Philosophy 12A Notes
1
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•
Ry Cooder
:
Chavez Ravine
•
Administrative Stuﬀ
–
There will be
no lecture on Wednesday
(10/15). And, Branden will
not have oﬃce hours on Wednesday either (I’ll be at Caltech).
– The Take-Home Mid-Term has been posted.
It’s due on Friday,
with resubmissions due the following Friday (just like a HW).
+
“Extra Credit Option”: for each validity problem in (3), you must
chose
one
way of proving it — for
both
initial
and
re-submission.
•
Today: Chapter 4 — Natural Deduction Proofs for LSL
–
Our natural deduction system for LSL.
*
How to deduce a conditional: the introduction rule for
→
.
*
Proofs by contradiction (
reductio ad absurdum
), and the
∼
rules.
*
Proof strategy — working backward and forward.
–
MacLogic
— a useful (free!) computer program for proofs.
UCB Philosophy
Chapter
4
10/13/08
Branden Fitelson
Philosophy 12A Notes
2
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How to Deduce a Conditional: I
•
To deduce a conditional, we
assume
its antecedent and try to deduce its
consequent from this assumption. If we are able to deduce the
consequent from our assumption of the antecedent, then we
discharge
our assumption, and infer the conditional.
•
To implement the
→
I rule, we will ﬁrst need a reﬁned Rule of
Assumptions that will allow us to assume arbitrary formulas “for the
sake of argument”, later to be discharged after making desired
deductions. Here’s the reﬁned rule of Assumptions:
•
Rule of Assumptions
(ﬁnal version): At any line j in a proof, any
formula
p
may be entered and labeled as an assumption (or premise,
where appropriate). The number j should then be written on the left.
Schematically:
j
(j)
p
Assumption (or: Premise)
UCB Philosophy
Chapter
4
10/13/08
Branden Fitelson
Philosophy 12A Notes
3
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$
%
How to Deduce a Conditional: II — The
→
I Rule
•
Now, we need a formal Introduction Rule for the
→
, which captures the
intuitive idea sketched above (
i.e.
, assuming the antecedent,
etc
.):
•
Rule of
→
-Introduction
: For any formulae
p
and
q
, if
q
has been
inferred at a line k in a proof and
p
is an assumption or premise
occurring at line j, then at line m we may infer
[
p
→
q
±
, labeling the line
‘j, k
→
I’ and writing on the left the same assumption numbers which
appear on the left of line k, except that we
delete
j if it is one of these
numbers. Note: we may have j
<
k, j
>
k,
or
j
=
k (
why
?). Schematically:
j
(j)
p
Assumption (or: Premise)
.
.
.
a
1
,. . . , a
n
(k)
q
.
.
.
{a
1
,. . . , a
n
}/j
(m)
p
→
q
j, k
→
I
UCB Philosophy
Chapter
4
10/13/08
Branden Fitelson
Philosophy 12A Notes
4
'
$
%
Using The
→
I Rule: An Example
•
Let’s do a deduction of:
A
→
(B
→
C)
∴
(A
→
B)
→
(A
→
C)
1
(1)
A
→
(B
→
C)
Premise
2
(2)
A
→
B
Assumption
3
(3)
A
Assumption
2, 3
(4)
B
2, 3
→
E
1, 3
(5)
B
→
C
1, 3
→
E
1, 2, 3
(6)
C
4, 5
→
E
1, 2
(7)
A
→
C
3, 6
→
I
1
(8)
(A
→
B)
→
(A
→
C)
2, 7
→
I
F
UCB Philosophy