Download overhead 7/conditional proof [ov]

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Some example proofs:
(a) 1. (G  H) v (I  ~J)
2. G  ~H
3. I  (J v A)
4.
5.
6.
7.
8.
9.
10. A
Pr
Pr
Pr / A
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(b) 1. (A  B)  ~(C  D)
2. ~(A v F)
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13. ~(D v F)
Pr
Pr / ~(D v F)
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(c) 1. B  (C  E)
2. E  ~(J v H)
3. ~S
4. J v S
5.
6.
7.
8.
9.
10.
11.
12.
13. B  ~C
Pr
Pr
Pr
Pr / B  ~C
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(d) 1. I  (B  M)
2. M  T
3. ~T  B
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14. ~I
Pr
Pr
Pr / ~I
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Conditional Proof
CONDITIONAL PROOF is a rule of logic
that allows you to prove a conditional by
using the ASSUMPTION of the
conditional's antecedent to get its
consequent.
----->
simple example:
1. N  O
2. N  P
3.
N
4.
O
5.
P
6.
OP
7. N  (O  P)
Pr
Pr / N  (O  P)
Assp (CP)
MP 1, 3
MP 2, 3
Conj 4, 5
CP 3-6
ASSUMING the conditional's antecedent
is different from having the antecedent as
a premise:
- the assumption is TEMPORARILY
made for the SPECIFIC purpose of
getting the conditional's consequent
- once the assumption has served this
specific purpose, it can't be used in the
justifications of further steps
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So why is assuming the conditional's
antecedent legitimate at all?
- to prove a conditional, all you have to
show is that IF the antecedent is true, then
the consequent is true
-----> take our simple example again:
1. N  O
Pr
2. N  P
Pr / N  (O  P)
3.
N
Assp (CP)
4.
O
MP 1, 3
5.
P
MP 2, 3
6.
OP
Conj 4, 5
7. N  (O  P)
CP 3-6
- to prove N  (O  P) follows, all you have
to show is that IF N is true, then (O  P) is
true (using rules of logic and prior lines of
the proof as your resources)
- the assumption on line 3. in effect says
"If N is true..."; of course, this doesn't
mean anything by itself, but it is
completed on line 6 with "then (O  P)."
Then on line 7., the results of lines 3-6 are
summarized as
N  (O  P); this summary, which is part of
the main proof, discharges the assumption.
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Some terminology:
-----> take our simple example again:
1. N  O
Pr
2. N  P
Pr / N  (O  P)
3.
N
Assp (CP)
4.
O
MP 1, 3
5.
P
MP 2, 3
6.
OP
Conj 4, 5
7. N  (O  P)
CP 3-6
- lines 3-6 are called a SUBPROOF; the
first line of the subproof is the assumed
antecedent of the conditional you are
proving and the last line is its consequent
- the SCOPE of the assumption on line 3. is
the subproof (the idea is: the scope of an
assumption encompasses the lines where
the assumption is operative)
- the arrow and vertical line marking the
scope of the assumption is called a SCOPE
MARKER
- note that on line 7., the justification
includes the line numbers for the
WHOLE SUBPROOF
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Three restrictions on CP to keep it legitimate:
1. Every assumption must be discharged by
the end of the proof.
Remember that in using CP
assumptions are temporarily made for
the specific purpose of proving a
conditional. So you CAN'T leave an
assumption undischarged--that would
be to treat it as an additional premise.
2. Once an assumption has been discharged,
neither it nor any step in its scope can be
used in the proof again.
Remember that the lines in the scope
of the assumption are DEPENDENT on
the assumption. So you CAN'T use
them in justifications AFTER the
assumption has been discharged--that
would be to treat them as
INDEPENDENT of the assumption.
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3. Assumptions inside the scope of other
assumptions must be discharged in the
reverse order in which they were made.
Numerous uses of CP can be made
within a single proof; in fact a use of
CP can be made WITHIN another use.
When a use of CP is made within
another use, the scope of one must be
COMPLETELY within the other--scope
markers cannot cross.
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Some example proofs using the rule of
Conditional Proof:
(a)
1. (A  B)  (C  B) Pr / (A v C)  B
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(b)
1. (A  B) v (~A  ~B) Pr / A  B
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(c)
1. A  D
2. (B  D)  ~C
Pr
Pr / A  (B  ~C)
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Indirect Proof
INDIRECT PROOF is a rule of logic that
allows you to prove a negation by using
the ASSUMPTION of the OPPOSITE of
the negation to get a contradiction, and
then inferring the negation.
----->
example:
1. N  O
2. (N  O)  P
3. P  ~O
4.
N
5.
O
6.
NO
7.
P
8.
~O
9.
O  ~O
10. ~N
Pr
Pr
Pr / ~N
Assp (IP)
MP 1, 4
Conj 4, 5
MP 2, 6
MP 3, 7
Conj 5, 8
IP 4-9
AGAIN, as with CP, an ASSUMPTION is
different from a premise:
- with IP the assumption is
TEMPORARILY made for the SPECIFIC purpose of getting a contradiction
- once the assumption has served this
specific purpose, it can't be used in the
justifications of further steps
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How does Indirect Proof work?
- you can think of it this way: if you can
show that an assumption leads to a
contradiction, then you've shown that the
assumption leads to an absurd
consequence; so we can infer the negation
of the assumption
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- or you can think of it this way: treat the
subproof in IP as being like the subproof
in CP in the following way:
----->
take our example again:
1. N  O
2. (N  O)  P
3. P  ~O
4.
