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PHLB55H3 (1)


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University of Toronto Scarborough
Erin Webster

Fundamentals of logic Benj Hellie January 9, 2014 1 Murder most foul! upstairs The Duke has been murdered by a dastardly foe. heiress dutchess heir heeiesse dutchess heiirr Fortunately, Sherlock Holmes is on the scene, ac- companied by his trusty sidekick Watson. They know exactly six people were in the manor house at the time: cook footman butler coook footman buterre ▯ – the Heiress – the Dutchess – the Heir – the Cook – the Footman A square in white represents someone still under – the Butler consideration. Downstairs Alternatively, the first piece of evidence could be indicative of a ‘downstairs’ These six suspects fall into categories as follows: person—such as a feather duster. Holmes would 1.1 The first piece of evidence? ▯ – ‘Upstairs’ (aristocrats): the Heiress, the then eliminate everyone aside from the downstairs people, leaving the following suspects: Dutchess, the Heir From this point forward, there are a number of ways the investigation could go. – ‘Downstairs’ (servants): the Cook, the Footman, the Butler 1.1.1 Upstairs or downstairs? ▯ – Men: the Heir, the Footman, the Butler heeiesse dutchess heiirr Upstairs For one thing, they could find – Women: the Heiress, the Dutchess, the Cook evidence indicative of an ‘upstairs’ person as the murderer—such as a diamond lying by the scene of ▯ – Young people: the Heiress, the Heir, the the crime. If so, Holmes would eliminate everyone Footman aside from the upstairs people. The remaining sus- coook footman buterre pects would be the following: – Old people: the Dutchess, the Cook, the Butler Holmes’s method is to eliminate suspects. Not downstairs rub them out, but eliminate them from considera- tion: to gradually narrow down the class of people who might have done it until only one person is left. 1.1.2 Man or woman? Then that person is the one who must have done it. Initially, Holmes starts out with a blank slate, Man Or perhaps the first piece of evi- where all six suspects still might have done it: dence could be indicative of a man—such as a large 1 footprint. That would eliminate everyone aside 1.2.1 Upstairs, and ... young from the men, leaving the following suspects: If Holmes finds first a diamond, he eliminates heiesse duucheess heeiirr the non-upstairs people, leaving the position in ( heeiesse dutchesss heiirr A man His second piece of evidence cook footmaan butterr could then be a man-sized footprint. Eliminating the non-men would leave the following suspect: coook footmann buterre upstairs heiesse dutchess heiirr Old And again, the first piece of evi- man dence could be indicative of an old person—such as polident dropped by the scene of the murder. Elim- inating all but the old people would leave the fol- cook ootmaan buterre Woman Alternatively, the first piece of lowing suspects: evidence could be indicative of a woman—such as a small footprint. Eliminating everyone aside from the women would leave the following suspects: old man heiesse duucheess heeiirr woman The Heir did it! Holmes is done. heeiesse dutchesss heiirr A woman But he is not yet done if his second piece of evidence is a woman-sized foot- cook footmaan butterr print: coook footmann buterre upstairs woman heiesse dutchess heiirr 1.2 First and second pieces of evidence If Holmes locates a second piece of evidence, he cook buterre 1.1.3 Young or old? then eliminates suspects incompatible with it that ootmaan remain after his first piece of evidence. Let us see Young Still further, the first piece of ev- how that plays out if his first piece of evidence con- idence could be indicative of a young person—such as tickets to the country ball dropped by the scene cerns upstairs/downstairs status. of the murder. Eliminating everyone aside from the young people would leave the following suspects: For both the Heiress and the Dutchess remain under consideration. 2 Young He is also not done yet if his sec- A man His next piece of evidence might young ond piece of evidence is tickets to the country ball: allow him to eliminate the non-men: heiesse dutchess heiirr upstairs young heeiesse dutchesss heiirr heiesse duucheess heeiirr cook ootmaan buterre coook footmann buterre cook footmaan butterr downstairs Voila! It was the Footman. man downstairs Old But not so if the new evidence elim- For both the Heiress and the Heir remain under con- In which case he is not yet done, because it remains inates the non-old: sideration. to distinguish between the Footman and the Butler. Old But finding Polident would wrap old A woman But if his evidence allows the case for him: him to eliminate