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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? - Principal Principle Simplified: an agent’s subjective probability ought to match the relevant objective probability - (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) --------------------------------------------------------------------------------------------------------- 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: - 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 Week 8 2 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’. - 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) - 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: - 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 3 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 - 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’ - 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 - 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 pq Philosophical Devices - 4 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