Laws of Logic and Logical Reasoning
Section 3.3
Using Laws of
Logic
Using contrapositives
The negation of a
hypothesis
or of a conclusion is
formed by denying the original hypothesis or
conclusion.
Statement Symbol Negation Symbol
The weather is good. P The weather is not good. ~p
I will go swimming. q I will not go swimming. ~q
The
inverse
of the conditional statement
p
q is
~p
~q.
The
contrapositive
of the conditional statement p
q is ~q
~p.
Examples:
Original Statement:
p
q
If Polly says “Hello”, then Paul says “Hello.”
Hypothesis, p
conclusion, q
Converse:
q
p
If Paul says “Hello’” then Polly says “Hello.”
Inverse:
~p
~q
If Polly does not say “Hello’” then Paul does not say
“Hello.”
Contrapositive:
~q
~p
If Paul does not say “Hello’” then Polly does not say
“Hello.”
Ex:
Statement: Tomorrow is Friday, if today is
Thursday
Conditional
: If today is Thursday, then
tomorrow is Friday (True)
Converse
: If tomorrow is Friday, then
today is Thursday. (True)
Inverse
: If today is not Thursday, then
tomorrow is not Friday. (True)
Contrapositive
: If tomorrow is not Friday,
then today is not Thursday. (True)
Ex:
Statement: A figure is a parallelogram if it is a
square
Conditional:
If a figure is a square, then it is a
parallelogram.(True)
Converse:
If a figure is a parallelogram, then it
is a square.(False)
Inverse:
If a figure is not a square, then it is not
a parallelogram.(False)
Contrapositive
: If a figure is not a
parallelogram, then it is not a square.(True)
Summary
**
If the original statement is TRUE, the contrapositive is TRUE.
If the original statement is FALSE, the contrapositive is FALSE.
They are said to be
logically equivalent
.
Laws of Logical Reasoning
Law of Syllogism (like transitive property)
“If p then q, if q then r therefore if p then r”
p
q
q
r
Therefore,
p
r
Ex:
If today is Tuesday, then I have gym.
If I have gym, then I wear my sneakers.
Conclusion using law of syllogism:
If today is Tuesday, then I wear my sneakers.
Laws cont.
Law of Detachment
(ordering
steps to reach conclusion)
p
q
P is true
Therefore, q is true
Ex: Given <1 = 50°
A) if <1= 50°, then <2= 40°
B) if <3= 40°, then <4= 140°
C) if <4= 140°, then <5= 140°
D) <1 = 50°
E) if <2= 40°, then <3= 40°
Solution: order of steps D,A,E,B,C
Conclusion using law of detachment: <5 = 140°
Laws of logic play a crucial role in reasoning and making deductions. This comprehensive guide explains the use of contrapositives, examples of conditional statements, and the significance of laws like the Law of Syllogism. Understanding these principles helps in effectively analyzing statements and drawing logical conclusions. The concepts of contrapositives, inverse, converse, and logical equivalences are elucidated, giving insight into how logical reasoning operates.
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Presentation Transcript
Section 3.3 Using Laws of Logic
Using contrapositives The negation of a hypothesis or of a conclusion is formed by denying the original hypothesis or conclusion. Statement Symbol Negation Symbol The weather is good. P The weather is not good. ~p I will go swimming. q I will not go swimming. ~q The inverse of the conditional statement p q is ~p ~q. The contrapositive of the conditional statement p q is ~q ~p.
Examples: Original Statement: p If Polly says Hello , then Paul says Hello. q Hypothesis, p conclusion, q Converse: q If Paul says Hello then Polly says Hello. Inverse: ~p ~q If Polly does not say Hello then Paul does not say Hello. Contrapositive: ~q If Paul does not say Hello then Polly does not say Hello. p ~p
Ex: Statement: Tomorrow is Friday, if today is Thursday Conditional: If today is Thursday, then tomorrow is Friday (True) Converse: If tomorrow is Friday, then today is Thursday. (True) Inverse: If today is not Thursday, then tomorrow is not Friday. (True) Contrapositive: If tomorrow is not Friday, then today is not Thursday. (True)
Ex: Statement: A figure is a parallelogram if it is a square Conditional: If a figure is a square, then it is a parallelogram.(True) Converse: If a figure is a parallelogram, then it is a square.(False) Inverse: If a figure is not a square, then it is not a parallelogram.(False) Contrapositive: If a figure is not a parallelogram, then it is not a square.(True)
Summary Statement If p, then q. Converse If q, then p. Inverse If not p, then not q. Contrapositive If not q, then not p. An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement. **If the original statement is TRUE, the contrapositive is TRUE. If the original statement is FALSE, the contrapositive is FALSE. They are said to be logically equivalent.
Laws of Logical Reasoning Law of Syllogism (like transitive property) If p then q, if q then r therefore if p then r p q q r Therefore, p r Ex: If today is Tuesday, then I have gym. If I have gym, then I wear my sneakers. Conclusion using law of syllogism: If today is Tuesday, then I wear my sneakers.
Laws cont. Law of Detachment(orderingsteps to reach conclusion) p q P is true Therefore, q is true Ex: Given <1 = 50 A) if <1= 50 , then <2= 40 B) if <3= 40 , then <4= 140 C) if <4= 140 , then <5= 140 D) <1 = 50 E) if <2= 40 , then <3= 40 Solution: order of steps D,A,E,B,C Conclusion using law of detachment: <5 = 140
