The puzzle Two senses of know Resolving the

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Knowledge and The Sure Thing Principle Anna Mahtani, LSE

The puzzle Two senses of ‘know’ You (Alice) Resolving the Puzzle Bob 2 out of 3 will be executed. Applications Carol

The puzzle Two senses of ‘know’ You (Alice) survive) Cr(Alice survives) = Resolving the Puzzle Bob survives 1/ Applications Carol survives 3 Cr(Alice survives/Carol doesn’t survive) = 1/2 Cr(Alice survives/Bob doesn’t survive) = 1/2 Thus, your credence that you survive should be 1/2.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Open me, and your credence in P will rationally become v Evidence E

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning (generalized) My credence in P should equal the expectation of my credence in P were I to come to learn E.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v We have seen that this reasoning is incorrect, because it requires you to have a credence of 1/2 that you will survive in the prisoner case – and this is counterintuitive. In case anyone does not share this intuition – or thinks that we should not rely on mere intuition on this topic. . .

The puzzle Two senses of ‘know’ You (Alice) David Resolving the Puzzle Bob Edward 4 out of 6 will be executed. Applications Carol Fiona

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications AB AC AD AE AF BC BD BE BF CD CE CF DE DF EF You (Alice) survive) = 1/3 Cr(Alice survives/Bob dies) = 2/5 Cr(Alice survives/Carol dies) = 2/5 Cr(Alice survives/David dies) = 2/5 Cr(Alice survives/Bob survives) = 1/5 Cr(Alice survives/Carol survives) = 1/5 Cr(Alice survives/David survives) = 1/5 Cr(Alice survives/Edward survives) = 1/5 Cr(Alice survives/Fiona survives) = 1/5 Open me, and your credence that you will survive will rationally become 2/5 Open me, and your credence that you will survive will rationally become 1/5

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v The appealing reasoning is appealing But it seems to give false results How can we reconcile these facts?

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Two senses in which you might ‘know’ E Amelia knows Bob Amelia knows how to tie her shoelace Amelia knows that Julius Caesar died in 44 BC Let E denote the fact that Julius Caesar died in 44 BC. Amelia knows E Amelia knows the fact which Peter has forgotten. ‘. . . knows. . . ’ is followed by a name, definite description, or variable – that denotes (or ranges over) propositions/facts/pieces of evidence.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Two senses in which you might ‘know’ E To basic-know E is simply to know the proposition denoted by E. To super-know E is to know which proposition E denotes (i. e. To recognize that proposition as E). Illustrations: • The school motto • Pythagoras’ Theorem • what your friend has decided to tell you

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v If we interpret the word ‘know’ (circled) as basic-know, then the reasoning is incorrect. If we interpret the word ‘know’ (circled) as super-know, then the reasoning is correct.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v If I know that there exists some piece of evidence E, such that if I were to come to basic-know just E then my credence in P would be v. . . Here ‘E’ is a variable. It does not name or denote any particular piece of evidence. If we take ‘E’ as a variable here, then the only interpretation of ‘know’ open to us is basic-know. To super-know E would involve knowing which piece of evidence ‘E’ denotes – and if ‘E’ is a variable then it does not denote.

The puzzle Two senses of ‘know’ You (Alice) survive) Resolving the Puzzle Bob survives Applications Carol survives Cr(Alice survives) = 1/3 Cr(Alice survives/Carol does not survive) = 1/2 Cr(Alice survives/Bob does not survive) = 1/2 I know that there exists some piece of evidence E, such that if I were to come to basic-know just E then my credence that Alice survives would be 1/ 2. The appealing reasoning leads us astray here.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v I know that there if I were exists to come someto piece basic-know of evidence E, then E, such my credence that if I were in P to wouldto come (rationally) basic-know be just v. E then my credence in P would be v. . . Here ‘E’ is a name, that must denote a particular proposition.

