Philosophy 140A TakeHome MidTerm
Branden Fitelson
03/20/07
You are to answer all six (6) exercises on this takehome exam. Your solutions are
due on Tuesday, April
10 at 4pm
. You may work in groups on this exam (with the usual rules and procedures for group work).
1
Formalizing Some of the Metatheory of
P
in
Q
1.1
The Formal System
PS
0
for
P
Consider the following formal system for
P
, which I will call
PS
0
. The system
PS
0
has the same three axiom
schemata (PS1)–(PS3) that Hunter’s formal system
PS
has, and it has the following single rule of inference:
(MP
0
) From
‘
PS
0
A
and
‘
PS
0
A
⊃
B
, infer
‘
PS
0
B
.
So, the only difference between
PS
and
PS
0
is that the (MP) rule of
PS
does
not
require its premises to be
theorems of
PS
, whereas the (MP
0
) rule of
PS
0
does
require its premises to be theorems of
PS
0
.
Exercise #1
. Explain why
PS
and
PS
0
have exactly the same set of theorems.
Exercise #2
. Explain why
PS
and
PS
0
are (nonetheless)
not
the same formal system.
1.2
Formalizing Some of the Metatheory of
PS
0
in
Q
Consider the following four universally quantified WFFs of
Q
:
(1)
V
x
0
V
x
00
F
*0
f
**0
x
0
f
**0
x
00
x
0
(2)
V
x
0
V
x
00
V
x
000
F
*0
f
**0
f
**0
x
0
f
**0
x
00
x
000
f
**0
f
**0
x
0
x
00
f
**0
x
0
x
000
(3)
V
x
0
V
x
00
F
*0
f
**0
f
**0
f
*0
x
0
f
*0
x
00
f
**0
x
00
x
0
(4)
V
x
0
V
x
00
(F
*0
x
0
⊃
(F
*0
f
**0
x
0
x
00
⊃
F
*0
x
00
))
Now, consider the following interpretation
I
of
Q
.
1
The domain
D
of
I
is the set of WFFs of
P
.
“
F
*0
” gets interpreted by
I
as the (metatheoretic) property “is a theorem of
PS
0
” (
i.e.
,
‘
PS
0
).
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 Spring '07
 FITELSON
 Philosophy, Logic, Propositional calculus, P S

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