106
Shortcut validity tests
A. Partial truth table method
B. Short truth table method
107
Shortcut validity tests
A. Partial truth table method
----->
REMEMBER example:
Premise 1 Bush will be elected only if
Gore doesn't carry Wisconsin but
Nader does.
Premise 2 Gore doesn't carry Wisconsin
but Nader does.
Conclusion Bush will be elected.
108
Symbolized in sentential logic:
P1 B  (~G  N)
P2 ~G  N
C B
- dictionary:
B  Bush will be elected.
G  Gore carries Wisconsin.
N  Nader carries Wisconsin.
Detailed argument form for this argument
instance:
P1 p  (~q  r)
P2 ~q  r
C p
109
STEP ONE write BASE COLUMNS:
(a) Write row listing all variables from
detailed argument form (in alphabetical
order).
(b) Write out all possible combinations of
truth values under variables. The value of
2n (where n is the number of variables)
gives you the number of possible
combinations of truth values.
Then alternate T, F under variable furthest
to the right; then alternate double the
number of Ts and Fs under next variable;
then again alternate double the number of
Ts and Fs under next variable; and so on
until columns under all variables are
filled.
110
----->
our example detailed argument form
has three variables, p, q, r, so the base
columns are:
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
111
STEP TWO:
(a) Write detailed argument form to the right
of the list of variables.
Our example detailed argument form is:
P1 p  (~q  r)
P2 ~q  r
C p
- so we have:
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
p  (~q  r), ~q  r / p
112
SO FAR, this is just the SAME as constructing
truth tables using full truth table method. But
at this point, things change:
(b) For each combination of truth values
(that is, for each line of the table) for ONLY
THE CONCLUSION, calculate truth values
for each operator symbol, going from less
complex subformulas to more complex
formulas until you calculate truth value for
the conclusion's major operator. Then
mark the column under the conclusion's
major operator.
113
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r p  (~q  r), ~q  r / p
T________________________________
F________________________________
T________________________________
F________________________________
T________________________________
F________________________________
T________________________________
F________________________________
(c) Now, consider ONLY the lines where
the conclusion is FALSE (for only these
lines can be counterexamples, that is,
instances where all the premises are true
and conclusion false). Calculate the truth
values for the PREMISES on these lines.
Now, check to see if there is a
counterexample. If there is a
counterexample, then the argument is
invalid; otherwise it is valid.
The basic idea behind the PARTIAL truth
table method: only write out the lines of the
truth table where a counterexample is
possible.
114
B. Short truth table method
The basic idea behind the SHORT truth table
method is to systematically search for
counterexamples, but without even the help of
a base column.
Our objective: to see if we can assign truth
values to the variables in the argument
form which:
(a) make the conclusion false, and
(b) make all the premises true
- if we CAN do this, we've found a
counterexample, and the argument form is
invalid
- if we CANNOT do this, then there is no
counterexample, and the argument form is
valid
115
----->
example:
[(p v q)  ~r], [(r  s) v t], [t  (w  s)] / (p  s)
STEP ONE:
Assign truth values to variables in the
argument form which make conclusion false.
NOTE: there may be MORE THAN ONE
WAY to do this (if there is you will have to
REPEAT this two step process for each way).
- with this example, there is only ONE way to
make conclusion false: we must assign p true
and s false
116
[(p v q)  ~r], [(r  s) v t], [t  (w  s)] / (p  s)
STEP TWO:
Now, using THESE assignments of truth
values for variables in the conclusion, and
assigning truth values to any ADDITIONAL
variables in the premises, see whether it's
possible to make all the premises true. (Be
sure that your assignment of truth values is
SYSTEMATIC.)
We've been able to assign truth values to the
variables in the argument form which:
(a) make the conclusion false, and
(b) make all the premises true
- so we've found a counterexample, and the
argument form is invalid
NOW, mark down the truth values of the
variables for the counterexample:
or
p
T
q
T
r
F
s
F
t
T
w
F
T
F
F
F
T
F
117
----->
another example:
(p  q), (~q  ~p) / (p  q)
STEP ONE:
Assign truth values to variables in the
argument form which make conclusion
false.
- with this example, there are TWO
ways of making the conclusion false, so
you have to go through the two step
process twice. We'll first assign p true
and q false.
