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PHIL 102 FALL 2004 — QUIZ TWO PRACTICE: Translation
and Truth Tables
• Name:
• TA:
• Section Time:
- Attempt as much as you possibly can: there
is partial credit, and no negative credit.
- Read questions carefully: answer all parts.
- Write answers clearly and neatly: if it
can't be read, it can't receive any credit.
- Assign time carefully: you have the whole
period. Leave quietly if you finish early.
- Write in the spaces provided. Use the back
of the sheet or note paper if needed.
- Write in pen.
- If you have questions raise your hand and
wait for us to come to you.
1. Translate the following into the language of TW or colloquial
English
as appropriate:
- a is not in front of b
- e
is in front of c and d; however it is behind f
- ¬(Tet(a) v
Small(a))
- a is a small cube to the left of d
- ¬Adjoins(a,b) ^
(Adjoins(a,c) v Adjoins(a,d))
- a is not larger than either b or c
- a is the same shape and same size as b and c
- a is either a large tetrahedron or a large cube
- (Between(c,d,e) v
Between(c,d,f)) ^ (Adjoins(a,b) v Adjoins(a,f))
- Neither a nor b is a small dodecahedron
2. Give an instance of each of the
following logical forms in
the language of TW – do not replace C
with an atomic sentence.
- ¬(A v (A ^ B))
- (A v (B ^ C)) v ¬A
3. This question refers to the
following three worlds:
- Each of the worlds A-C corresponds to a row
of the following truth table. Indicate which by writing the appropriate
letters next to the rows in question
- What is the truth value of ¬(Larger(a,b) ^ Cube(c) ^ SameCol(a,d) ^
RightOf(c,d))
in World A?
in World B?
in World C?
- Briefly (one sentence will suffice)
explain the relationship between a TW world and a row of the truth
table.
- ¬(Larger(a,b)
^ Cube(c) ^ SameCol(a,d) ^ RightOf(c,d)) is not
a logical consequence of SameCol(a,d) and
RightOf(c,d). Show this by drawing a (clear) counter-example world.
4.
(a) Complete the following truth table (indicate the final column
clearly):
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A
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B
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¬ ( ¬ ( A ^ B )
v ¬ ( A v B ) )
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(b)
Complete
the following truth table (indicate the final column clearly):
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A
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B
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C
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( ¬ A ^ ¬ B )
v ¬ ( A ^ C )
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5. This question refers to the following truth table:
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Cube(c)
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Tet(c)
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¬(Cube(c) ^ Tet(c)) v
¬Cube(c)
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Tet (c) v ¬(Cube(c) ^
Tet(c))
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T
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T
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F
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T
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T
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F
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T
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T
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F
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T
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T
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T
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F
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F
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T
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T
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- Is Tet(c) v ¬(Cube(c) ^ Tet(c)) a tautology? What property of its truth table tells
you this?
- Explain why every tautology is a logical truth.
- Explain
why ¬(Cube(c) ^ Tet(c)) v ¬Cube(c) and Tet(c) v
¬(Cube(c) ^ Tet(c)) are logically
equivalent even though they are not tautologically equivalent.
- Is ¬(Cube(c) ^ Tet(c)) v ¬Cube(c) a tautological consequence of Tet(c) v ¬(Cube(c) ^ Tet(c))? What property of their truth table tells you this?
6. Put the following into Negation Normal
Form, citing as a justification for each substitution either De
Morgan's laws or the Double Negation rule:
¬(¬(¬A v B) v C)
7. Give examples of predicates that are (one of each is enough):
- Transitive:
- Reflexive:
- Symmetric,
Transitive and Reflexive:
8.
(a) Give an infomal proof involving proof by cases of the
following (you may use the language of TW where appropriate): Larger(a,
d), a=b v a=c therefore Larger(b, d) v Larger(c, d).
(b) Give a formal proof of the following, involving Ana Con and Taut Con: ¬(¬Larger(a,
b) v ¬Larger(b, c)) therefore Smaller(c,
a).
(c) Explain how indirect
proof is used in the following argument: Suppose for the sake of argument that God
exists. Then by definition an all-powerful, all-good being exists. But
if an all-powerful, all-good being existed, then there would be no
evil. But there is, so God does not exist. How might you rebut
the argument?
Solutions available by Friday
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