Philosophy & Reason
NATURAL DEDUCTION
CLASSIC VALID ARGUMENTS
You must be able to recognise these and be able to NAME them.
1.
AFFIRMING THE ANTECEDENT
(A.A.)
6.
(PvQ)
~P
Q
(PQ)
P
Q
2.
DENYING THE CONSEQUENT
(D.C.)
7.
(PR)
CONJUNCTION (Con)
8.
P
Q
( P & Q)
4.
REDUCTIO AD ABSURDUM (RAS)
P  ( Q & ~Q )
~P
P  ~P
~P
SIMPLIFICATION (Simp)
(P & Q)
P
5.
CHAIN ARGUMENT (Ch.A)
(PQ)
(QR)
(PQ)
~O
~P
3.
DENYING A DISJUNCT (Den.D)
9.
COMPLEX CONSTRUCTIVE
DILEMMA (C.C.D.)
(PR)
(QS)
(PvQ)
(RvS)
DISJUNCTIVE ADDITION (D.A.)
P
( P v Q)
1
TAUTOLOGIES
These are transitive, in that each side is equivalent to the other. You must be able to recognise
them and be able to NAME them.
1.
6.
DOUBLE NEGATION (D.N.)
(P
P  ~~P
2.
7.
COMMUTATION (Comm)
8.
DeMORGAN’S LAW (De.M.)
EXPORTATION / IMPORTATION
ASSOCIATION (Assoc.)
( P & ( Q & R))  ((P & Q) & R)
(P v (Q v R))  ((P v Q ) v R)
( P & Q )  ~( ~P v ~Q )
( P v Q )  ~( ~P & ~Q )
9.
4.
Q )  ~( ~P  Q )
(( P & Q )  R)  ( P  (Q  R))
(P&Q)(Q&P)
(PvQ)(QvP)
3.
EXCLUSIVE DISJUNCTION (Ex.Disj.)
DISTRIBUTION 9Dist.)
( P & ( Q v R))  ((P & Q) v (P & R)
(P v (Q & R))  ((P v Q ) & (P v R)
CONTRAPOSITION (Contra)
( P  Q )  ( ~Q  ~P )
5.
IMPLICATION (Imp)
( P  Q )  ~( ~P v ~Q )
NAME THE FOLLOWING TAUTOLOGIES:
1. ~ (p & q)  ~ ~ ~ (p & q)
2. (~p & (q v r) )  ((q v r) & ~p
3. ((p & ~q)  r)  ~(( p & ~q)  r)
4. (q v ~ r)  (~ q  ~ r)
5. (((p  q) v r) v s)  ((p  q) v (r v s))
6. (~p v q)  ~ (p & ~q)
7. (~p & (q v r))  (~p & q) v (~p & r)
8. ~~(p & (q v r)  (p & (q v r) )
9. ~(~p v q)  (p & ~q)
10. ( (p v q)  r)  ~r  ~(p v q)
2
11. ~p & (q  r)  (q  r) & ~p
12. (~ (p v q) & r)  ~( ( p v q) v ~r)
13. (~p v (q & ~r) )  (~ p v q) & (~ p v ~ r)
14. ( ( p v q)  (r v p) )  ~ (r v p)  ~ (p v q)
15. ( ( p & q)  ~ r)  (~(p & q) v ~ r)
16. ( ( p & q) & (~ r v s) )   ( ( r v s) & (p & q) )
17. ( ( q &~ p)  r)  (q  (~ p  r ) )
18. ( ( p  ~q) r)  ~ (p  ~ q)  r)
19. ( (p v ~ q) v ~ r)  ~ (~ (p v ~ q) & r)
20. ( ( ~ p & ~ q)  r)  ~ p  (~ q  r)
NAME THE FOLLOWING ARGUMENTS
1.
2.
(A v B)
~A
B
5.
3.
(C  (D & ~D))
~C
6.
7.
N  ~N
~N
M&N
M
9.
14.
8.
11.
((M  N) v (N  C))
~(M  N)
(N  C)
15.
(P  Q) &R
(P  Q)
(A v B)
~C
((A v B) & ~C)
13.
((K v L)  ~(K v L))
~(K v L)
D
(D v E)
MN
M
N
A
B
(A&B)
10.
((A & B)  C)
~C
~(A & B)
(A  B)
(B  C)
(A  C)
4.
(M  ~N)
(M  ~N) v N
12.