118
(p  q), (~q  ~p) / (p  q)
STEP TWO:
Now, using THESE assignments of
truth values for variables in the
conclusion, and assigning truth values
to any ADDITIONAL variables in the
premises, see whether it's possible to
make all of the premises true.
Assigning p true and q false, we CANNOT
make all of the premises true. So we have not
yet found a counterexample.
But there's second way of making the
conclusion false: assign p false and q true.
119
SECOND PROCESS
(p  q), (~q  ~p) / (p  q)
STEP ONE:
Assign truth values to variables in the
argument form which make conclusion
false.
- this time assign p false and q true
120
(p  q), (~q  ~p) / (p  q)
STEP TWO:
Now, using THESE assignments of
truth values for variables in the
conclusion, and assigning truth values
to any ADDITIONAL variables in the
premises, see whether it's possible to
make all of the premises true.
This time we have been able to assign truth
values to the variables in the argument form
which:
(a) make the conclusion false, and
(b) make all the premises true
- so we've found a counterexample, and the
argument form is invalid
NOW, mark down the truth values of the
variables for the counterexample:
p
F
q
T
121
If you CANNOT assign truth values to the
variables which make the conclusion false
and all of the premises true, then there is
no counterexample, and the argument form
is valid.
BUT REMEMBER:
- there may be MORE THAN ONE
assignment of truth values that makes the
conclusion false
- EACH assignment of truth values that
makes the conclusion false COULD be a
counterexample
- so to prove that an argument form is
VALID you have to go through the two
step process for EACH assignment of truth
values that makes the conclusion false
122
YOU MUST KNOW THESE VERY WELL:
p
~p
T
F
F
T
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
F
p
T
T
F
F
q
T
F
T
F
pvq
T
T
T
F
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
T
123
Further applications of truth tables
A. Tautologies, contradictions, and
contingencies
B. Logical equivalence and logical
implication
C. Inconsistency and consistency
124
First, some background (ignoring, for the
moment, the difference between statement
instances and statement forms):
- tautologies, contradictions, and
contingencies are SINGLE STATEMENTS
- logical equivalence and logical implication
are (usually) defined as relations between
TWO STATEMENTS
- inconsistency and consistency are relations
among TWO OR MORE STATEMENTS
125
A. Tautologies, contradictions, and
contingencies
1. TAUTOLOGIES are statement forms for
single statements which are true for every
substitution instance. (Instances of these
forms are called tautologous instances.)
----->
example:
Take the following English statement
instance:
Either it is raining or it is not raining.
Symbolize this statement instance in
sentential logic:
- dictionary:
R  It is raining.
- we get
R v ~R
126
Now write a detailed statement form for this
statement instance:
p v ~p
And now construct a truth table for this
statement form:
p
T
F
p v ~p
AGAIN: Tautologies are statement forms for
single statements which are true for every
substitution instance. That means that the
column under the major operator is ALL T's.
Since truth tables list ALL POSSIBLE truth
values for the statement, there are NO
POSSIBLE instances that are FALSE.
127
2. CONTRADICTIONS are statement forms
for single statements which are false for every
substitution instance. (Instances of these
forms are called contradictory instances.)
----->
example:
Take the following English statement
instance:
It is raining and it is not raining.
Symbolize this statement in sentential logic:
- dictionary:
R  It is raining.
- we get
R  ~R
128
Now write a detailed statement form for this
statement instance:
p  ~p
And now construct a truth table for this
statement form
p
T
F
p  ~p
AGAIN: Contradictions are statement forms
for single statements which are false for every
substitution instance. That means that the
column under the major operator is ALL F's.
Since truth tables list ALL POSSIBLE truth
values for the statement, there are NO
POSSIBLE instances that are TRUE.
129
3. CONTINGENCIES are statement forms for
single statements which are true for some
substitution instances and false for other
substitution instances. (Instances of these
forms are called contingent instances.)
----->
example:
Take the following English statement
instance:
It is raining and IBM's stock went up.
Symbolize this statement in sentential logic:
- dictionary:
R  It is raining.
I  IBM's stock went up.
- we get
RI
130
Now write a detailed statement form for this
statement instance:
pq
And now construct a truth table for this
statement form:
p
T
T
F
F
q
T
F
T
F
p  q
AGAIN: Contingencies are statement forms
for single statements which are true for some
substitution instances and false for other
substitution instances. That means that the
column under the major operator is a mix of Ts
and Fs.
