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1. METHODS OF
DEDUCTION

2. METHODS OF DEDUCTION
Formal Proof of Validity
The Elementary Valid Argument Forms
Formal Proofs of Validity Exhibited
Constructing Formal Proofs of Validity
Constructing More Extended Formal Proofs
Expanding the Rules of Inference:
Replacement Rules
The System of Natural Deduction

3. NATURAL DEDUCTION
 Method of proving the validity of a
deductive argument
 Rules of inference

4. RULES OF INFERENCE
 Permit valid inferences from statements assumed
as premises. 23 rules of inference:
a) 9 elementary valid argument forms
b) 10 logical equivalences
c) 4 rules governing instantiation and generalization

5. FORMAL PROOF OF
VALIDITY
 Sequence of statements:
a) Premise; or
b) Preceding statements by an elementary
valid argument or logical equivalence

6. I. FORMAL PROOF
OF VALIDITY
 The last statement is the conclusion of
argument- validity is proved

7. EXAMPLE
If Anderson was nominated, then she went to
Boston.
If she went to Boston, then she campaigned there.
If she campaigned there, she met Douglas.
Anderson did not meet Douglas.
Either Anderson was nominated or someone more
eligible was selected.
Therefore, someone more eligible was selected
And ● Or V If-then ‫כ‬ Not ~ Conclusion ∴

8. If Anderson was nominated, then
she went to Boston.
If she went to Boston, then she
campaigned there.
If she campaigned there, she met
Douglas.
Anderson did not meet Douglas.
Either Anderson was nominated or
someone more eligible was selected.
Therefore, someone more eligible
was selected
A ‫כ‬ B
B ‫כ‬ C
C ‫כ‬ D
~D
A v E
∴ 𝐄

9. II. 9 ELEMENTARY
VALID ARGUMENT
FORMS
 Rules of Inference
 Used in constructing a valid
proof of validity

10. 9 ELEMENTARY VALID
ARGUMENT FORMS
1. Modus Ponens (MP)
1st Premise: p ‫כ‬ q
2nd Premise: p
Conclusion: ∴ q

11. 9 ELEMENTARY VALID
ARGUMENT FORMS
2. Modus Tollens (MT)
1st Premise: p ‫כ‬ q
2nd Premise: ~ q
Conclusion: ∴ ~ p

12. 9 ELEMENTARY VALID
ARGUMENT FORMS
3. Hypothetical Syllogism (HS)
1st Premise: p ‫כ‬ q
2nd Premise: q ‫כ‬ r
Conclusion: ∴ p ‫כ‬ r

13. 9 ELEMENTARY VALID
ARGUMENT FORMS
4. Disjunctive Syllogism (DS)
1st Premise: p v q
2nd Premise: ~ p
Conclusion: ∴ q

14. 9 ELEMENTARY VALID
ARGUMENT FORMS
5. Constructive Dilemma (CD)
1st Premise: (p ‫כ‬ q ) ● ( r ‫כ‬ s )
2nd Premise: p v r
Conclusion: ∴ q v s

15. 5. CONSTRUCTIVE DILEMMA
1st Premise: If we send more troops to Afghanistan,
the population will hate us even more as occupiers
of their country; whereas, if we do not send more
troops to Afghanistan, the Taliban will most likely
take over the country.
2nd Premise: We either send more troops or
we do not.
Conclusion: So, either the population will hate us
even more as occupiers of their country or the
Taliban will most likely take over the country.

16. If we send more troops to
Afghanistan, the population
will hate us even more as
occupiers of their country;
whereas,
if we do not send more
troops to Afghanistan, the
Taliban will most likely
take over the country.
We either send more
troops or we do not.
So, either the population will hate
us even more as occupiers of their
country or the Taliban will most
likely take over the country.
S ‫כ‬ H ●
~S ‫כ‬ T
S v ~S
∴ H v T

17. 9 ELEMENTARY VALID
ARGUMENT FORMS
6. Absorption (Abs.)
1st Premise: p ‫כ‬ q
Conclusion: ∴ p ‫כ‬ (p ● q)

18. 6. ABSORPTION (ABS)
1st Premise: If I know that passing the final is
sufficient for John passing the course,
Conclusion: then I know that if John passed
the final, he passed both the final and
the course.

19. If I know that passing
the final is sufficient for
John passing the
course,
then I know that if John
passed the final, he passed
both the final and the
course.
F ‫כ‬ C
∴F ‫כ‬ (F ● C)

20. 9 ELEMENTARY VALID
ARGUMENT FORMS
7. Simplification (Simp.)
1st Premise: p ● q
Conclusion: ∴ p

21. 7. SIMPLIFICATION (SIMP)
1st Premise: If we know that John passed the
final and the course,
Conclusion: then we know that John passed the
final.
1st Premise: If we know two things,
Conclusion: then we know one thing.

22. If we know that John
passed the final and the
course,
then we know that John
passed the final.
F ● C
∴ F∴ C
then we know that John
passed the course.

23. If we know two things,
T ● T
∴ Tthen we know one thing.

24. 9 ELEMENTARY VALID
ARGUMENT FORMS
8. Conjunction (Conj.)
Two propositions that are true
1st Premise: p
2nd Premise: q
Conclusion: p ● q

25. 8. CONJUNCTION (CONJ.)
1st Premise: Pizza is my favorite snack
2nd Premise: Blue is my favorite color.
Conclusion: Pizza is my favorite snack and
blue is my favorite color.

26. Pizza is my favorite
snack.
Pizza is my favorite
snack and blue is my
favorite color.
P
Blue is my favorite color. B
∴ P ● B

27. 9 ELEMENTARY VALID
ARGUMENT FORMS
9. Addition (Add)
 Either of the two propositions is
true
1st Premise : p
Conclusion: ∴ p v q

28. 9. ADDITION (ADD)
Suppose I am walking down a street with a friend
who is a doctor. We come across a person lying
flat out on the ground. His breathing is quite
labored and he looks terrible. The color of his
skin is not right.
1st Premise: I say to my doctor friend, “This
guy looks very drunk.”
Conclusion: My doctor friend says, “Either he
is very drunk or he is suffering insulin shock
(the symptoms are the same).”

29. This guy looks very
drunk.
Either he is very drunk or
he is suffering insulin
shock
D
∴ D v S