These answers, minus a few comments I've made, are a model for what you
should write on your quizes on Monday: no working unless asked, but enough
to make your answer clear. 6(b), 7 and 9 could have other answers, but these
seem the most obvious to me.
1.
(a) Predicate: __ is a satelite of __ Arity:
2 Names: the Moon, the Earth
(b) Predicate: __, __, __, __ and
__ were in a pop group Arity: 5
Names: Jackie, Jermaine, Marlon, Michael, Tito
(c) Predicate: __, __, __ and __ were in a pop group
Arity: 4 Names: John, Paul, George,
Ringo
So (b) and (c) are not the same predicate!
2. (a) F (b) F (c) T
(d) F (e) T (f) T (g)
F
(h) T
3.
(a) | All dogs prefer Bone–O | All Corgis are dogs
| All Corgis prefer Bone–O
Valid
(b)
| a
is in front of b | a is in front
of c
| b is in the same row as c
Invalid
4. (a) Invalid: B is a counter-example.
(b) Valid: It is sound
in C.
6.
(a)
'___ scores no more than ___ on the test': TRAN
REF
'___ scores less than ___ on the test': TRAN
'___ and ___ are taking the same course': SYM
REF – can you see an example that shows it's not transitive?
SameShape(_, _):
SYM TRAN REF
EQ
(b) E.g., '__ is older than __' and '__ is younger than __',
or '__ is the parent of __' and '__ is the child of __', or '<'
and '>' etc.
7.
(a) Superman is faster than Batman, so, since Bruce Wayne is Batman,
by II (the indiscernibility of identicals) Superman is faster than Bruce Wayne.
But then it follows that Bruce Wayne is slower than Superman, since the predicates
'faster than' and 'slower than' are inverses.
(b) Since c is in front of d and b is in front of
c, it follows that b is in front of d, by transitivity
of 'in front of'. But a is in front of b, so a must also
be in front of d, again by transitivity of 'in front of'. Hence d
is in back of a, because 'in front of' and 'in back of' are inverses.
8. | 1. SameShape(c, f)
| 2. SameShape(e, d)
| 3. Tet(c)
| 4. f = e
|–
| 5. SameShape(c, e)
= Elim 1, 4
| 6. SameShape(c, d)
Ana Con 2, 5
| 7. Tet(d)
Ana Con 3,6
9. (a) Smaller(f, c)
(b) (Dodec(b) ^ Dodec(d))
v (Tet(b) ^ Tet(d)) v (Cube(b) ^ Cube(d))
(c) Sam and Billy are siblings and Sam is female. (Billy
is Sam's brother is wrong because (i) it doesn't tell us that Sam
is female, and (ii) the given sentence could be true even if Billy is Sam's
sister – given names are always gender ambiguous, as is clear in this
case.)
(d) ¬Large(a) v ¬Dodec(a) by De Morgan
(e) I am not French and I am not Scottish.