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Glances Ahead: More to Think About
I. Introduction to Symbolic Logic- the Use of the Truth Table
for Determining Validity
-Truth tables are useful formal tools for determining validity
of arguments because they specify the truth value of every premise in
every possible case
-Truth tables are constructed of logical symbols used to represent
the validity- determining aspects of an argument
-Symbols:
the dot (.) is used
to represent any word that joins two conjuncts (ex. 'and', 'moreover',
'furthermore', 'but', 'yet', 'still', 'however', 'also', 'nevertheless',
'although')
the wedge (v) is
used to represent any word that joins two disjuncts, most frequently representing
the word “or” in an inclusive sense (that is, the inclusive
“or” asserts that at least one disjunct is true, while the
exclusive “or” asserts that at least one disjunct is true,
but not both are true) the
tilde (~) is used the represent the negation of any simple statement
(ex. p= lead is heavy; ~p= lead is not heavy)
the horseshoe (>) is used
to represent the equivalent of ~(p . ~q); it is used for any conditional
statement; for any conditional “if, then” statement
to be true, p>q, the negation of the conjunction of its antecedent
with the negation of its consequent, must be true also
Examples:
Truth Table for: p . q
p q
p . q
T T
T
T F
F
F T
F
F F
F
Truth Table for: p v q
p q
p v q
T T
T
T F
T
F T
T
F F
F
Truth Table** for: A > (B v C)
B
Therefore, A > ~C
A B
C B v C A
> (B v C) ~C
A > ~C
T T
T T
T
F F
T T
F T
T
T T
T F
T T
T
F T
T F
F F
F
T T
F T
T T
T
F T
F T
F T
T
T T
F F
T T
T
F T
F F
F F
T
T T
**This argument is shown to be INVALID by the above truth table
because, in the first row, both premises are true, but the conclusion
is false.
Either Atlanta wins their conference championship and Baltimore
wins their conference championship or Chicago wins the superbowl.
Translation: (A . B) v C
Truth table:
A B
C A . B
(A . B) v C
T T
T T
T
T T
F T
T
T F
T F
T
T F
F F
F
F T
T F
T
F T
F F
F
F F
T F
T
F F
F F
F
The truth of the statement can be assessed, but, because this is
not an argument, validity cannot be assessed.
Exercises
Determine the validity of the following arguments by first translating
them into symbols and then constructing truth tables.
* Mary has brown hair and James has black hair. Therefore,
Mary has brown hair.
* It is not the case that humans have ten toes and humans do not
have either blond hair or brown hair. Humans have blond hair. Therefore,
It is not the case that humans have ten toes and have brown hair.
* Mars is uninhabited by human life or is made of solidified soda.
Mars is uninhabited by human life. Therefore, mars is not made of
soda.
Solutions
* first argument is valid and the second two are invalid