the non-women: heiesse dutchess heiirr upstairs old woman heeiesse dutchesss heiirr heiesse duucheess heeiirr cook ootmaan buterre coook buterre footmann cook footmaan butterr downstairs Both the Cook and the Butler remain as unelimi- nated suspects. downstairs Eliminating the remaining non-old people leaves 1.3 A third piece of evidence only the Dutchess as a possibility. Then he can point to the Cook as the culprit. In several of the cases discussed, two pieces of ev- idence were not enough to settle the mystery. One 1.2.2 Downstairs, and ... Young Alternatively, his evidence was ( One possibility after that is to find a Suppose Holmes’s first piece of evidence is a might allow elimination of the non-young: small footprint, which would single out the Heiress: feather duster; he eliminates from consideration the non-downstairs people, reaching the position in (1.2.2). 3 woman 2.1.1 What Holmes believes, disbelieves, and is ▯ In (2), in none of the possibilities he takes upstairs young uncertain about seriously (namely, none (neither) of cook or heiiess dutchess heiirr butler): (a) upstairs (after all, the Butler is Consider what he believes, disbelieves, and is un- not in upstairs and neither is the Cook); (b) certain about in ( young (after all, the Butler is not in young 1. Holmes believes each of the following: and neither is the Cook); (c) no one did the murder (after all, if the Butler did it, some- cookk ootmaan butterr (a) downstairs one did, and same for the cook); and so forth (b) old ▯ In (3), in only some of the possibilities he (c) someone did the murder takes seriously (namely, in only some (ex- actly one, not both) of cook and butler): (a) (d) either the Cook or the Butler did it (e) it wasn’t the Dutchess man (after all, the Butler is in man but the Cook is not); (b) woman (after all, the Cook The other possibility here, of course, is to find evi- (f) not-young is in woman but the Butler is not); (c) the dence implicating a man, singling out the Heir. 2. Holmes disbelieves (rejects, denies) each of Butler did it (after all, that is so if the Butler Also in (, two pieces of evidence did it but not if the Cook did it); and so forth weren’t enough. One possibility after that is to find the following: a large footprint, singling out the Butler: (a) upstairs 2.2 Defining belief (b) young old (c) no one did the murder We can generalize: (d) it was the Footman heiiess dutchess heiirr ▯ A person believes P (e) not-old , 3. Holmes is uncertain about each of the fol- In every possibility they take seriously, lowing: P cookk ootmaan butterr (a) man ▯ A person disbelieves P (b) woman , (c) the Butler did it (d) the Cook did it In no possibility they take seriously, P man downstairs ▯ A person is uncertain about P The other possibility, of course, is to find a small 2.1.2 Connection to possibilities Holmes takes , seriously In exactly some possibilities they take footprint, singling out the Cook. These are connected to possibilities taken seriously seriously, P in the following way: 2 Belief ▯ In (1), in every possibility he takes seriously 2.3 Exercise 2.1 Belief and possibilities taken seri- (namely, every one (both) of cook and but- ously ler): (a) downstairs (after all, the Butler is in ▯ For other bodies of evidence, determine what downstairs and so is the Cook); (b) old after Holmes believes, disbelieves, and is uncer- Holmes’s beliefs are fixed by the possibilities (in all, the Butler is in old and so is the Cook; (c) tain about this case, suspects) he takes seriously. someone did the murder (after all, the Butler is someone and so is the Cook); and so forth 4 2.4 Inconsistent beliefs ▯ At the first possibility Watson takes all throughout section 1. But if I believe everything, seriously—hey, wait, Watson takes no pos- any evidence I get will not change that. Believing 2.4.1 Watson goes inconsistent sibilities seriously. So we’re done! There everything is a trap we can’t get out of. Suppose that after collecting the evidence in aren’t any counterexamples. So: at every possibility Watson takes seriously, woman., Watson thinks he sees a small footprint near 3 Logical relationships the murder site and convinces himself of woman. So Watson believes woman. The result would be the following: What else does Watson believe? Let’s ask: does The central notion of logic is entailment. It is a▯l- Watson believe man? iated with equivalence, nonentailment, and incon- woman young sistency. ▯ Here again, there are no counterexamples, heeiesse dutchess heiirr possibilities Watson takes seriously where not-man. 3.1 Entailment So Watson believes man. The general idea of entailment is that of compul- He also believes young, and old, and upstairs, sion: a set of claims entail Q when belief in the set and downstairs, ... and anything else you could compels belief in Q. coook footman buterre come up with—such as not-woman, and not-man, and ... 