Branden Fitelson
Philosophy 12A Notes
1
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•
TV On the Radio
:
Dear Science
•
Administrative Stuﬀ
–
HW #3 will be returned Wed. Resubs due Friday (as usual).
+
When you turn in resubmissions, make sure that you
staple them to your original homework submission
.
– TakeHome MidTerm will be posted on Friday.
•
Today: Chapter 3, Finalé — Brief Final Words on LSL Semantics
–
An actual
LSAT
problem.
[LSAT logic problems are LSL
problems!]
–
Some famous logic puzzles (again, these are just LSL problems).
•
Then: Chapter 4 — Natural Deduction Proofs for LSL
–
Validity (
±
)
vs
Proof (
`
) — some introductory remarks.
–
Our natural deduction proofs system for LSL.
–
Soon —
MacLogic
— a useful computer program for proofs.
+
Natural deductions are the most challenging topic of the course.
UCB Philosophy
Chapter
3
Finalé, Chapter
4
Intro
10/06/08
Branden Fitelson
Philosophy 12A Notes
2
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$
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Expressive Completeness: Rewind, and More ExtraCredit
•
Q
. How can we deﬁne
↔
in terms of

?
A
. If you naïvely apply the
schemes I described last time, then you get a
187 symbol monster
:
[
p
↔
q
± ,
A

A
, where
A
is given by the following
93 symbol
expression:
(((p

(q

q))

(p

(q

q)))

((p

(q

q))

(p

(q

q))))

(((q

(p

p))

(q

(p

p)))

((q

(p

p))

(q

(p

p))))
•
There are
simpler
deﬁnitions of
↔
using

.
E.g.
, this
43 symbol
answer:
[
p
↔
q
± ,
((p

(q

q))

(q

(p

p)))

((p

(q

q))

(q

(p

p)))
•
I oﬀered extracredit for a shorter solution. One of the students —
while still in class! — came up with the following 19symbol solution:
[
p
↔
q
± ,
((p

p)

(q

q))

(p

q)
•
I had a hunch this might be
the
shortest possible
solution. And, in fact,
it is
! I wrote a computer program to check all the shorter candidates
(there are actually only several hundred possibilities one has to check).
•
More E.C.
Find the shortest possible deﬁnitions of (1)
[
p
→
q
±
, (2)
[
p
∨
q
±
, and (3)
[
∼
p
∼
q
±
in terms of
p
,
q
, and the NAND operator

.
UCB Philosophy
Chapter
3
Finalé, Chapter
4
Intro
10/06/08
Branden Fitelson
Philosophy 12A Notes
3
'
$
%
LSL and LSAT Logic Problems: A Sample Question
A university library budget committee must reduce exactly
five of eight areas of expenditureG, L, M, N, P, R, S,
and Win accordance with the following conditions:
If both G and S are reduced, W is also reduced.
If N is reduced, neither R nor S is reduced.
If P is reduced, L is not reduced.
Of the three areas L, M, and R, exactly two are reduced.
Which one of the following could be a complete and accurate
list of the areas of expenditure reduced
by the committee?
(A) G, L, M, N, W