### Introduction to Truth-Functional/Propositional Logic: Deduction Part II

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All are….
None are….
Some are….
Some are not….
All human beings are mortal.
(All things identical with) LaRissa
is a human being.
Therefore, (all things identical
with) Larissa is mortal
All human beings are
mortal.
(All things identical with)
LaRissa is a human
being.
Therefore, (all things
identical with) LaRissa is
mortal.
All H are M.
L is H.
Therefore, L is M.
• Contraposition (switch sub and
pred terms & replace both with
complementary terms): All
things that are immortal are
things unidentical with Larry.
• Obversion (move horiz across
square & replace pred term
with complementary term): No
things identical with Larry are
things that are immortal. And
then…..
Therefore, (all things
identical with) Larry is
mortal
• Conversion (switch position of
sub and pred terms): No things
that are immortal are things
identical with Larry.
Proposition is a statement
with a truth value of true
or false.
Logic of sentences or
propositions or claims
(not categories).
We are in HUM 106 at this
moment.
We are in HUM 106 at this
moment, and we are
smoking salmon.
If we are in LRC 105, at this
moment, then we are
studying material related to
PHIL 1.
We are in HUM 106 at this
moment (P). Equivalent to P.
We are in HUM 106 at this
moment (P), and we are
smoking salmon (S).
Equivalent to P & S.
If we are in HUM 106 at this
moment (P), then we are
smoking salmon (S).
Equivalent to P → S.
Either we are in HUM 105
at this moment (Q), or we
are in HUM 106 (P).
Equivalent to P ˇ Q.
We are in HUM 106 at this
moment (P).
P
T
F
Negation (“it is not the
case that”) symbolized
by ~
Thus, ~P = “It is not
the case that we
are in HUM 106 at
this moment.”
P
T
F
~P
F
T
P
T
F
~P
F
T
To “interpret” the truth
table is to add content.
It is raining outside.
Andy says he is feeling
chipper today.
Alyssa works at Ace
Hardware.
Creates a compound
claim from two or
more simpler claims.
P
T
T
F
F
And, while, but, even
though, etc.
We are in HUM 106
at this moment (P),
and we are smoking
salmon (S).
P&S
S
T
F
T
F
Both conjuncts must be true for the
claims to be true.
P
T
T
F
F
S
T
F
T
F
P&S
T
F
F
F
Also creates a compound claim
from two or more simpler
claims.
P
T
T
F
F
Or.
Either we are in HUM 105 at
this moment (Q), or we are in
HUM 106 (P).
PˇQ
Q
T
F
T
F
A disjunction is false if and
only if both disjuncts are false.
P
Q
T
T
F
F
T
F
T
F
PˇQ
T
T
T
F
P
T
T
F
F
“If….then…..”
If we are in LRC 106 at
this moment (P), then we
are smoking salmon (S).
P→S
• P = the “antecedent.”
• S = the “consequent.”
S
T
F
T
F
A conditional is false if and only if
its antecedent is true and its
consequent is false.
P
T
T
F
F
S
T
F
T
F
P→S
T
F
T
T
1. If Quincy learns to symbolize, Paula will be amazed.
2. Paula will teach him if Quincy pays her a big fee.
3. Paula will teach him only if Quincy pays her a big fee.
4. Only in Paula helps him will Quincy pass the course.
5. Quincy will pass the course if and only if Paula helps him.
1.
If Parsons signs the papers then Quincy will go to jail, and Rachel
will file an appeal.
2.
If Parsons signs the papers, then Quincy will go to jail and Rachel
will file an appeal.
3.
If Parsons signs the papers and Quincy goes to jail, then Rachel
will file an appeal.
4.
Parsons signs the papers and if Quincy goes to jail Rachel will file
an appeal.
5.
If Parsons signs the papers then if Quincy goes to jail Rachel will
file an appeal.
6. If Parsons signs the papers Quincy goes to jail, and if
Rachel files an appeal Quincy goes to jail.
7. Quincy goes to jail if either Parsons signs the papers or
Rachel files an appeal.
8. Either Parsons signs the papers or, if Quincy goes to jail,
then Rachel will file an appeal.
9. If either Parsons signs the papers or Quincy goes to jail
then Rachel will file an appeal.
10. If Parsons signs the papers then either Quincy will go to
jail or Rachel will file an appeal.
If you have
satisfactorily
fulfilled the
requirements
described in the
course syllabus,
then you will earn a
passing grade.
