Today’s Topics
Review Logical Implication & Truth Table
Tests for Validity
 Truth Value Analysis
 Short Form Validity Tests
 Consistency and validity (again)
 Substitution instances (again)

Logical Implication
One statement logically implies another if,
but only if, whenever the first is true, the
second is true as well
 If a statement, S1, implies S2 then the
conditional (S1  S2) will be a tautology
 Implication is the validity of the conditional.

Determining whether S1
Logically Implies S2
Construct a truth table with columns for S1
and S2.
 If there is no row in which S1 is true and S2
false, then S1 implies S2.
 If there is no row in which S2 is true and S1
is false, then S2 implies S1.

NOTE: Logical Equivalence is
Mutual Implication

Equivalence is the validity of the biconditional
Truth Table Tests for Validity
(and Non-validity)
Construct a column for each premise in the
argument
 Construct a column for the conclusion
 Examine each row of the truth table. Is
there a row in which all the premises are
true and the conclusion is false. If so, the
argument is non-valid. If not, then the
argument is valid.

When using a truth table test for
validity, one is looking for an
Invalidating Row (or a CounterExample Row). Failure to find an
invalidating row shows that the
argument is valid.
Test the following argument for
validity: P ▼Q, P, ~Q
Testing for Validity
P
Q
PQ
P
 ~Q
T
T
T
T

T

T
T
T

T
T






T
Verdict:
NOT VALID, row 1
Test the following argument for
validity:
(P ● Q)  P, ~P, Q P
Testing for Validity
P
T
T
F
F
Q
T
F
T
F
(P  Q)  P
T T
F T
F T
F T
~P
F
F
T
T
Q
T
F
T
F
P
T
T
F
F
Verdict: NON VALID! In ROW 3 all the
premises are true and the false conclusion

Test the following argument for
validity:
(P  Q), ~ Q ~P
Testing for Validity
P
Q
PQ
~Q
 ~P
T
T
T


T

T
T


T
T

T



T
T
Verdict: VALID, no invalidating rows
Truth Value Analysis




Sometimes we can know the truth value of a
compound statement without knowing the truth
values of each component simple statement.
Sometimes we don’t need a full truth table.
Since truth tables get very large very quickly (e.g.,
8 variables produces 256 rows) this is good news.
Download the Handout on Truth Value Analysis
and read it.
Examples




We know that a conditional with a false antecedent
is true, so, if ‘P’ is false, then
P  (Q v (R  S)) is TRUE, no matter what the
truth values of ‘Q,’ ‘R,’ and ‘S’ happen to be!
Similarly, since a conjunction with a false
conjunct is false, if any one of ‘P,’ ‘Q,’ ‘R,’ or ‘S’
is false, then
P  (Q  (R  S)) is FALSE no matter what the
truth values of the others.
Rules for truth value analysis
A conjunction with a false conjunct is false
 A disjunction with a true disjunct is true
 A conditional with a false antecedent or a
true consequent is true
 A biconditional with a true component has
the same truth value as the other component
 A biconditional with a false component has
a truth value opposite the other component

Try a few exercises
Download the Handout Truth Value
Analysis Exercises and determine whether
each formula is true, false or undecided give
the assumptions. I call this a resolution of
the truth value of a statement.
 Discuss your answers via the bulletin board.

Short Form Validity Tests (Truth
Value Analysis of Validity)
When using a truth table test for
validity, one is looking for an
Invalidating Row (or a CounterExample Row). Failure to find an
invalidating row shows that the
argument is valid.
In an invalidating row, the
conclusion must be false:
We can skip constructing ANY rows in
which the conclusion is true.
 Assume the conclusion to be false, and
assign truth values to the simple statements
in it accordingly.
 Using those assignments, try to make all the
premises true.

 If
you succeed, if it is possible to
make all the premises true while
the conclusion is false, the
argument is non-valid.
 If you fail, if it is impossible to
make the premises true after
making the conclusion false, the
argument is valid.
If making the conclusion false
forces at least one premise to be
false, then the argument is valid.
NOTE: If more than one
assignment of truth values makes
the conclusion false, you MUST
test each assignment. ANY
combination of truth values that
results in true premises and a
false conclusion invalidates the
argument
NOTE: This method is most
valuable when the conclusion is
falsified by only one or two
combinations of truth values.
Hence, it is most valuable when
the conclusion is either a
conditional or a disjunction.
Try a few on your own
Download the Handout Truth Value
Analysis Validity Tests and read the
explanation. Now read it again.
 Now work the problems and discuss your
answers via the bulletin board

Testing for Consistency
A set of statements is consistent if, but only
if, it is possible for all of the members of the
set to be true.
 If there is ANY row in a truth table for a set
of statements in which each of the
statements is true, then the set is consistent.
 If there is NO such row, then the set is
inconsistent.

Consistency and Validity (Again)
Consistency is closely related to validity
 If the premises of a argument are consistent
with the negation of the conclusion, then the
argument is non-valid.
 If the premises of a argument are
inconsistent with the negation of the
conclusion, then the argument is valid.

Statement Forms and
Substitution Instances
A statement form is a mix of sentential
variables and logical operators (which
remain constant)
 Every WFF’s is a substitution instances of
a basic statement form
 WFF’s are also substitution instances of
other (non-basic) statement forms

Substitution Instance
WFF Fis a substitution
instance of the statement form Yif, but
only if, Fcan be obtained by
replacing each sentential variable in Y
with a WFF, using the same WFF for
the same sentential variable
throughout.
 A compound
For example:

~(~A  B) is a substitution instance of
p, ~p, ~(p  q), and ~(~p  q)
 However, while ‘~~A’ is a substitution
instance of ‘~~p,’ ‘A’ is not, even
though ‘A’ and ‘~~A’ are logically
equivalent

Logical Form and Logical
Equivalence are not the same
 Understanding
the difference between
sentences and sentence forms and
between variables and constants is
crucial to understanding logic
Variables and Constants
In statement forms, the lower case letters are
sentential variables, they stand for complete
statements but are not themselves statements
 The logical operators in statement forms are
constants, they do not change in the instances of
the form
 Every substitution instance of a statement form
has the same dominant operator as the form

Argument Forms and
Substitution Instances
Each and every legitimate use of a rule of
inference or equivalence involves a
substitution instance (or instances) of the
statement form(s) that occur in the rule
 A rule can be applies only to substitution
instances of the forms that occur in the rule

Let’s try to determine which
WFFs are instances of which
statement forms
For each statement form in the left hand
column, determine whether or not each
WFF in the right hand column is an instance
of it.
 Discuss your answers, questions on the
bulletin board.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1.
2.
3.
p
~p
pvq
4.
5.
pq
~(p  q)
6.
7.
~p  q
~p  (q v r)
8.
9.
10.
11.
(p v q)  r
pq
~(p  q)
~p ( q v r)
A.
~[(P  Q)  R]
B. ~(Q v R)  ~(R  S)
Key Ideas
 Logical
implication & truth table tests
Truth Value Analysis shortcuts constructing
full truth tables by ignoring rows that could
not be invalidating rows.
 Testing for consistency, using a consistency
test to test for validity
 Constants and variables in statement forms

Thus endeth the first unit


Download the Sample Exam for Sample Exam #
1. Take the exam, give yourself 50 minutes. Early
Wednesday I will post a key to the sample exam.
We can have a review for the exam via the bulletin
board.
Honor system, no collaborating on the exam (and,
since the person you cheat off of might be more
clueless than you, it REALLY isn’t a good idea in
logic).