131
B. Logical equivalence and logical implication
1. Two (or more) statement forms are
LOGICALLY EQUIVALENT if and only if the
truth table columns under their major
operators are the same.
----->
example:
a. Take the following English statement
instance:
It is not the case that cigarette
manufacturers are either honest or socially
responsible.
Symbolize this statement instance in
sentential logic:
- dictionary
H  Cigarette manufacturers are honest.
S  Cigarette manufacturers are socially
responsible.
- we get:
~(H v S)
132
Now write a detailed statement form for this
statement instance:
~(p v q)
b. Take the following English statement
instance:
Cigarette manufacturers are not honest and
they are not socially responsible.
Symbolize this statement instance in
sentential logic:
- same dictionary
- we get:
~H  ~S
Now write a detailed statement form for this
statement instance:
~p  ~q
133
And now construct joint truth table for these
two statement forms:
p
T
T
F
F
q
T
F
T
F
~(p v q)
~p  ~q
Again: two (or more) statement forms are
logically equivalent if and only if the truth
table columns under their major operators are
the same.
Two statement forms are logically
equivalent if and only if the result of
joining them with a triple bar is a
tautology. Why?
134
2. Statement form 1 LOGICALLY IMPLIES
statement form 2 if and only if there is no row
in their joint truth table in which statement
form 1 comes out true and statement form 2
comes out false.
----->
p
T
T
F
F
example:
q
T
F
T
F
(1)
~p
F
F
T
T
(2)
~ (p  q)
F
T
T
F
T
F
T
F
Statement form 1 logically implies statement
form 2. (But statement form 2 does not
logically imply statement form 1.)
Two statement forms are logically
equivalent if and only if they logically
imply EACH OTHER. Why?
135
C. Inconsistency and consistency
1. A set of statement forms is
INCONSISTENT if and only if there is NO
ROW in their joint truth table in which they
all come out true at once.
----->
example:
Take the following English statement
instances:
a. Bill likes Al.
b. If Bill likes Al, then he likes Tipper.
c. Bill doesn't like Tipper.
Symbolize these statement instances:
- dictionary
A  Bill likes Al
T  Bill likes Tipper
- so we get:
a. A
b. A  T
c. ~T
136
Now write detailed statement forms for these
three statement instances:
a. p
b. p  q
c. ~q
And now construct joint truth table for these
three statement forms:
p
T
T
F
F
q
T
F
T
F
p
pq
~q
A set of statement forms is inconsistent if and
only if the conjunction of all the statement
forms is a contradiction. Why?
p
T
T
F
F
q
T
F
T
F
p  [(p  q)  ~q]
137
2. A set of statement forms is CONSISTENT if
and only if there is a row in their joint truth
table in which they all come out true at once.
So a set of statement forms is consistent if
and only if the conjunction of all the
statement forms is NOT a contradiction.
There is at least one row in the truth table
for this conjunction which comes out true
under the major operator.
138
Review of applications of truth tables:
We've used
truth tables with: to determine:
arguments
validity/invalidity
- invalid just in case
counterexample
- valid just in case no
counterexample
single
statements
tautology/contradiction/
contingency
- tautology just in case
column under major
operator is all Ts
- contradiction just in case
column under major
operator is all Fs
- contingency just in case
column under major
operator is mix of Ts and Fs
139
We've used
truth tables with: to determine:
pairs of
statements
logical equivalence/
logical implication
- statement forms are
logically equivalent just in
case same truth values under
major operator
- a statement form 1 logically
implies statement form 2
just in case there is no row
in the table where 1 is T and
2 is F
sets of 2 or more
statements
inconsistency/consistency
- statement forms are
inconsistent just in case
there is no row in table in
which they all come out true
- statements are consistent
just in case there is a row
in table in which they all
come out true
140
Consider:
If you negate a tautology, you get a _____?
If you negate a contradiction, you get a _____?
If you join two logically equivalent statement
forms with a triple bar you get a _____?
If you join a set of inconsistent statement
forms with dots, you get a _____?
141
Also consider:
For an ARGUMENT FORM: if the conditional
formed by taking
- the conjunction of the premises as the
antecedent, and
- the conclusion as the consequent
is a tautology, then the argument is valid.
Why?
p
T
T
F
F
q
T
F
T
F
p  q,
T
F
T
T
p
T
T
F
F
q
T
F
T
F
[(p  q)  ~q]  ~p
~q
F
T
F
T
/
~p
F
F
T
T