truth only if ( P is a logical truth.
If an argument is not sound, then it is not valid.
The following is a logical truth: if there are at least three objects, then there are at least two.
The following argument is invalid: (x(y Likes(x, y), so (x (y Likes(x,y).
The sentence, (x (Cube(x) ( (yCube(y)), is a logical truth.
(xTet(b) is logically equivalent with (yTet(b).
A contradiction entails any sentence.
10. The following sentence is a tautology: (x Cube(x) ( ((x Cube(x) ( (xTet(x)).
Conceptual problems / Each is worth five points.
1. Identify a sentence in disjunctive normal form that is logically equivalent
with ((P ! Q). Be sure to simplify your answer as much as possible.
2. Validity and Logical truth: Give an informal proof of the following. For any
argument P1, & , Pn, C, the argument is valid if and only if (P1 ( & ( Pn)(C is
a logical truth.
III. Translation. Each is worth two points.
Using the scheme of translation below, give translations into logical notation for the English sentences that follow.
Domain: blocks
Dictionary:
Cube(x)= x is a cube
Dodec(x) = x is a dodecahedron
Tet(x) = x is a tetrahedron
Small(x)= x is small
BackOf(x,y) = x is in back of y
LeftOf(x,y)= x is left of y
Larger(x,y) = x is larger than y
Between(x,y,z) = x is between y and z
x=y = x is the same block as y
a, b, etc., are names of blocks
English sentences to be translated
No dodecahedron to the left of a cube has anything in back of it.
Only cubes are larger than everything else.
Nothing but a cube is between two other blocks.
There are at most two tetrahedra.
a is left of the cube (on the Russellian analysis).
Using the same scheme of translation as above, translate the following sentences into colloquial English.
(x(Between(x, d, c) ( x=b)
(y(x ((Tet(x) ( Small(x))( x=y)
8. (x ((x = a ( x = d) ! (y (z Between(x, y, z))
9. (x((y BackOf(x,y) ( (x=a ( x=b ( x=c))
10. (x (y ((Tet(x) ( Small(x) ( Tet(y) ( Small(y)) ! x = y)
IV. For each of the following arguments, determine whether or not the conclusion is a firstorder logical consequence of the premises. If so, give a formal proof of this using some standard deduction system (e.g., natural deduction or truth trees). If you remember the name of the author of the textbook where you learned this technique, please give it. If the conclusion is not a firstorder logical consequence of the premises, construct a counterexample to the argument that shows this. Briefly explain how what you have done demonstrates that the argument is invalid. Each is worth 15 points.
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