Midterm
PHIL 151
Feb. 5, 2010
Apart from the natural deduction proofs you are asked to provide in problem
6, you may in your proofs use any lemma or theorem from the van Dalen
text.
Problem 1
. Suppose we have a set of symbols that includes: a propositional
symbol
p
i
for each natural number
i
, symbols
→
and
⊥
; and parentheses (
and ). For this problem take
PROP
to be the smallest set such that
•
p
i
is in
PROP
for each natural number
i
.
• ⊥
is in
PROP
.
•
If
ϕ
and
ψ
are in
PROP
, then so is (
ϕ
→
ψ
).
Part a)
(4points) Deﬁne the function
length
from
PROP
to the natural num
bers recursively such that
length
(
ϕ
) is the number of symbols in
ϕ
(counting
parentheses as well as the other symbols).
Part b)
(4 points) Suppose
ψ
is a proposition in
PROP
. Deﬁne the function
sub
ψ,p
0
:
PROP
→
PROP
recursively, so that
sub
ψ,p
0
(
ϕ
) is the result of
substituting
ψ
for
p
0
in
ϕ
.
Part c)
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 Winter '09
 Mathematical Induction, Prime number, Proposition, φ, maximally consistent set

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