Natural Deduction:

1
Natural Deduction Using simple valid argument
forms as demonstrated by truth-tablesas rules
of inference. A rule of inference is a rule
stating that whenever premises of certain forms
occur, conclusions of a certain form follow
necessarily.
First, we need to focus on the main operator of
the statements we encounter, so we can be on the
lookout for FORMS of valid inferences.
2
p . q
p gt q
(A v B) . (C v D)
__ gt __
1 gt 2
(S v B) gt W
((W . Y) v X) . (H gt B)
(K gt P) . N gt (A . C)
F . (G v B) . (J gt I)
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Modus Ponens MP
p gt q p ----- q
(B v N) gt M (B v N) ------------- M
A gt C A ------ C
(L gt N) gt (B gt V) L gt N ---------
(H . (J v N)) gt (N . L) H . (J v N) --------
N . L
B gt V
4
Modus Tollens MT
p gt q q ----- p
F gt H H ----- F
(J v L) (M . B) gt (J v L) ------- ( M . B)
G v (B . O) gt (D . V) . R (D . V) .
R -------
G v (B . O)
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Disjunctive Syllogism DS
p v q p ----- q
(K v L) v O (K v L) ------ O
(A v (B . D)) A ----- B . D
(T v Y) gt (L . D) v (F . T) v O gt W (T v
Y) gt (L . D) ------
(F . T) v O gt W
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Hypothetical Syllogism HS
p gt q q gt r ------- p gt r
(N . B) gt G G gt ( L v E) ------------- (N . B)
gt (L v E)
M gt (B gt G) (B gt G) gt (F . T) -----------
V gt (O . C) gt P (K v L) gt V -----------------
M gt (F . T)
(K v L) gt (O . C) gt P
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Simplification SM
p . q ------ p
(B v N) . (C gt L) ----- B v N
A . (B gtC) ------ A
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Conjunction CN
p q --- p . q
(M . N) v W D gt K --- ((M . N) v W) . (D gt
K)
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p gt q p / q MP
p gt q q / p MT
p v q p / q DS
p gt q q gt r / p gt r HS

1. A gt B
2. A gt (C v D)
3. B
4. C / D

1,3 MT
5. A
2, 5 MP
6. C v D
7. D
4,6 DS
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MP MT DS HS p gt q p gt q p v q p gt q p / q
q / p p / q q gt r / p gt r

1. E gt (K gt L)
2. F gt (L gt M)
3. G v E
4. G
5. F / K gtM

6. E
3, 4 DS
7. K gt L
1, 6 MP
8. L gt M
2, 5 MP
7,8 HS
9. K gt M
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MP MT DS HS p gt q p gt q p v q p gt q p / q
q / p p / q q gt r / p gt r

1. J gt (K gt L)
2. L v J
3. L / K
4. J
5. K gt L
6. K

2,3 DS
1, 4 MP
5, 3 MT
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MP MT DS HS p gt q p gt q p v q p gt q p / q
q / p p / q q gt r / p gt r
1. (S ? T) gt (P gt Q) 2. (S ? T) gt P 3. P
/ Q 4. (S ? T) 5. P gt Q 6. Q
2,3 MT
1, 4 MP
3,5 MP
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MP MT DS HS p gt q p gt q p v q p gt q p / q
q / p p / q q gt r / p gt r

• H gt E gt (C gt D)
• D gt E
• E v H
• E / C
• H
• 6. E gt (C gt D)
• 7. C gt D
• 8. C gt E
• 9. C

3,4 DS
5,1 MP
4, 6 MP
7, 2 HS
4, 8 MT
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Addition AD
p --- p v q
Constructive Dilemma
(p gt q) . (r gt s) p v r ----- q v s
T v F T
A conjunction of conditionals, plus the
disjunction of their antecedents yields the
disjunction of their consequents.
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Rules of inference (8) MP p gt q / p //
q MT p gt q / q // p HS p gt q / q
gt r // p gt r DS p v q / p // q SM
p . q // p CN p / q // p. q AD p
// p v q CD (p gt q) . (r gt s) / p v r //
q v s
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Rules of inference (mt, mp, ds, etc.) are one
way rules.
Rules of equivalence are two way rules,
allowing substitution of a statement form for an
equivalent statement form.
Rules of equivalence are written using to
indicate two expressions are equivalent to one
another.
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Double Negation p p
Two tildes can be added or deleted from any
statement with no effect on the truth-value.
1.A gt (B . C) 2. B . C

1. ( H v K)
2. H

3. (B . C) 2 DN 4. A 1,3 MT
3. H v K 1 DN 4. K 2 ,3 DS
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(p v q)
p . q

neither
not this and not that

(p . q)
p v q
not both
either not this or else not that
DM DeMorgans Theorem
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CM Commutation (p . q) (q . p) (p v q)
(q v p)
The order of statements around a dot or wedge is
of no consequence to the truth-value of the
statement
1. (J gt N) v (C v D) 2. C 3. D / J gtN

• 4. C . D CONJ 2,3,
• 5. (C v D) DM 4
• 6. J gt N COMM, DS 1,5

6. (C v D) v (J gtN) COMM, 1 7. J gt N DS 5, 6
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Association AS p . (q . r) (p . q) .
r p v (q v r) (p v q) v r
The grouping of simple statements around dots
and wedges is of no consequence for
truth-values
1. A . (B . C) / C
2. (A . B) . C AS 1 3. C . (A . B) CM 2 4. C
SM 3