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Logic Unit 4: Semantics
©
Niko Scharer
1
UNIT 4
SEMANTICS:
FUN WITH TRUTHTABLES
4.3 EG1
~R
∨
S
↓
R
S
~
R
∨
S
T
T
F
T
T
T
T
F
F
T
F
F
←
F
T
T
F
T
T
F
F
T
F
T
F
A contingent sentence.
~(R
∨
S)
↓
R
S
~
( R
∨
S )
T
T
F
T
T
T
←
T
F
F
T
T
F
←
F
T
F
F
T
T
←
F
F
T
F
F
F
A contingent sentence.
4.3
EG2
Let’s do a truthtable for the sentence:
(Q
∨
~R)
∧
~ (P
→
Q)
Since there are 3 atomic sentences, there will be 2
3
possible TVA’s.
That’s 8 rows.
Truthtable for:
( Q
∨
~ R )
∧
~
(P
→
Q )
*
↓
*
P
Q
R
( Q
∨
~
R)
∧
~
( P
→
Q )
T
T
T
T
T
F
T
F
F
T
T
T
T
T
F
T
T
T
F
F
F
T
T
T
T
F
T
F
F
F
T
F
T
T
F
F
T
F
F
F
T
T
F
T
T
T
F
F
←
F
T
T
T
T
F
T
F
F
F
T
T
F
T
F
T
T
T
F
F
F
F
T
T
F
F
T
F
F
F
T
F
F
F
T
F
F
F
F
F
T
T
F
F
F
F
T
F
Not a contradiction.
It’s contingent, as we can see from the fourth row.
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View Full Document Logic Unit 4: Semantics
©
Niko Scharer
2
4.4 EG1:
Let’s try it out.
Are the following sentences tautologous, contradictory or contingent?
a)
~P
∨
Q
↔
( P
→
Q )
: removed parentheses for informal notation
TAUTOLOGY
*
↓
*
P
Q
( ~
P
∨
Q )
↔
(P
→
Q )
T
T
F
T
T
T
T
T
T
T
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
T
T
F
T
T
F
F
T
F
T
F
T
F
T
F
b)
~ ( ~ P
∨
~
Q)
∧
(Q
→
~ P)
CONTRADICTION
*
↓
*
P
Q
~
( ~
P
∨
~
Q )
∧
( Q
→
~
P )
T
T
T
F
T
F
F
T
F
T
F
F
T
T
F
F
F
T
T
T
F
F
F
T
F
T
F
T
F
T
F
T
F
T
F
T
T
T
F
F
F
F
T
F
T
T
F
F
F
T
T
F
c)
~ (P
↔
(P
→
Q))
Contingent.
↓
P
Q
~
( P
↔
( P
→
Q ))
T
T
F
T
T
T
T
T
T
F
T
T
F
T
F
F
F
T
T
F
F
F
T
T
F
F
T
F
F
F
T
F
4.4 EG2
a)
Are these sentences equivalent?
YES.
(
P
∨
~Q)
( Q
→
P )
↓
↓
P
Q
(P
∨
~
Q)
(Q
→
P)
T
T
T
T
F
T
T
T
T
T
F
T
T
T
F
F
T
T
F
T
F
F
F
T
T
F
F
F
F
F
T
T
F
F
T
F
Logic Unit 4: Semantics
©
Niko Scharer
3
b)
Are these sentences consistent?
yes
~
(
P
∨
Q)
( P
↔
Q )
P
Q
~
(P
∨
Q)
(P
↔
Q)
T
T
F
T
T
T
T
T
T
T
F
F
T
T
F
T
F
F
F
T
F
F
T
T
F
F
T
F
F
T
F
F
F
F
T
F
←
c)
Are these sentences consistent, equivalent or neither?
neither.
(
~
P
∧
Q)
( Q
→
P )
NEITHER, INCONSISTENT
↓
↓
P
Q
(~
P
∧
Q)
(Q
→
P)
T
T
F
T
F
T
T
T
T
T
F
F
T
F
F
F
T
T
F
T
T
F
T
T
T
F
F
F
F
T
F
F
F
F
T
F
4.4 EG3:
Is this argument valid?
(P
∧
~Q)
∨
R .
~ R
∨
Q.
∴
~P
→
Q
Does { ((P
∧
~Q)
∨
R ),
(~ R
∨
Q)
} tautologically imply (~P
→
Q)?
VALID , TAUTOLOGICAL IMPLICATION
↓
↓
↓
P
Q
R
((P
∧
~
Q)
∨
R)
∧
(~
R
∨
Q)
→
(~
P
→
Q)
T
T
T
T
F
F
T
T
T
T
F
T
T
T
T
F
T
T
T
Y
T
T
F
T
F
F
T
F
F
F
T
F
T
T
T
F
T
T
T
T
F
T
T
T
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
F
T
T
T
F
T
F
T
T
F
T
F
T
F
T
T
F
Y
F
T
T
F
F
F
T
T
T
T
F
T
T
T
T
T
F
T
T
Y
F
T
F
F
F
F
T
F
F
F
T
F
T
T
T
T
F
T
T
F
F
T
F
F
T
F
T
T
F
F
T
F
F
T
T
F
F
F
F
F
F
F
F
T
F
F
F
F
T
F
T
F
T
T
F
F
F
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View Full DocumentLogic Unit 4: Semantics
©
Niko Scharer
4
4.4 E1:
Construct a full truthtable for each of the following sentences.
Determine whether
each sentence is a tautology, a contradiction or a contingent sentence.
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This note was uploaded on 02/01/2012 for the course PHL PHL245 taught by Professor Scharer during the Winter '11 term at University of Toronto Toronto.
 Winter '11
 Scharer

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