PHIL 102 FALL 2004 — QUIZ TWO PRACTICE SOLUTIONS: Translation and Truth Tables

Note: I've use italics to indicate an explanatory remark rather than an answer.

1. Remember, any logically equivalent sentence is also answer –
  1.  ¬FrontOf(a,b)
    2.
  2.  FrontOf(e,c) ^ FrontOf(e,d) ^ BackOf(e,f)
  3.  a is neither a tetrahedron nor small
  4.  Small(a) ^ Cube(a) ^ LeftOf(a,d)
  5.  a does not adjoin b, but it does adjoin either c or d
  6.  ¬(Larger(a,b) v Larger(a,c))
  7.  SameShape(a,b) ^ SameShape(a,c) ^ SameSize(a,b) ^ SameSize(a,c)
  8.  (Large(a) ^ Tet(a)) v (Large(a) ^ Cube(a))
  9.  c is between d and either e or f, while a adjoins either b or f
  10.  ¬(Small(a) ^ Dodec(a)) ^ ¬(Small(b) ^ Dodec(b))
    11.
2. Note: in (a) I choose to replace A with Cube(e) and B with Tet(e) throughout, but I could have choosen any sentences – Between(a,b,c) for A, or ¬Small(f) for B, or (SameRow(e,f) v SameCol(e,f)) for either or both, etc. Similar remarks apply to (b).
  1. ¬(Cube(e) v (Cube(e) ^ Tet(e)))
  2. (Large(f) v (Adjoins(a,b) ^ (Tet(a) v Dodec(a)))) v ¬Large(f)
    3.

3.
(a) To answer this question, just evaluate the truth values of the atomic sentences, and indicate the row in which they take those values.

Counting the top row as 1 and the bottom one as 16, we find that A correponds to row 1 and both B and C to row 14.

(b) To answer this question, simply read off the truth value of the sentence in the rows you just picked – after all the truth table shows the truth value of a sentence for any assignment of truth values to its atoms.

World A: F
World B: T
World C: T

(c) Each row of a truth table shows what truth value a sentence has given an assignment of truth values to its atoms, while each world assigns truth values to the atoms (assuming that they are evaluable).

(d) We need a world in which the premises are true and the conclusion false – that is, a world corresponding to the first row of the truth table. You should draw your own, but World A will fit the bill.

4.
(a) 

    A

    B

    ¬ ( ¬ ( A  ^  B )  v   ¬ ( A  v  B ) )

    T
    T
    		  
                T

    T

    		 F             F

    F

    		 T             F

    F

    		 F             F

 (b)

    A

    B

    C

    ( ¬ A  ^  ¬ B )   v   ¬ ( A  ^  C )

    T
    T    		  
    T    		  
    F


    T
    T    		  
    F    		  
    T


    T
    F
    T
    		  
    F


    T
    F
    F    		  
    T


    F
    T
    T    		  
    T


    F
    T
    F
    T 


    F
    F
    T
    T 


    F
    F
    F 
    T 


5.
(a) Yes. Every row of its truth table contains a T.

(b) If a sentence is a tautology then it is true in every row of its truth table. But every world corresponds to some row of its truth table, so the sentence is true in every world, which is what it is to be a logical truth.

(c) They are not tautologically equivalent because they have different truth values in the first row. However, there are no worlds corresponding to the first row, because there are no worlds in which c is both a cube and a tetrahedron. Thus in every world corresponds to one of the other rows, and so is a world in which they have the same truth values, which is what it is for them to be logically equivalent.

(d) No. In the first row the former is F and the latter is T.

Of course, there are no worlds corresponding to the first row, so ¬(Cube(c) ^ Tet(c)) v ¬Cube(c) is a logical consequence of Tet(c) v ¬(Cube(c) ^ Tet(c)).


6. Pay attention to how the connectives change when you apply De Morgan's rule.

¬(¬(¬A v B) v C)

<=> ¬¬(¬A v B) ^ ¬C         by De M

<=> (¬A v B) ^ ¬C           by Double ¬


7. Give examples of predicates that are (one of each is enough):

(a) Larger, SameRow, __ has the same birthday as __, etc
(b) SameCol, __ is the same height as __, etc
(c) =, SameSize, __ is tautologically equivalent to __, __ has the same biological parents as __, etc

8.
(a) Suppose first that a=b; then, since Larger(a, d), we know Larger(b,d) by II. But if Larger(b,d) then its true that either b or c is larger than d: Larger(b,d) v Larger(c,d) (by v-Intro). Now suppose a=c; then from Larger(a,d) we infer Larger(c,d) by II, and so Larger(b,d) v Larger(c,d) by v-Intro. But a=b v a=c, so since we've seen Larger(b,d) v Larger(c,d) follows in either case, we can conclude Larger(b,d) v Larger(c,d) by proof by cases.

(b)
| ¬(¬Larger(a, b) v ¬Larger(b, c))
| Larger(a, b) ^ Larger(b, c)               Taut Con (DeM and then double ¬¬ twice)
| Larger(a,c)                                          Ana Con (transitivity of Larger)
| Smaller(c, a)                                       Ana Con (Larger and Smaller are inverses)

(c) The argument seeks to show that God does not exists, but starts by assuming the negation – that God does exist – and attempts to demonstrate that an absurdity follows – that there is no evil in the world. If the existence of God entails an absurdity, then the existence of God is itself absurd. This is the 'problem of evil' – you can think for yourself how to rebut it: is there for instance any reason that an all good being might permit evil? To allow some other good thing?


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