Philosophy 148 — Assignment #1
02/14/08
This assignment is due Thursday, 2/28/08. Answer all questions. If you work
in a group, list your group members at the top of your submitted work.
1
Problem #1
In this class, we take a probability function Pr
(
•
)
to be a realvalued function on
a (essentially, sentential) language
L
containing finitely many atomic sentences
satisfying the following three (
Kolmogorov
) axioms, for all sentences
p, q
∈ L
:
1. Pr
(p)
≥
0.
2. If
p
>
, then Pr
(p)
=
1.
3. If
p
&
q
⊥
, then Pr
(p
∨
q)
=
Pr
(p)
+
Pr
(q)
.
In Ch. 6 of his book
Choice and Chance:
An Introduction to Inductive Logic
(posted on website), Skyrms adopts the following six probability “rules” :
(
i
) If
p
>
, then Pr
(p)
=
1.
(
ii
) If
p
⊥
, then Pr
(p)
=
0.
(
iii
) If
p
q
, then Pr
(p)
=
Pr
(q)
.
(
iv
) If
p
&
q
⊥
, then Pr
(p
∨
q)
=
Pr
(p)
+
Pr
(q)
.
(
v
) Pr
(
∼
p)
=
1

Pr
(p)
.
(
vi
) Pr
(p
∨
q)
=
Pr
(p)
+
Pr
(q)

Pr
(p
&
q)
.
Here are the two problems you must solve:
(
a
) Prove Skyrms’s six rules from our axioms.
Specifically, prove all of his
six rules from our axioms (2) and (3) alone. You may not use any results
proved in class until you’ve proved them (yourself) as “lemmas” for the
purpose of this problem. You may prove Skyrms’s rules in any order, and
once you have proved a result you may use it in subsequent proofs.
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 Spring '08
 FITELSON
 Philosophy, Logic, Probability, Probability theory, Skyrms

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