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Philosophical Devices
Week 8 – Some Conditions May Apply
--------------------------------------------------------------------------------------------------------CONNECTION BETWEEN CHANCE & CREDENCE

Subjective Probability (credence): measures how strongly an individual believes a
certain proposition/outcome will come about

Objective Probability (chance): measures the real tendency or genuine physical
likelihood of some proposition/outcome
Question: how should the values we give our credences relate the values of chances?
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Principal Principle Simplified: an agent’s subjective probability ought to match the
relevant objective probability
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(Proper) Principal Principle: Let C be any reasonable initial credence function, t be
any time, x be any real number in the unit interval, Cht(p) = x be the proposition that
the chance, at time t, of p’s holding equals x, and E be any proposition compatible
with Cht(p) = x that is admissible at t. Then, C(p |Cht(p) = x & E) = x
The key notion here is that of admissibility. We can’t allow arbitrary evidence to be included,
since some E’s will make the advice the PP gives incorrect.
Admissible evidence includes
 Historical information [e.g., past observations about this coin]
 Theoretical information about the dependence of chance on history
 Conditionals with chance consequents
 Boolean combinations of the above
Inadmissible Evidence includes any information relevant to the occurrence of p, (e.g. after it
happens).
Note that PP is not a fully general epistemic principle – it is a restricted epistemic norm! This
is because it’s not telling you what your credence should be for arbitrary p relative to
arbitrary background knowledge. Instead, it merely links credence & chance values. One
general epistemic principle is:
Requirement of Total Evidence (RTE): C should take into account one’s total evidence
K – i.e., an agent’s credence for p (in a context C) should be Pr(p | K), for some Pr
and the agent’s total evidence K (in C)
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CONDITIONAL PROBABILITY
One issue I promised we’d come back to was how to handle conditionals in the probability
calculus. Well, here’s something like that:
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Conditional probability: the probability of p given q, Pr(p|q) = Pr(p&q)/Pr(q)
Philosophical Devices
The probability of drawing a Jack from a standard
deck is 4/52. But suppose you know the next card is a
face-card. If so, you have more information –
information which will change the probability.
There are only 12 face-cards (4 Jacks, 4 Queens and 4
Kings) in a deck, so the probability that the drawn
card is a Jack given that it is a face-card now
becomes 4/12, or 1/3.
To obtain the answer for P[Jack|Face-card], we
translate it into a ratio of the chances of both p and q
happening to the chances of q happening. In other
words, the conditional probability for our example is
given by the following equation:
P[Jack|Face-card] = P[of being a Jack and a facecard] / P[of being a face-card]
Which is P[Jack|Face-card] = (4/52)/(12/52) = 1/3
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Some Conditions May Apply
Given you’ve got an Ace as your first card, what is
the probability of being dealt a blackjack?
P[Blackjack|Ace dealt first] = P[Blackjack and Ace
dealt first]/P[Ace dealt first]
For independent events, P[A and B] = P[A] * P[B].
Further, the probability of being dealt blackjack is
(4/52)*(16/51) + (16/52)(4/51)
However, because we specified the Ace comes first,
we must exclude any blackjacks where a 10-valued
card comes first. Therefore, the blackjack is made up
of an Ace first and only then a 10-value card. This
means we can say that P[Blackjack and Ace dealt
first] = (4/52 * 16/51).
Therefore, P[Blackjack|Ace dealt first] = P[Blackjack
and Ace dealt first]/P[Ace dealt first] = (4/52 *
16/51)/(4/52) = 16/51!
--------------------------------------------------------------------------------------------------------CONDITIONALIZATION & BAYES’ THEOREM
An upshot of the above two sections is that, when it comes to fixing our credences, we ought
to update our degrees of belief relative to new information we receive. This is
‘conditionalization’.
