TruthfunctionAl
Valid ?
John’s young or he’s wise.
He’s not young.
So, he’s wise.
Truth-Functional Logic
John’s young or he’s wise.
He’s not young.
So, he’s wise.
He’s wise.
John is young He’s not
or he is wise. young.
Truth-Functional Sentence
Definition.* A truth-functional sentence is a
compound sentence whose truth value can
be determined from merely the truth values
of its simplest component sentences.
Example:
It squawks and it flys.
S&F
S&F
S
F
It squwaks.
It flys.
true
true
true
false
true
false
true
false
false
false
false
false
It squwaks and it flys.
Example of a Truth-Functional Sentence
S&F
S
F
It squwaks.
It flys.
true
true
true
false
true
false
true
false
false
false
false
false
It squwaks and it flys.
Which are Truth-Functional Sentences ?
No
a. It squawks before it flies.
No
b. It squawks.
No
c. It squawks because it flies.
Yes
d. It does not both squawk and fly.
Yes
e. It is not the case that it flies.
Sentence Constant
Upper-Case Roman Letter ( C, M, R, etc.)
Stands for an actual, given sentence.
Sentence Variable
Lower-Case Roman Letter ( p, q, r, etc.)
Stands for any sentence whatever.
John is young or he is wise.
He’s not young.
So, he’s wise.
Y or W
not Y____
W
p or q
not p____
q
She’s a yak or he’s an ox.
She’s not a yak.____
So, he’s an ox.
Y or O
not Y__
O
p or q
not p__
q
Is Every Argument Of This FormValid ?
Valid ?
p or q
not p_
q
A Formal Truth-Functional Language
Every sentence will be either:
1. Simple ( M, K, R, etc.) or
2. Compound as follows :
not M
Negation
L and R
Conjunction
R or K
Disjunction
Simple sentences.
Bert burped.
B
Wilma often laughed at the thought of Bert
burping during their wedding. L
Negations
Bert did not burp.
not B
It’s not the case that Wilma often laughed at the
thought of Bert burping during his speech.
not W
Truth-Functional Negation
Negation of a sentence reverses its truth value.
Roy runs. It’s not the case that Roy runs.
R
t
f
not R
f
t
p___not p
t
f
f
t
Valid Argument Forms for Negation
p
not not p
not not p
p
Example
It’s not the case that Roy doesn’t run.
Therefore, Roy runs.
not not R
R
Truth-Functional Conjunction
Maryland remained in the Union and Virginia
joined the Confederacy.
M and R
Form: p & q
A Conjunction Claims that Both are True
W
R
W&R
Wade wades. Roy roosts. Wade wades and Roy Roosts.
T
T
F
T
T
F
T
F
F
F
F
F
Valid Conjunctive Argument Forms
p&q
p
p&q
q
Not both p & q
p
not q
p
q
p&q
Not both p & q
q
not p
Truth-Functional Disjunction:
• True when either simple sentence (“disjunct”) is true.
• Claims that at least one of the disjuncts is true.
p
t
f
t
f
q p or q
t
t
t
t
t
f
t
f
Roy runs or Wade wades.
R W
t
f
t
f
t
t
f
f
R or W
t
t
t
f
Valid Disjunctive Argument Forms
p or q
not p
q
p or q
not q
p
p
p or q
Conditional (Hypothetical) Reasoning
•
•
•
•
“What if … ?” reasoning.
Fiction
Science
Common Sense ?
E
M
If the earth is round, then the mast will appear first.
antecedent sentence then ……
consequent sentence.
If ……
The Truth-Functional (“Minimal”) Conditional.
If you buy that car, then you’ll go broke.
If C then G.
Not both C and not G.
If she sings, then people will leave.
If S then L .
If p then q.
Not both S and not L.
Not ( p and not q ).
Truth Values for the Minimal Conditional
not q
p and not q
If p then q
not (p and not q)
p
q
t
t
f
f
t
f
t
f
f
t
t
f
t
t
f
f
f
t
f
t
False in only one situation (case): when antecedent is
true but consequent is false.
Valid Conditional Argument Forms
If p then q
p
If p then q
Not q
If p then q
If q then r
q
Not p
If p then r
Modes Ponens
Modes Tollens
Chain Argument
If p then not p
If p then both (q and not q)
Not p
Not p
Reductio Ad Absurdum
Truth Table for Modes Ponens:
If p then q
p
q
Premise
p
Conclusion
q
Premise
If p then q
1
t
t
t
2
f
t
3
t
f
t
f
4
f
f
t
Reductio ad Absurdum. (examples)
If Bob becomes the boss, then he won’t become the
boss (because everyone will quit). So, he won’t become
the boss.
If B then not B.
Not B
If there’s a greatest number ( “G”), then G+200 is less
than G (because G is the greatest) and G+200 is not less
than G (by simple arithmetic). So, there’s no greatest number.
If G, then both L and not L.
Not G
Reductio ad Absurdum. Form.
If p then (q and not q).
Not p
If p then not p.
Not p
Dilemma
Definition.
A dilemma is an argument
which claims that among the alternatives
presented, at least one must be
taken.
If p then q
If r then s
p or r
q or s
Truth Values for the Minimal Conditional
not q
p and not q
If p then q
not (p and not q)
p
q
t
t
f
f
t
f
t
f
f
t
t
f
t
t
f
f
f
t
f
t
False in only one situation (case): when antecedent is
true but consequent is false.
Converse and Contrapositive of the Truth-Functional Conditional
p
q
Not p Not q
t
t
f
f
t
t
t
f
t
t
f
t
f
t
t
f
f
t
f
t
f
f
f
t
t
t
t
t
If p then q
If q then p If not q then not p
Same values in each case (line).
The conditional and its converse are not related by implication
nor equivalence. (Though they are subcontraries.)
The conditional and its contrapositive are logically equivalent.
A Translation Rule for Conditionals
The sentence following the “if…” is
the antecedent, except when the “if…”
follows “only…”.
Examples:
A
H
She’s away if it’s a holiday.
She’s away only if it’s a holiday.
A
H
H
A
If it’s a holiday then she’s away.
If she’s away then it’s a holiday.
A
H
Further Examples for Translating Conditionals
M
F
It’s a mother only if it’s a female.
If M
then
F
It’s female if it’s a mother.
If
M then F
not H
W
He’s not happy if he’s wealthy.
If
W then not H
U
W
Only if he’s unhappy is he wealthy.
If
W
then U
Translating “…unless…”
p unless q
“says”
p if not q
Apply “if” Rule
If not q then p.
If not p then q.
(contrapositive)
Rule for translating
“unless” sentences:
Negate one of the two sentences,
put it in the antecedent position,
put the other in the consequent
position.
Example of Translating “…unless…”
He’s been married seven times! Why ?
He’s unhappy unless he’s married.
U unless M
U if not M
If not U then M.
If not M then U.
Biconditional
p if and only if q.
(p if q) and
(p only if q).
(If q then p) and
(If p then q).
Truth Values for the Biconditional
(p if q)
p
q
t
t
f
If q then p
&
(p only if q)
If p then q
P if and only if q
t
t
t
t
f
t
f
t
f
t
f
f
f
f
t
t
t
The biconditional is true when both truth values are the same,
otherwise it is false.