3.1.1 News that is no news He believes every claim. Suppose that in the case, Holmes is get- downstairs 2.4.3 That’s bad ting ready to clap the Dutchess in irons, when Wat- son rushes up with some news. He has found new After all, suppose Watson decided to clap someone evidence! A small footprint—the murderer is a At this point, Watson takes no possibilities seri- ously. in irons. The first person who comes along is the woman! Butler—a downstairs old man. Watson believes a Holmes, of course, will find this no surprise. He downstairs old man did it—so he claps the Butler in already has enough information to know that the 2.4.2 Watson believes every claim irons. Then the Heiress comes along—an upstairs murderer is a woman. Having learned the murderer After all, recall that a person believes P just if in young woman. Watson believes an upstairs young is an upstairs person and and old person, that settles woman did it—so he claps her in irons. But then— the Dutchess as the murderer. ‘every’ possibility they take seriously, P. Let’s ask: does Watson believe woman? wait—he also thinks it wasn’t a downstairs old man But that means the murderer is a woman. So who did it—so he sets the Butler free. But wait—it the ‘information’ Watson brought is no new infor- was a downstairs old man who did it—so he locks mation at all for Holmes. Holmes already had the ▯ Well, to check for whether every in possibil- ity someone takes seriously, P, we make sure the Butler back up again. information that the murderer was a woman. there are no counterexamples: no possibili- Or suppose Holmes decides to ask Watson’s Note—Holmes never got that piece of infor- opinion. ‘Was it a man?’ ‘Yes and no’, replies Wat- ties they take seriously where not-P. mation explicitly. It was implicit in his existing More explicitly: we start by checking the first son. ‘What do you mean?’ ‘I mean this: it was a information. man; and it was not a man’, replies Watson. ‘Uh, possiblity, then the second possibility, then let’s get you some rest’, replies Holmes. the third possibility ... and if we get to the end without having found one where not-P, Believing everything is disastrous. In order to commit to action, we need to settle one way or the the answer is yes. other. And in order to give useful information to Let’s assess whether there are no counterexamples others, we need to make up our minds. Believing everything is way worse than believ- where, at a certain possibility Watson takes seri- ously, not-P: ing nothing. If I believe nothing, I can get evi- dence and start believing something—as happened 5 3.1.2 Defining entailment 3.2.1 Defining equivalence 3.3.3 A caution about nonentailment In this case, ‘the murderer is an upstairs person’ Nonentailment is the lack of a requirement. It is and ‘the murderer is old’ entail ‘the murderer is a ▯ P and Q are equivalent not a forbiddance: woman’. , ▯ Despite (6), someone can accept all of the More generally: Anyone who believes P believes Q and claims in it: if someone thought the Heiress anyone who believes Q believes P is the murderer, they would accept upstairs, ▯ P 1 P2;::: entail Q , woman, and young , P entails Q and Q entails P Anyone who believes P ; P ;::: be- 1 2 3.3.4 Other examples lieves Q ▯ P a‘ Q abbreviates Here are some further examples of nonentailment: ▯ P 1 P2;::: ‘ Q P and Q are equivalent abbreviates 7. (a) upstairs;young 0 man (b) downstairs;man 0 young P 1 P2;::: entail Q (c) downstairs;old 0 woman ▯ We sometimes refer to P 1 P2;::: as the 3.2.2 Examples premisses and Q as the conclusion 5. (a) upstairs and man a‘ heir 3.4 Inconsistency (b) downstairs and woman a‘ downstairs and old and woman A notion closely related to entailment is inconsis- 3.1.3 Other examples tency. Here are some further examples of entailment: 3.3 Nonentailment 3.4.1 Crazy talk 4. (a) upstairs;man ‘ young Nonentailment is just absence of entailment. Suppose that in the case, Watson shows up (b) downstairs;young ‘ man and says ‘I know it was a woman!’ (c) downstairs;woman ‘ old 3.3.1 Defining nonentailment Holmes says ‘you agree it was a downstairs per- For nonentailment (absence of entailment), we son?’ Watson: ‘yes’. 3.2 Equivalence write: Holmes: ‘you agree it was a young person?’ Equivalence is mutual entailment: each entails the other. When claims are equivalent, they involve ex- ▯ P 1 P2;::: 0 Q Watson: ‘yes’. , Holmes: ‘so ... you agree no young downstairs actly the same information. person is a woman, right?’ Someone can believe P 1 P2;::: but not
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