Logical Inferences and Rules of Inference
Logical Inferences:
A set of premises accompanied by a suggested conclusion
regardless of whether or not the conclusion is a logical
consequence of the premises.
Hence it may be valid inference or faulty inference.
Inference is written as
(conjunction of premises)
→
(conclusion)
Inference is
valid
if the implication is
tautology
otherwise
invalid or faulty inference or fallacy
Tautology
valid inference
Not a Tautology
invalid inference (faulty)
Rules of Inference – valid
:
There are 4 fundamental rules
Fundamental rule 1
:
If the statement in
P
is assumed as
True
, and
the statement
P →Q
is accepted as
true
, then we must accept
Q
as
True. (Modus Ponens Rule)
Symbolically
P
P
→Q
therefore Q
Fundamental rule 2
:
Whenever two implications
P →Q
and
Q →R
are accepted as true then we must accept the implication
P →R
as true
(Hypothetical Syllogism or Transitive Rule)
Symbolically
P
→Q
Q →R
therefore P → R
Hypothesis / premises
Conclusion
Hypothesis / premises
Conclusion
Fundamental rule 3
:
DeMorgan’s Law
~( P V Q )
=
~P ^ ~Q
~( P ^ Q )
=
~P V ~Q
Fundamental rule 4
:
Law of Contra positive
P
→ Q
=
~ Q → ~ P
Rules of Inference – Invalid
Logical inference is
invalid
if the implication is not a
tautology
Also called as
faulty inference
or
fallacy
Inference is written as
(conjunction of premises)
→
(conclusion)
Tautology
valid
inference
Not a Tautology
Invalid
inference
(faulty)
Rule 1 (Fallacy 1)
The fallacy of denying the antecedent takes the form
P
→
Q
~P
therefore ~Q
Rule 3 (Fallacy 3)
Rule 2 (Fallacy 2)
Exercise Problems:
Determine the following arguments are valid or
invalid
1.
p → q
2. r → s
3. r → s
q → r
~s
p → q
r → s
−−−−−−
r V p
−−−−−−
~r
−−−−−−
p → s
s V q
Determine the following arguments are valid or
invalid
4. p → (r → s)
5. ~ r→(s → ~t)
6. p
~r → ~p
~r V w
p → q
p
~p → s
q → r
−−−−−−−
~w
−−−−−−−
s
−−−−−−−
r
t → p
Determine the following arguments are valid or
invalid
7. ~p
8. (p Ʌ q) → ~t
p → q
w V r
q → r
w → p
−−−−−−−
r → q
~r
−−−−−−−−−−−−
(w V r) → ~t
Determine the following arguments are valid or invalid
9.
If Tallahassee is not in Florida, then golf balls are not
sold in Chicago.
Golf balls are not sold in Chicago.
Hence, Tallahassee is in Florida.
10.
If a baby is hungry, then the baby cries.
If the baby is not mad, then he does not cry.
If a baby is mad, then he has a red face.
Therefore, if a baby is hungry, then he has a red face.
Determine the following arguments are valid or
invalid
11.
If Nixon is not reelected, then Tulsa will lose it air base.
Nixon will be re-elected iff Tulsa will vote for him.
If Tulsa keeps its air base, Nixon will be re-elected.
Therefore, Nixon will be reelected
.
Fill the blanks
12.
If today is Thursday, 10days from now will be
Monday.
Today is Thursday.
Hence,________________
13.
If today is Sunday, then I will go to church.
______________________
Therefore, I will go to church.
Fill in the blanks for conformity with transitive
rule.
14. Triangle ABC is equilateral implies triangle
ABC is equiangular.
Triangle ABC is equiangular implies angle A=60.
Hence, _______________________
Determine whether the argument is valid or
invalid
15. If today is David’s Birthday, then today is
January 24
th
.
Today is January 24
th
.
Hence, today is David’s Birthday
Use a contradiction argument to verify the
following valid inferences.