((A v B)  C)
(A v B)
C
NAME THE FOLLOWING VALID ARGUMENTS
A.
1.
2.
B.
1.
2.
C.
1.
D.
1.
2.
(p v q)  s
~m  (p v q)
~MS
H.
1.
2.
3.
4.
P
P
I.
1.
pvq
P
~p  q
P
 (p v q) & (~p  q)
(p  ~q)  ~(p  ~q)
 ~(p  ~q)
J.
1.
2.
P
K.
1.
2.
A v ~B
P
CB
P
 (A v ~B) v C
3
(A  B)  C
(B v C)  ~B
~A B
(A  B) v (B v C)
 C v ~B
P
P
P
P
~A  ((C v B) & ~(C v B))
 ~~A
(~p & ~q)  (r v s)
~( r v s)
 m (~p & ~q)
(p v ~q) v (r  p)
~(p v ~q)
 ~( r  p)
E.
1.
2.
F.
1.
2.
3.
G.
1.
2.
(A  B) v ~(~A  B)
~(A  B)
 ~( ~A  B)
P
P
A  (B v C)
BC
~(B v C)
 ~A
P
P
P
(~A  B)  ~(A  B)
~A  B
 ~(A  B)
P
P
L.
Fill in the steps of the following using valid
arguments and tautologies.
1.
p & q Pr
2.
rvs
Pr
3.
(p&q)&(r v s) 1,2_____________________
4.
((p&q) & r) v ((p&q)& s) 3.______________
5.
p
4._________________________
6.
p v (r v s) 5._________________________
7.
(p v r) v s 6._________________________
NATURAL DEDUCTIONS
A natural deduction is a method of proof that moves in a series of continuous steps from the
premises of an argument to its conclusion. Each step is either a valid argument or a tautology.
Each step is justified because it is a valid inference. All of the deductions in this book are valid.
1.
2.
3.
4.
5.
6.
7.
etc.
B  (O v F) Premise
(O v F)  P Premise
~ (P & C) Premise
B
Premise
/ ~C Conclusion
______________
______________
______________
In order to do this deduction you must write in each step (5 6 7 etc). These must come from the
steps ABOVE the step you are doing. E.g.
1.
2.
3.
4.
5.
6.
7.
8.
9.
B  ( O v F) Pr.
(O v F)  P Pr.
~ (P&C) Pr.
B Pr.
B  P 1,2 Chain Arg.
P
5,4 Affirm.Ant.
~P v ~C 3. De.Morg.
~~P 6. D.N.
~C
7,8 Den.Disj.
WRITE IN THE NAMES OF THE STEPS OF THE FOLLOWING DEDUCTIONS.
THEY ARE TAUTOLOGIES OR VALID ARGUMENTS.
A.
1.
2.
3.
4.
((AvB) (C&D))
(~Cv~D)
~(C&D)
~(AvB)
B.
1.
2.
3.
4.
(~AC)
(A~A)
~A
C
C.
1.
2.
3.
4.
5.
(A v (B&C)) pr.1
~B
pr.2
((AvB)&(AvC)) 1.
(AvB)
3.
A
4.2
pr.1
pr.2
2.
1.3
pr.1
pr.2
2.
1.3
4
D.
1.
2.
3.
4.
(~AB)
~B
(~BA)
A
pr.1
pr.2
1.
3.2
E.
1.
2.
3.
4.
~C
pr.1
(AvB)
pr.2
(~C&(AvB)) 1.2
(~C&A)v(~C&B) 3.
F.
1.
2.
3.
4.
5.
6.
(AB)
(BC)
~C
(AC)
~A
(~AvM)
pr.1
pr.2
pr.3
1.2
4.3
5
G.
1.
2.
3.
4.
5.
(AB)
(CD)
A
(AvC)
(BvD)
pr.1
pr.2
pr.3
3.
1.2.4
H.
1.
2.
3.
4.
5.
((A&B)C)
A
(A(BC))
(BC)
(~BvC)
pr.1
pr.2
1.
3.2
4
I.
1.
2.
3.
4.
5.
~(~AvB)
(A&~B)
A
(Av~C)
(~A~C)
pr.1
pr.2
2.
3
4
J.
1.
2.
3.
4.
((BvA)vC)
~B
(Bv(AvC))
(AvC)
pr.1
pr.2
1
3.2
WRITE IN THE NAMES OF THE STEPS
AND STEP NUMBERS USED FOR THE FOLLOWING:
A.
