142
Proof Method
A. Background
B. Introduce 8 rules of inference
C. Recognizing symbolized instances of 8
rules of inference
D. An example proof
143
A. Background
The PROOF METHOD is a technique for
proving that an argument is VALID by
deducing the conclusion of the argument from
the premises by way of a set of rules of logic.
Take the following valid English
argument:
If the safe was opened, it must have been
opened by Smith, with the assistance of
Brown or Robinson. None of these three
could have been involved unless he was
absent from the meeting. But we know
that either Smith or Brown was present at
the meeting. Furthermore, we know that
the safe was opened. Therefore it must
have been Robinson who helped open it.
144
With this argument, the following are given as
premises:
(P1)
If the safe was opened, it must have
been opened by Smith, with the
assistance of Brown or Robinson.
(P2)
None of these three could have been
involved unless he was absent from the
meeting.
(P3)
Either Smith or Brown was present at
the meeting.
(P4)
The safe was opened
The following is given as the conclusion:
(C)
Robinson helped open the safe.
But how do we get from the premises to the
conclusion???
145
(P1)
If the safe was opened, it must have
been opened by Smith, with the
assistance of Brown or Robinson.
(P2)
None of these three could have been
involved unless he was absent from the
meeting.
(P3)
Either Smith or Brown was present at
the meeting.
(P4)
The safe was opened
(5)
Smith opened the safe and either
Brown or Robinson helped (from (P1)
and (P4).
(6)
Smith was absent from the meeting
(from (5) and (P2)).
(7)
Brown was present at the meeting
(from (6) and (P3)).
(8)
Brown did not help to open the safe
(from (7) and (P2)).
(C)
Robinson helped open the safe (from
(8) and (5))
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We get from the premises to the conclusion by
a series of subconclusions, each deduced by a
rule of logic.
- in the part of the course focusing on Units 7 9, we will learn a set of rules of logic for
making deductions
- we will continue to focus on logical
relations among whole sentences; that is,
we'll continue to do sentential logic
- the proof method is a method for proving
validity; it cannot be not used to prove
invalidity
- ADVANTAGES of the proof method
compared to truth table method:
less clumsy to write out, and similar
to ordinary reasoning
- DISADVANTAGES of the proof method
compared to truth table method:
unable to prove invalidity, and not
mechanical--it requires some
creativity
147
REMEMBER: argument forms are a series of
statement forms related as premises and
conclusion
----->
example:
pq
p
q
can be used to stand for:
GH
G
H
as well as
(G v ~H)  (K  L)
G v ~H
KL
both are instances of this
argument form
148
----->
example:
pvq
~q
p
can be used to stand for:
GvH
~H
G
as well as
(G  H) v ~(K  L)
~~(K  L)
GH
both are instances of this
argument form
149
B. Introduce 8 rules of inference
The 8 RULES OF INFERENCE are just basic
VALID argument forms. The first 3 are:
Modus Ponens (MP):
pq
p
q
Modus Tollens (MT):
pq
~q
~p
Hypothetical
Syllogism (HS):
pq
qr
pr
150
EXAMPLE INSTANCES of these rules
Modus Ponens (MP)
If insurance companies contribute millions of
dollars to political campaigns, then
meaningful insurance reform is impossible.
Insurance companies contribute millions of
dollars to political campaigns. Therefore,
meaningful insurance reform is impossible.
(P1) If insurance companies contribute
millions of dollars to political
campaigns, then meaningful insurance
reform is impossible.
(P2) Insurance companies contribute
millions of dollars to political
campaigns.
(C) Meaningful insurance reform is
impossible.
symbolized as:
(P1) I  R
(P2) I
(C) R
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What about:
(P1) Insurance companies contribute
millions of dollars to political
campaigns.
(P2) If that is so, then meaningful
insurance reform is impossible.
(C) Meaningful insurance reform is
impossible.
symbolized as:
(P1) I
(P2) I  R
(C) R
- this is still an instance of MP; the order of
the premises DOES NOT matter
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Modus Tollens (MT)
If TV viewing provides genuine relaxation,
then TV enhances the quality of life. TV does
not enhance the quality of life. So TV viewing
does not provide genuine relaxation.