The puzzle Two senses of ‘know’ You (Alice) survive) Resolving the Puzzle Bob survives Applications Carol survives Cr(Alice survives) = 1/3 Cr(Alice survives/Carol does not survive) = 1/2 Cr(Alice survives/Bob does not survive) = 1/2 E = The true first true proposition expressed by ‘Bob does not survive’ or ‘Carol and ‘Carolnot does survive’. not survive’ arranged in alphabetical order. I know that if I were to come to basic-know E then my credence that Alice survives would (rationally) be 1/2. The appealing reasoning leads us astray here.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v If we interpret the word ‘know’ (circled) as basic-know, then the reasoning is incorrect If we interpret the word ‘know’ (circled) as super-know, then the reasoning is correct.

The puzzle Two senses of ‘know’ You (Alice) survive) Resolving the Puzzle Bob survives Applications Carol survives E = The first true proposition expressed by ‘Bob does not survive’ and ‘Carol does not survive’ arranged in alphabetical order. I know that if I were to come to super-know the piece of evidence E then my credence that Alice survives would (rationally) be 1/2. Cr(Alice survives/E is that Bob does not survive) = 1/2 Cr(Alice survives/E is that Carol does not survive) = 0 The appealing reasoning does not lead us astray here.

Two senses of ‘know’ Resolving the Puzzle Applications HEADS (Carol) TAILS (Bob) The puzzle You (Alice) survive) Bob survives Carol survives E = The proposition selected at random from those true propositions expressed by ‘Bob does not survive’ and ‘Carol does not survive’. I know that if I were to come to super-know the piece of evidence E then my credence that Alice survives would (rationally) be 1/2. Cr(Alice survives/E is that Bob does not survive) = 1/ 3 Cr(Alice survives/E is that Carol does not survive) = 1/3 The appealing claim does not lead us astray here.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The appealing reasoning If I know that there is a piece of evidence E, such that if I were to come to know (just) E, then my credence in P would (rationally) be v. . . Then my credence in P should be v If we interpret the word ‘know’ (circled) as basic-know, then the reasoning is incorrect If we interpret the word ‘know’ (circled) as super-know, then the reasoning is correct.

The puzzle Two senses of ‘know’ Connections • The Monty Hall Problem • The Boy-Girl Problem • The Sure-Thing Principle • The Reflection Principle Resolving the Puzzle Applications

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The Sure-Thing Principle “. . . Suppose a physician now knows that his patient has one of several diseases for each of which the physician would prescribe immediate bed rest. We assert that under this circumstance the physician should and, unless confused, will prescribe immediate bed rest whether he is now, later, or never, able to make an exact diagnosis. ” (Friedman and Savage, 1952) The person either has flu or bronchitis. If you knew he had flu. . . BEDREST If you knew he had bronchitis. . . BEDREST

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications The Sure-Thing Principle “. . . Suppose a physician now knows that his patient has one of several diseases for each of which the physician would prescribe immediate bed rest. We assert that under this circumstance the physician should and, unless confused, will prescribe immediate bed rest whether he is now, later, or never, able to make an exact diagnosis. ” (Friedman and Savage, 1952) “. . . on its face, the reasoning leading to the conclusion appears no less compelling than before” (Aumann, Hart & Perry, ‘Conditioning and the Sure-Thing Principle’)

Cr(patient has flu) = 0. 7 Cr(patient has both)= 0. 4 Cr(patient has bronchitis) = 0. 7 The doctor thinks bed rest is needed only if the patient has both flu and bronchitis. Iff Cr(BOTH)>0. 5, the doctor will prescribe bedrest.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Open me, and you will decide to prescribe bedrest Evidence E You know that there is some piece of evidence E, and that if you were to come to know E, then you would decide to prescribe bed-rest. . . Therefore, you should decide to prescribe bed-rest now. The reasoning is appealing. But it is incorrect. How can we reconcile these facts?