What’s the symbolic
expression?
• P→S
What’s the truth table for P and S?
P
S
T
T
T
F
F
T
F
F
A conditional is false if and only if its
antecedent is true and its consequent is
false. So what’s the truth table for P → S?
P
T
T
F
F
S
T
F
T
F
P→S
T
F
T
T
P
T
T
F
F
If scientist do not find a cure for
baldness, John will always have a
shiny head.
~P → S.
To generate truth table, begin
with simple claims.
P=Scientists find a cure for
baldness.
S=John will always have a shiny
head.
S
T
F
T
F
Now add ~P
P
T
T
F
F
S
T
F
T
F
~P
Now apply the rule of
conditionals to generate
~P → S: A conditional is
false if and only its
antecedent is true and its
consequent is false.
P
T
T
F
F
S
T
F
T
F
~P
F
F
T
T
~P → S
P
T
T
F
F
S
T
F
T
F
~P
F
F
T
T
~P → S
T
T
T
F
If the McKay tract is approved, then
The truth table must list all
the number of homeless camping in
possible truth values for P, Q,
the forest will increase and the natural and R.
environment will be damaged.
Building a truth table is
• P → (Q & R) vs. (P → Q) & R
getting tough!
Simple claims:
• P = The McKay tract is approved.
• Q = The number of homeless
camping in the forest will increase.
• R = The natural environment will be
damaged.
Claims with 1 letter have two
possible combinations of truth
values.
Claims with 2 letters have four
possible combinations of truth
values.
So….every time we add a letter,
the number of T and F
combinations doubles and so,
therefore, the number of rows in
the truth table doubles.
So….r = 2n
Then, alternate T’s and F’s in the
right-most row until you have
the correct number of rows.
Then, alternate pairs of T’s and
F’s in the next row to the left.
Then, alternate sets of four T’s
and four F’s in the next row to
the left, and so on and so on.
The left-most row will always be
half T’s and half F’s.
If the McKay tract is approved, then the number of
homeless camping in the forest will increase and the
natural environment will be damaged.
P → (Q & R)
If r = 2n, then how many
rows will the truth table
have?
1st: Alternate T’s and F’s in
the right-most row until you
have the correct number of
rows.
P
Q
R
T
F
T
F
T
F
T
F
Then, alternate pairs of T’s
and F’s in the next row to
the left.
Then, alternate sets of four
T’s and four F’s in the next
row to the left, and so on
and so on.
P
Q
R
T
F
T
F
T
F
T
F
Both conjuncts
must be true for
the claims to be
true.
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
Q&R
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
Q & R•
T
F
F
F
T
F
F
F
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
Q&R
T
F
F
F
T
F
F
F
P → (Q&R)
Now apply the
rule of
conditionals: A
conditional is
false if and only
if its antecedent
is true and its
consequent is
false.
If the McKay tract is approved, then the number of
homeless camping in the forest will increase and the
natural environment will be damaged.
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
Q&R
T
F
F
F
T
F
F
F
P → (Q&R)
T
F
F
F
T
T
T
T
Now apply the
rule of
conditionals: A
conditional is
false if and only
its antecedent is
true and its
consequent is
false.
Generates all possible
combinations of truth values
for statements and
combinations of statements.
Allows you to test for validity
with certainty.
Validity: It is impossible for
the conclusion to be false
and premises to be true.
An invalid argument is an
illogical argument.
An illogical argument is not
a good argument.
Use a truth table to determine the validity or invalidity of this argument:
“If building the bookshelf requires a screwdriver then I will
not be able to build it. After reading the directions I see that a
screwdriver is needed. So, I can’t build it.”
First, translate this argument into
standard form
Now into symbols
If S then not B
S
_
Not B
S → ~B
S
_
~B
27
S → ~B
S
_
~B
Now, build a truth table.
We have two claim variables, “S” and “~B”
that each need a column.
We need a column for each
premise and the conclusion.
S
T
T
F
F
~B
T
F
T
F
S → ~B ~B
28
S → ~B
S
_
~B
Now, fill in the truth values for the first
premise based on the rule of the conditional:
A conditional is false if and only if its
antecedent is true and its consequent is false.
We’re done. Our truth table now
tells us whether or not the
argument is valid.
What do you think?
S
T
T
F
F
~B
T
F
T
F
S → ~B
T
F
T
T
29
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