Branden Fitelson
Philosophy 12A Notes
1
Announcements & Such
•
Miles Davis (with Coltrane)
:
So What
•
Administrative Stuff
–
HW #3 resubs due today (usual drill).
When you turn in resubmissions, make sure that you
staple them to your original homework submission
.
– The TakeHome MidTerm has been posted.
It’s due next Friday,
with resubmissions due the following Friday (just like a HW).
–
Please submit any extracredit solutions with (any one of) your
homework assignment(s). There is no deadline for extracredit.
•
Today: Chapter 4 — Natural Deduction Proofs for LSL
–
Our natural deduction system for LSL.
*
Learning the Intro. and Elim. rules for each connective.
*
Proof strategy — working backward and forward.
–
MacLogic
— a useful (free!) computer program for proofs.
UCB Philosophy
Chapter
4
10/10/08
Branden Fitelson
Philosophy 12A Notes
2
An Example of a Natural Deduction Involving
&
and
→
•
The following is a valid LSL argument form:
A
&
B
C
&
D
(A
&
D)
→
H
∴
H
•
Here’s a (7line) natural deduction proof of the sequent
corresponding to this argument:
A
&
B, C
&
D, (A
&
D)
→
H
H
.
1
(1)
A
&
B
Premise
2
(2)
C
&
D
Premise
3
(3)
(A
&
D)
→
H
Premise
1
(4)
A
1
&
E
2
(5)
D
2
&
E
1, 2
(6)
A
&
D
4, 5
&
I
1, 2, 3
(7)
H
3, 6
→
E
UCB Philosophy
Chapter
4
10/10/08
Branden Fitelson
Philosophy 12A Notes
3
The Rule of Assumptions (Preliminary Version)
•
Rule of Assumptions
(preliminary version): The premises of an
argumentform are listed at the start of a proof in the order in which
they are given, each labeled ‘Premise’ on the right and numbered with
its own line number on the left. Schematically:
j
(j)
p
Premise
•
We can see that our example proof begins, as it should, with the three
premises of the argumentform, written as follows:
1
(1)
A
&
B
Premise
2
(2)
C
&
D
Premise
3
(3)
(A
&
D)
→
H
Premise
UCB Philosophy
Chapter
4
10/10/08
Branden Fitelson
Philosophy 12A Notes
4
The Rule of
&
Elimination (
&
E)
•
Rule of
&
Elimination
: If a conjunction
p
&
q
occurs at line j, then at
any
later
line k one may infer either conjunct, labeling the line ‘j
&
E’ and
writing on the left all the numbers which appear on the left of line j.
Schematically:
a
1
,. . . , a
n
(j)
p
&
q
a
1
,. . . , a
n
(j)
p
&
q
.
.
.
OR
.
.
.
a
1
,. . . , a
n
(k)
p
j
&
E
a
1
,. . . , a
n
(k)
q
j
&
E
•
We can see that our example deduction continues, in lines (4) and (5),
with two correct applications of the &Elimination Rule:
1
(4)
A
1
&
E
2
(5)
D
2
&
E
UCB Philosophy
Chapter
4
10/10/08
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Branden Fitelson
Philosophy 12A Notes
5
The Rule of
&
Introduction (
&
I)
•
Rule of
&
Introduction
: For any formulae
p
and
q
, if
p
occurs at line j
and
q
occurs at line k then the formula
p
&
q
may be inferred at line
m, labeling the line ‘j, k
&
I’ and writing on the left all numbers which
appear on the left of line j
and
all which appear on the left of line k.
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 Spring '08
 FITELSON
 Philosophy, Logic, Deduction, Natural deduction, Branden Fitelson, Philosophy 12A Notes

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