1.
2.
3.
4.
5.
6.
7.
8.
9.
(FG)
G (H&J)
F
G
H&J
G&(H&J)
(G&H)&J
G&H
~(~Gv~H)
pr.1
pr.2
pr.3
5
B.
1.
2.
3.
4.
5.
6.
7.
8.
9.
PQ
(Q&R)S
~S
Q(RS)
(P(RS)
(P&R)S
~(P&R)
~Pv~R
P~R
pr.1
pr.2
pr.3
C.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
A(BvC)
(BvC)~C
C
A~C
~~C
~A
~(BvC)
~B&~C
~A&(~B&~C)
(~A&~B)&~C
C~A
~AvC
AC
A~A
~A
pr.1
pr.2
pr.3
D.
1.
2.
3.
4.
5.
6.
7.
(P~Q)&P
RQ
P
P~Q
~Q
~Q~R
~R
pr.1
pr.2
E.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A(B&C)
C(D&~D)
AvB
~(B&C)~A
~C
~Cv~B
~Bv~C
~(B&C)
~A
B
pr.1
pr.2
pr.3
F.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
~((SQ)v(PvR ))
SR
~(SQ)&~(PvR)
~(PvR)
~P&~R
~SvR
~R
~S
~P
~P&~S
~(PvS)
~(PvS)vR
(~P&~S)vR
Rv(~P&~S)
(Rv~P)&(Rv~S)
pr.1
pr.2
6
A.
1.
2.
3.
4.
5.
6.
7.
Pv(RS)
PQ
~Q
~P
RS
~S~R
Sv~R
pr.1
pr.2
pr.3
B.
1.
2.
3.
4.
5.
6.
7.
8.
9.
PQ
(Q&R)S
~S
(Q(RS))
P(RS)
(P&R)S)
~(P&R)
~Pv~R
P~R
pr.1
pr.2
pr.3
C.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
A(B&C)
C(D&~D)
AvB
~(B&C)~A
~C
~Cv~B
~Bv~C
~(B&C)
~A
~AB
B
pr.1
pr.2
pr.3
D.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
A(BvC)
(BvC)~C
C
A~C
~~C
~A
~(BvC)
~B&~C
~A&(~B&~C)
C~A
~AvC
AC
A~A
~A
pr.1
pr.2
pr.3
E.
1.
2.
3.
4.
5.
6.
7.
8.
9.
P(QS)
P&Q
~Sv(QR)
(P&Q)S
S
~~S
QR
Q
R
pr.1
pr.2
pr.3
7
F.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
~Pv~P
(SvR)(P&Q)
QvS
P~P
~P
~Pv~Q
~(P&Q)
~(SvR)
~S&~R
~S
Q
pr.1
pr.2
pr.3
A.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
A(C&D)
~BF
BA
~BvA
Av~B
(C&D)vF
Fv(C&D)
(FvC)&(FvD)
FvC
~FC
~CF
pr.1
pr.2
pr.3
B.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
((PvQ)vR)(S&T)
P&(S(T~R))
(~PvR)vQ
P
PvQ
(PvQ)vR
S&T
S(T~R)
(S&T)~R
~R
P&~R
~(~PvR)
~(~PvR)Q
Q
pr.1
pr.2
C.
1.
2.
3.
4.
5.
6.
7.
8.
~D(RvW)
RS
WL
~S
~L
~R
~W
~R&~W
pr.1
pr.2
pr.3
pr.4
pr.5
D.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
~((P&Q)v(RvS))
(~PR)&((S&T)Q)
~(P&Q)&~(RvS)
~(RvS)
~R&~S
~R
~PR
~~P
P
~(P&Q)
pr.1
pr.2
pr.3
8
E.
11.
12.
13.
14.
~Pv~Q
P~Q
Q~P
~Q
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
~((SQ)v(PvR))
SR
~(SQ)&~(PvR)
~(PvR)
~P&~R
~SvR
~R
~S
~P
~P&~S
~(PvS)
~(PvS)vR
(PvS)R
pr.1
pr.2
JUSTIFY THE CONCLUSION OF THE ARGUMENTS BELOW, USING
THE RULES FOR TAUTOLOGIES OR VALID ARGUMENTS FOR YOUR STEPS.
A.
1.
2.
3.
PQ
P
RS
(Q&~R)v(Q&S)
F.