(P1) If TV viewing provides genuine
relaxation, then TV enhances the quality of
life.
(P2) TV does not enhance the quality of
life.
(C) TV viewing does not provide genuine
relaxation.
symbolized as:
(P1) R  Q
(P2) ~Q
(C) ~R
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Hypothetical Syllogism (HS)
If sea levels rise twenty feet worldwide, then
coastal cities from New York to Sydney will be
inundated. If the ice sheets on Antarctica slip
into the sea, then sea levels will rise twenty
feet worldwide. Consequently, if the ice
sheets on Antarctica slip into the sea, then
coastal cities from New York to Sydney will be
inundated.
(P1) If sea levels rise twenty feet
worldwide, then coastal cities from
New York to Sydney will be
inundated.
(P2) If the ice sheets on Antarctica slip
into the sea, then sea levels will rise
twenty feet worldwide.
(C) If the ice sheets on Antarctica slip
into the sea, then coastal cities from
New York to Sydney will be
inundated.
symbolized as:
(P1) S  C
(P2) I  S
(C) I  C
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5 more rules of inference:
Simplification (Simp):
pq
p
pq
q
Conjunction (Conj):
p
q
pq
Disjunctive
Syllogism (DS):
pvq
~p
q
pvq
~q
p
Addition (Add):
p
pvq
q
pvq
Dilemma (Dil):
(p  q)
(r  s)
pvr
qvs
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EXAMPLE INSTANCES of these rules
Simplification (Simp)
(P1) Main floor seats cost $6 and
balcony seats cost $4.50.
(C) Main floor seats cost $6.
symbolized as:
(P1) M  B
(C) M
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Conjunction (Conj)
(P1) Main floor seats cost $6.
(P2) Balcony seats cost $4.50.
(C) Main floor seats cost $6 and balcony
seats cost $4.50.
symbolized as:
(P1) M
(P2) B
(C) M  B
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Disjunctive Syllogism (DS)
(P1) Either the 10 or the 15 goes to Las
Vegas.
(P2) The 10 doesn't go to Las Vegas.
(C) The 15 goes to Las Vegas.
symbolized as:
(P1) E v I
(P2) ~E
(C) I
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Addition (Add)
(P1) Los Angeles is on the Pacific
Ocean.
(C) Either Los Angeles is on the Pacific
Ocean or Denver is on the Pacific
Ocean.
symbolized as:
(P1) L
(C) L v D
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Dilemma (Dil)
(P1) If we choose nuclear power, then
we increase the risk of nuclear accident.
(P2) If we choose conventional power,
then we add to the greenhouse effect.
(P3) We must either choose nuclear
power or conventional power.
(C) We either increase the risk of
nuclear accident or add to the
greenhouse effect.
symbolized as:
(P1) N  I
(P2) C  G
(P3) N v C
(C) I v G
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C. Recognizing symbolized instances of 8
rules of inference
M v ~B
~M
~B
~B  ~L
G  ~B
G  ~L
P v ~S
~~S
P
K  ~C
~~C
~K
SF
F  ~L
S  ~L
161
~N  T
~N
T
(R  H)  (S  I)
RH
~J  [~A  (D  A)]
~J
~A  (D  A)
~F v (R  L)
~~F
RL
RE
HR
HE
(R  F)  [(R  ~G)  (S  Q)]
~[(R  ~G)  (S  Q)]
~(R  F)
162
~D  ~F
~~F
~~D
(B  D)
(E  C)
BvE
DvC
(~H  ~L)  (R v S)
~(R v S)
~(~H  ~L)
H v ~L
~H
~L
(~M  Q)
(R  ~T)
(~M  Q)  (R  ~T)
163
D. An example proof
Proofs use rules of inference (and other
rules that we will learn later) to derive a
conclusion from a set of premises:
-----> for example:
1.
2.
3.
4.
5.
6.
7.
8.
9.