The puzzle Two senses of ‘know’ Evidence E Resolving the Puzzle Applications E is the claim that the patient has flu if that is true – otherwise it’s the claim that the patient has bronchitis. You know that there is some piece of evidence E, and that if you were to come to know E, then you would decide to prescribe bed-rest. . . Therefore, you should decide to prescribe bed-rest now. If taken to mean basic-know, then the reasoning is incorrect. If taken to mean super-know, then the reasoning is not (shown to be) incorrect.

The puzzle Two senses of ‘know’ Resolving the Puzzle Evidence E Applications E is the claim that the patient has flu if that is true – otherwise it’s the claim that the patient has bronchitis. If you come to basic-know E, you will have learnt either that: The patient has flu The patient has bronchitis Not a partition (because they can both be true)

The puzzle Two senses of ‘know’ Resolving the Puzzle Evidence E E is the claim that the patient has flu if that is true – otherwise it’s the claim that the patient has bronchitis. If you come to super-know E, you will have learnt either that: The patient has flu and this is E The patient has bronchitis and this is E Applications A partition

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Group Reflection (restricted version) Cr. A Cr. B Gain in evidence You know that: • B knows all that you (A) know • B knows some true proposition E that you do not • B is rational – and has responded to the evidence just as you would (and should) have done • B’s credence in P is v Therefore, you (A) should have a credence of v in P.

The puzzle Two senses of ‘know’ Resolving the Puzzle Group Reflection (restricted version) The principle is appealing But it seems to give false results The Story of the Hats The Sleeping Beauty Problem The Prisoner Problems The Cable Guy Paradox Applications

Two senses of ‘know’ Resolving the Puzzle Applications HEADS (Carol) TAILS (Bob) The puzzle You (Alice) survive) Bob survives Carol survives E = The proposition selected at random from those true propositions expressed by ‘Bob does not survive’ and ‘Carol does not survive’. F = Whichever proposition is both true, and not E, out of ‘Bob does /does not survive’, and ‘Carol does/does not survive’.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Group Reflection (restricted version) E) ( nce Ga You (A) de i v ne i n i The ‘E person’ Cr. E (A survives) = 1/2 Gain in evidence (F) The ‘F person’ Cr. F (A survives) = ?

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Group Reflection (restricted version) The ‘E person’ E You know that: Cr. E (A survives) = 1/2 You (A) • The E person knows all that you (A) know • The E person knows E (and you do not). • The E person is rational – and has responded to the evidence just as you would (and should) have done • The E person’s credence in that you survive is 1/2 Therefore, you (A) should have a credence of 1/2 that you survive. Group Reflection gives us a false result here. But only when we interpret ‘knows’ as basic-knows

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Group Reflection (restricted version) Cr. A Cr. B Gain in evidence You know that there is something that B knows and you don’t (so you should defer to B). Call this - what B knows. But does B know what B knows? The mark of the cases where Group Reflection fails is that B does not super -know what she knows – usually because she doesn’t know that she is B

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Single-person Reflection (restricted version) Cr 0 Cr 1 Gain in evidence You know that there is something that your future self at t 1 knows and you (at t 0) don’t, (so you should defer to your future self at t 1). Call this – what you learnt by t 1. But does your future self at t 1 know what you learnt by t 1? The mark of the cases where Reflection fails is that the future-self at t 1 does not super-know what she learns by t 1 – usually because she doesn’t know that she is at t 1. This explains why the ‘stopping time restriction’ (Seidenfeld, Schervish & Kadane) works.

The puzzle Two senses of ‘know’ Resolving the Puzzle Applications Conclusion By distinguishing two senses of ‘know’, we can explain why it is that the appealing reasoning is both incorrect (on one reading) and correct on another. This can help us clarify the motivation for the Sure-Thing Principle – and why it holds only when the possible pieces of information form a partition. This can also help us refine the Reflection Principle to ensure that it does not lead us astray – and explain why the ‘stopping time restriction’ holds.