16. q → t
17. ~p → (q →~w)
s → r
~s → q
q V s
~t
________
~p V t
t V r
_______
w → s
Use contrapositive argument to verify the
following valid inference
18.
w → (r →s)
____________
(w Ʌ r) → s
Logical inferences involve drawing conclusions from premises, which can either be valid or invalid based on the rules of inference. This includes Modus Ponens, Hypothetical Syllogism, DeMorgan's Law, and Law of Contrapositive. Invalid inferences result in fallacies like denying the antecedent. Exercise problems help in testing the validity of arguments using these rules.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Logical Inferences: A set of premises accompanied by a suggested conclusion regardless of whether or not the conclusion is a logical consequence of the premises. Hence it may be valid inference or faulty inference. Inference is written as (conjunction of premises) (conclusion) Not a Tautology Tautology valid inference invalid inference (faulty) Inference is valid if the implication is tautology otherwise invalid or faulty inference or fallacy
Rules of Inference valid: There are 4 fundamental rules Fundamental rule 1: If the statement in P is assumed as True, and the statement P Q is accepted as true, then we must accept Q as True. (Modus Ponens Rule) Symbolically P P Q Hypothesis / premises Fundamental rule 2: Whenever two implications P Q and Q R are accepted as true then we must accept the implication P R as true (Hypothetical Syllogism or Transitive Rule) Symbolically P Q Q R therefore P R therefore Q Conclusion Hypothesis / premises Conclusion
Fundamental rule 3: DeMorgans Law ~( P V Q ) ~( P ^ Q ) = = ~P ^ ~Q ~P V ~Q Fundamental rule 4: Law of Contra positive P Q = ~ Q ~ P
Rules of Inference Invalid Logical inference is invalid if the implication is not a tautology Also called as faulty inference or fallacy Inference is written as (conjunction of premises) (conclusion) Tautology Not a Tautology valid inference Invalid inference (faulty)
Rule 2 (Fallacy 2) The fallacy of denying the antecedent takes the form P Q ~P therefore ~Q Rule 3 (Fallacy 3)
Exercise Problems: Determine the following arguments are valid or invalid 1. p q q r r s p s 2. r s ~s ~r 3. r s p q r V p s V q
Determine the following arguments are valid or invalid 4. p (r s) ~r ~p p s 5. ~ r (s ~t) 6. p ~r V w ~p s ~w t p p q q r r
Determine the following arguments are valid or invalid 7. ~p 8. (p q) ~t p q w V r q r w p r q ~r (w V r) ~t
Determine the following arguments are valid or invalid 9. If Tallahassee is not in Florida, then golf balls are not sold in Chicago. Golf balls are not sold in Chicago. Hence, Tallahassee is in Florida. 10. If a baby is hungry, then the baby cries. If the baby is not mad, then he does not cry. If a baby is mad, then he has a red face. Therefore, if a baby is hungry, then he has a red face.
Determine the following arguments are valid or invalid 11. If Nixon is not reelected, then Tulsa will lose it air base. Nixon will be re-elected iff Tulsa will vote for him. If Tulsa keeps its air base, Nixon will be re-elected. Therefore, Nixon will be reelected.
Fill the blanks 12. If today is Thursday, 10days from now will be Monday. Today is Thursday. Hence,________________ 13. If today is Sunday, then I will go to church. ______________________ Therefore, I will go to church.
Fill in the blanks for conformity with transitive rule. 14. Triangle ABC is equilateral implies triangle ABC is equiangular. Triangle ABC is equiangular implies angle A=60. Hence, _______________________
Determine whether the argument is valid or invalid 15. If today is David s Birthday, then today is January 24th. Today is January 24th. Hence, today is David s Birthday
Use a contradiction argument to verify the following valid inferences. 16. q t s r q V s ________ t V r 17. ~p (q ~w) ~s q ~t ~p V t _______ w s
Use contrapositive argument to verify the following valid inference 18. w (r s) ____________ (w r) s