1.
2.
3.
~(F&G)
G
HF
~H
B.
1.
2.
PvQ
(QR)&~P
R
G.
1.
2.
~(MvP)
NP
~N
C.
1.
2.
3.
H.
1.
2.
AB
(B&C)D
C(AD)
D.
1.
(PvQ)vR
(PvQ)S
~S
TvR
~P&(Pv~Q)
~(QvP)
I.
1.
2.
3.
Av(B&C)
AD
DC
~C(B&C)
E.
1.
P&Q
PvQ
J.
1.
2.
~A~(B&C)
B
~C v A
WORKSHEET NO.9
Symbolize the following arguments and name the argument or tautology that appears. Are there
any fallacies here?
1. If Peter hits John, then John will hit Mary. So, if Peter hits John then Mary hits Tim, because
when John hits Mary, Mary hits Tim.
2. Whenever a cow gets knighted it gets pompous. There is a pompous cow. I bet it’s a knight.
9
3. I’ll get married only if the right man asks me. So, I’ll get married only if the right man asks me or
I’ll try to swim the Pacific Ocean.
4. If you pass the logic test then I’m the front end of a Hippo and I’m not the front end of a Hippo.
So, you’ve got no chance.
5. If you eat nothing for two weeks you’ll get slimmer. Well, I’m not going to eat nothing for that
long, so I guess I’ll just have to stay the weight I am.
6. You are either a raving beauty or you like kissing boys. You’re no raving beauty, so I guess you
like kissing boys.
7. I’m clever and handsome, so I guess I’m clever.
8. If I study then I’ll do well. If I’m smart I’ll do well without studying. So, if I’m either smart or I
study then I’ll do well or do well without study.
9. It’s not that Fred is both a drunk and a dope taker. So, Fred must be either not a drunk or not a
dope taker.
10. If you get the giggles your face will go green. Ha, ha, your face is green. So, you must have
giggled.
11. Your logic teacher will love you only if you get 100%. So, for heaven’s sake don’t get 100% and
you’ll escape this terrible fate.
FORMAL FALLACIES
Affirming the Consequent
(P  Q )
Q
P
Denying Antecedent
(PQ)
~P
~Q
The Fallacy of Inconsistent Premises
(P&Q)
~P
Q&T
A fallacy is a MISTAKE in reasoning. These can NOT be used as valid steps in a deduction as they
are invalid.
10
PRACTICE TEST
1.
a)
b)
c)
d)
e)
Distribution
De Morgans Law
Reductio Ad Absurdum
Complex Constructive Dilemma
Chain Argument
2.
Name the following valid arguments and tautologies:
a)
1. M&N
2. L v C
(M&N) & (LvC)
3.
Name the steps in the following deductions:
A.
1.
2.
3.
4.
5.
6.
7.
8.
9.
~((A&B)v(CvD))
(~AC)&((D&E)B)
~(A&B)&~(CvD)
~(CvD)
~C&~D
~C
~AC
~~A
A
pr
pr.
B.
1.
2.
3.
4.
5.
6.
7.
8.
p(r&s)
~qU
qp
~q v p
p v ~q
(r&s) v U
U v (r&s)
(U v r) & (U v S)
pr
pr.
pr.
C.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
AB
(B&C)D
~D
B  (CD)
A(CD)
(A&C) D
~(A&C)
~A v ~C
A ~C
~D v A
DA
D  ~C
pr
pr.
pr.
b)
1. (AvB)C
2. A v B
C
11
c)
1. (CD)(LvM)
2. ~(LvM)
 ~(CD)
PRACTICE TEST
1.
Name the following as Tautologies, Valid Arguments or Fallacies.
(a)
I don’t like the heat. I’m going to work in Malaysia and luckily I like the heat. So, I’ll enjoy
the experience.
(b)
If I go to snooker then I’ll relax. So, either I won’t go to snooker or I’ll relax.
(2 marks)
2.
Fill in the justification steps and lines of the following deduction:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
p  (q & r)
Pr
~s  t
Pr
sp
Pr
~s v p
t v (q & r)
(T v q)&(t v r)
tvq
~p  ~s
~p  t
pvt
tvp
(t v p) & (t v q)
t v (p&q)
(10 marks)
3.
Fill in the justification steps and lines for the following deduction:
(i)
(ii)
~q  ~p
(q & r)  t /  r  (q  t)
(4 marks)
12