N
5.
O
6.
NO
7.
P
8.
~O
9.
O  ~O
10. ~N
Pr
Pr
Pr / ~N
Assp (IP)
MP 1, 4
Conj 4, 5
MP 2, 6
MP 3, 7
Conj 5, 8
IP 4-9
- the assumption on line 4. in effect says
"If N is true..." and this is completed on
line 9. with "then O  ~O." So we've gotten
the conditional N  (O  ~O). But we
know that O  ~O is false (it's a
contradiction). So we know ~(O  ~O). So
we can use MT to conclude ~N. (Note that
the use of MT ISN'T explicitly stated.)
- then on line 10., the results of lines 4-9
are summarized as ~N; this summary,
which is part of the main proof, discharges
the assumption.
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All the terminology (subproof, scope, scope
marker) that applies to CP applies to IP as
well.
Three restrictions on CP apply to IP as well.
1. Every assumption must be discharged by
the end of the proof.
Remember that in using IP
assumptions are temporarily made for
the specific purpose of proving a
contradiction. So you CAN'T leave an
assumption undischarged--that would
be to treat it as an additional premise.
2. Once an assumption has been discharged,
neither it nor any step in its scope can be
used in the proof again.
Remember that the lines in the scope
of the assumption are DEPENDENT on
the assumption. So you CAN'T use
them in justifications AFTER the
assumption has been discharged--that
would be to treat them as
INDEPENDENT of the assumption.
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3. Assumptions inside the scope of other
assumptions must be discharged in the
reverse order in which they were made.
Numerous uses of IP can be made
within a single proof; in fact a use of IP
can be made WITHIN another use (or
within a use of CP). When a use of IP
is made within another use, the scope
of one must be COMPLETELY within
the other--scope markers cannot cross.
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An example proof using the rule of Indirect
Proof:
1. B  (A  ~B) Pr / ~B
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An example proof using both Conditional
Proof and Indirect Proof:
1. (A v B)  ~C
2. D  (~F  ~G)
Pr
Pr / (A v D)  ~(C  F)
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Theorems
Theorems are statement forms that can be
proven without any PREMISES; rather
theorems are proven with
ASSUMPTIONS using CP or IP.
- the idea is that theorems follow from
the rules of logic ALONE
- every theorem is a tautology, and
every tautology is a theorem.
(REMEMBER: TAUTOLOGIES are
statement forms which are true for
every substitution instance).
----->
example using IP:
(a) prove: p v ~p
1.
~(p v ~p)
2.
~p  ~~p
3. ~~(p v ~p)
4. p v ~p
Assp (IP)
DeM 1
IP 1-2
DN 3
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----->
examples using CP:
(b) prove: p  (q  p)
1.
p
2.
q
3.
pvp
4.
p
5.
qp
6. p  (q  p)
Assp (CP)
Assp (CP)
Add 1
Dup 3
CP 2-4
CP 1-5
(c) prove: p v ~p
1.
p
Assp (CP)
2.
pvp
Add 1
3,
p
Dup 2
4. p  p
CP 1-3
5. ~p v p
CE 4
6. p v ~p
Comm 5
- in proving a theorem, NEVER assume the
theorem ITSELF
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-----> REVIEW three points:
(i) when doing proofs, the arguments are
valid; the goal is just to prove that they are
valid
- the system of 20 logical rules that you've
learned (8 rules of inference, 10
replacement rules, CP, and IP) can prove
ALL AND ONLY valid arguments
- proving ALL valid argument
indicates this system of rules is
COMPLETE
- proving ONLY valid arguments
indicates that this system of rules is
CONSISTENT
(ii) proofs CANNOT be used to demonstrate
INVALIDITY
- but you can demonstrate invalidity using
truth tables
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(iii) proofs CANNOT be used to demonstrate
SOUNDNESS
- to demonstrate soundness, you have to
show that premises are true: this goes
BEYOND LOGIC
- think of a premise such as: all whales
are mammals--this is a question of
biology, not logic
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Predicate logic
REMEMBER this example from the second
lecture:
Premise 1 All whales are mammals.
Premise 2 All mammals are warm blooded
animals.
Conclusion All whales are warm blooded
animals.
This is a valid argument. But symbolize
this argument in sentential logic.
- set up dictionary:
W  All whales are mammals.
M  All mammals are warm blooded
animals.
H  All whales are warm blooded animals.
- we get:
argument form:
P1 W
P2 M
C H
P1 p
P2 q
C r
- THAT'S not a valid argument form (you
can show that it isn't using a truth table)
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- in order to understand the validity of:
Premise 1 All whales are mammals.
Premise 2 All mammals are warm blooded
animals.
Conclusion All whales are warm blooded
animals.
we need to represent the logical structure
INTERNAL to simple sentences
(REMEMBER: a simple sentence is one
that does not contain any other sentence as
a component--for example, dictionary
entries used for symbolization in
sentential logic are always simple
sentences).
PREDICATE LOGIC represents the logical
structure internal to simple sentences.
The FORM of the argument instance above is:
Premise 1 All A's are B's.
Premise 2 All B's are C's.
Conclusion All A's are C's.
- in this argument form, A, B, and C are
variables for class terms