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Principle of Conditionalization: If your old conditional credence Pr(p|q)=k, and you
come to know q, your credence for Pr(p) should equal k
If one begins with prior probabilities Pi, and one acquires new evidence which can be
represented as becoming certain of an evidentiary statement E (assumed to state the totality of
one's new evidence and to have initial probability greater than zero), rationality requires
transforming one's initial probabilities to generate final or posterior probabilities Pf by
conditionalizing on E.
(Note the connection between this principle, PP, and RTE)
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Bayes’ Theorem: Pr(h|e) = (Pr(h) x Pr(e|h)/Pr(e)
Where the final probability of a hypothesis h is generated by conditionalizing on evidence e,
Bayes' Theorem provides a formula for the final probability of h in terms of the prior or
initial likelihood of h on e (Pi(e|h)) and the prior or initial probabilities of h and e:
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Better Principle of Conditionalization: Prf(h) = Pri(h|e) = Pri(e|h) × Pri(h)/Pri(e)
Solving the Monty Hall Problem
What we want to discover is the probability that the unselected & unopened door contains
the prize given the choice and the opening of door:
Philosophical Devices
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Some Conditions May Apply
Week 8
We now calculate the various components:
because the host never selects the door you
choose or the one with the prize
Meanwhile,
because the position of the prize had an
initial probability of 1/3 and this doesn’t change based on the choice
Finally,
open either B or C
because, given the choice of A, Monty can
Plug these in:
So the probability that the prize is behind the door that has not been chosen or opened is 2/3.
This means you should change your choice – after all, 2/3>1/3!
--------------------------------------------------------------------------------------------------------THREE (FOUR?) KINDS OF CONDITIONALS
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Everyday Conditional: In English, the typical ‘If…, then…’ construction. Typically,
there is an assumption that there is a relevant connection between the antecedent and
the consequent
 Good: ‘If it is raining, then I’ll wear my jacket’
 Bad: ‘If it is raining, then 2*2=4’
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Indicative Conditional: An ‘if…,then…’ claim make in the indicative mood
 Indicative mood: a category of verb forms used to state facts – e.g. ‘It is
raining outside’, ‘We'll be home by ten’
 If he works hard, he will finish on time
 If Martin Luther didn’t nail his 95 Theses to the door in Wittenberg, then
someone else did
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Subjunctive Conditional: An ‘if…, then…’ claim make in the subjunctive mood
 Subjective mood: a category of verb forms used to express things that
are not facts (wishes, possibilities, doubts, suggestions, etc) – e.g. ‘I
suggest Robert wait’, ‘It might snow soon’, ‘I wish it were summer’
 If he had worked harder, he would have finished on time
 If Martin Luther hadn’t nailed his 95 Theses to the door in Wittenberg, then
someone else would have
p
q
pq
Philosophical Devices
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Some Conditions May Apply
Week 8
Material Conditional: expressed using ‘’ or ‘’,
this is the operator that you learn about in Formal
Logic. Does not require there be any sort of
relevance connecting the antecedent and the
consequent, other than conforming to the
following truth-table:
T
T
F
F
T
F
T
F
T
F
T
F
Subjunctive & Indicative conditionals are different!
Analysis of Subjunctive conditionals: David Lewis modeled/analysed counterfactuals using
possible world semantics! The semantics of (A ⟥⟶ B) are given by a function on the
relative closeness of (i) worlds where A is true and B is true and (ii) worlds where A is true
but B is not, on the other:
 (A ⟥⟶ B) is vacuously true iff there are no worlds where A is true (e.g. A is
impossible)
 (A ⟥⟶ B) is non-vacuously true iff, among the worlds where A is true, some worlds
where B is true are closer to the actual world than any world where B is false;
 (A ⟥⟶ B) is false otherwise
(K) If I kangaroos didn’t have tails, they would topple over
The truth of (K) consists in the fact that amongst the worlds where kangaroos lack tails there
is at least one world where they fall over and this world is closer to the actual world than any
world where kangaroos lack tails but remain upright