~A  (B  ~C)
~D  (~C  A)
D v ~A
~D
~A
B  ~C
~C  A
BA
~B
Pr
Pr
Pr
Pr / ~B
________
________
________
________
________
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The 8 rules of inference:
Modus Ponens
(MP):
pq
p
q
Dilemma
(Dil):
(p  q)
(r  s)
pvr
qvs
Modus Tollens
(MT):
pq
~q
~p
Disjunctive
Syllogism
(DS):
pvq pvq
~p
~q
q
p
Hypothetical
Syllogism
(HS):
pq
qr
pr
Addition
(Add):
p
q
pvq pvq
Simplification
(Simp):
pq pq
p
q
Conjunction
(Conj):
p
q
pq
165
In doing PROOFS in Unit 7, you are looking for
instances of the 8 rules of inference
- what are proofs?
- a proof is a derivation (a sequence of
justified steps) in which the last step is
the conclusion of an argument (Klenk, p.
128)
- the point of a proof is to demonstrate
that an argument is VALID; we do this by
JUSTIFYING the subconclusions and
conclusion that make up the steps of a
proof
- rules of inference are one SUBGROUP
of rules of logic (you will learn other
subgroups of rules of logic) used to
justify the steps of a proof
- REMEMBER: rules of inference are
valid argument forms
166
----->
three things to keep in mind:
- when doing proofs, the arguments
are valid; the goal is just to
demonstrate that they are valid
- proofs cannot be used to
demonstrate invalidity
- proofs cannot be used to
demonstrate soundness
167
----->
how is a proof constructed?
- you are given an argument either in
English or symbolized in sentential
logic
- if the argument is in English,
symbolize it
----->
example of symbolized argument:
1. ~G  [G v (S  G)]
2. (S v L)  ~G
3. S v L
Pr.
Pr.
Pr. / L
- use rules of logic to justify
subconclusions that will eventually
take you to the conclusion
168
----->
1.
2.
3.
4.
5.
6.
7.
8.
example:
~G  [G v (S  G)]
(S v L)  ~G
SvL
~G
G v (S  G)
SG
~S
L
Pr.
Pr.
Pr. / L
________
________
________
________
________
- you must justify EACH STEP of the
proof--that is, you must justify each of
the subconclusions and the
conclusion
- this justification involves
writing the name of the
rule and the line numbers
of instances of the
premise(s) of the rule
----->
the result is that you demonstrate that the
argument is VALID
169
A formal definition of "justified step" (for
logical rules in Unit 7):
A justified step is either a premise, or a
step that FOLLOWS FROM previous
steps according to one of the given rules
of inference (Klenk, p. 118).
A formal definition of "follows from" (for
logical rules in Unit 7):
A step S follows from previous steps
P1...PN according to a given rule of
inference R just in case S is a substitution
instance of the conclusion of the rule R
and P1...PN are the corresponding
substitution instances of the premises of
rule R (Klenk, p. 118).
1.
2.
3.
4.
5.
6.
7.
8.
~G  [G v (S  G)]
(S v L)  ~G
SvL
~G
G v (S  G)
SG
~S
L
Pr.
Pr.
Pr. / L
________
________
________
________
________
170
-----> another example:
1.
2.
3.
4.
5.
6.
7.
8.
9.
~A  (B  ~C)
~D  (~C  A)
D v ~A
~D
~A
B  ~C
~C  A
BA
~B
Pr.
Pr.
Pr.
Pr. / ~B
________
________
________
________
________
In these examples, we've been GIVEN the
subconclusions, and we only had to provide the
justifications. Normally, only the premises and
conclusion are given, and you have to figure out
subconclusions that take you to the conclusion.
171
- some basic points for doing proofs:
- there's more than one possible route to the
conclusion by way of a series of justified
steps
- the ORDER of the steps does not matter
- the NUMBER of steps does not matter;
it's ok if you make steps that end up
being dead ends
- it's ok to use premises to justify MORE
THAN ONE step
- it's ok if some premises AREN'T USED
AT ALL to justify steps
- the point is to get to the conclusion by way
of a series of justified steps; exactly how you
do this does not matter