PL 120-740 Symbolic Logic I Online Summer 2009 Exam # 1 partial answer key
Question # 2. If an argument is valid, then its conclusion must be true. It is not true that valid
argument must have a true conclusion. Deductive validity is a characteristic of
arguments in which the truth of the premises guarantees the truth of the conclusion. It
is impossible for both the premises of a valid argument to be true and the conclusion to
be false. Validity does NOT guarantee the truth of the conclusion. It is possible for the
conclusion of a valid argument to be false. If this is the case, then at least one premise
must be false. The following argument is VALID:
•
All trout are mammals
All mammals have wings
SO, all trout have wings
This argument is valid because IF the premises were true (which they aren’t) THEN the
conclusion WOULD HAVE TO be true (even though it isn’t). This holds even though the
premises are in fact false.
Question # 3. If the falsity of the conclusion of an argument is inconsistent with the
truth of its premises, then the argument is valid. True. The defining condition of nonvalidity is the possibility or true premises with a false conclusion. If the falsity of the
conclusion is inconsistent with the truth of the premises, that is, if it is impossible for the
conclusion to be false while the premises are true, then the condition of non-validity is
not met, so the argument is valid.
Question # 4. The English phrase 'neither...nor' is equivalent to the English phrase 'not
both'. False. “Not both” means either on or the other of a pair is missing, and it is
possible that both are missing. “Neither nor” means both members of a pair are
missing. Contrast “Jon and Sue were not both in class last Friday” with “Neither Jon nor
Sue was in class last Friday.”
Question # 5. An argument whose premises are contradictory must be valid. True.
Question # 6. Justification for # 5. An argument with contradictory premises is guaranteed
to be valid, no matter what its conclusion. Validity is binary—either an argument is valid
or it is not. The condition of non-validity is defined as the possibility of all the premises
being true while the conclusion is false. Well, if the premises are contradictory, then
they cannot all be true (that’s just what contradictory means) so they can’t all be true
while the conclusion is false (the necessary condition for non-validity). So the argument
cannot be non-valid, it must be valid. Thus an argument with contradictory premises is
valid.
Question # 7. Suppose I gave you a consistency checker for a set of statements, that
is, a machine that would tell you whether any set of statements is consistent or
inconsistent. Devise a way to use this machine to determine whether an argument is
valid or not. What, exactly, do you test and why? A set of statements is consistent if,
but only if, it is possible for all of the statements in the set to be true. A set of statements
is inconsistent if, but only if, it is impossible for all of the statements in the set to be true.
To test an argument for validity using a consistency checker, negate the conclusion of
the argument and then ask whether the set of statements consisting of the premises
and the negation of the conclusion is consistent. If yes, then the argument is NONVALID, because you have shown that it is possible for all the premises to be true while
the conclusion is false. If no, if that set is inconsistent, then the argument is VALID. In
essence you are asking if an invalidating row is possible, if the set of statements that
makes up an invalidating row is consistent.
Question # 16. Construct a truth table to determine whether the following formulas are
equivalent and then choose the most appropriate response from the options below:
'~(P  Q)' and '(P  ~Q)'
P
T
T

F
Q
T

T

~(P  Q)
 T
T 
 T
 T
(P  ~Q)
 
T T
 
 T
Valid, no row where the premise is true and the conclusion false.
Question # 18. Assuming 'P' to be true, 'Q' false, and 'R' unknown, determine whether
the following formulae are true, false, or undecided. [(P  R)  ~Q]  (R  ~P) .
Undecided.
[(P  R)  ~Q]  (R  ~P)
[(T  ?)  ~]  (?  ~T)
(T  T) (? )
T ?
?
Question # 20. Assuming 'P' to be true, 'Q' false, and 'R' unknown, determine whether
the following formulae are true, false, or undecided. (P  Q)  (~R  ~P) True.
(P  Q)  (~R  ~P)
(T)  (~?  ~T)
  ?
T A conditional with a false antecedetn is true.