WEBVTT 00:00:00.250 --> 00:00:04.363 Could two situations be different, yet look exactly the same through the 00:00:04.384 --> 00:00:06.594 observations we are allowed to make? 00:00:07.393 --> 00:00:10.391 Imagine a card hidden inside an envelope. 00:00:10.587 --> 00:00:14.224 It might have a dot, a stripe, both, or neither. 00:00:15.217 --> 00:00:17.268 We want to work out which card it is. 00:00:17.892 --> 00:00:21.408 But what we can discover will depend on what evidence we are 00:00:21.408 --> 00:00:22.704 allowed to use. 00:00:23.614 --> 00:00:25.655 Let’s give two statements short names. 00:00:26.291 --> 00:00:28.678 X means “the card has a dot.” 00:00:29.309 --> 00:00:31.605 Y means “the card has a stripe.” 00:00:31.927 --> 00:00:33.588 There are four possibilities. 00:00:34.722 --> 00:00:36.807 Card A has a dot only. 00:00:37.293 --> 00:00:39.300 Card B has both features. 00:00:39.910 --> 00:00:42.113 Card C has a stripe only. 00:00:42.722 --> 00:00:44.554 Card D has neither. 00:00:45.417 --> 00:00:50.583 Each possibility is a state: a complete answer to both questions about 00:00:50.583 --> 00:00:51.249 the card. 00:00:51.692 --> 00:00:56.648 On our diagram, the first bit answers X, the dot question, and 00:00:56.648 --> 00:00:59.884 the second answers Y, the stripe question. 00:01:00.367 --> 00:01:03.924 One means true and zero means false. 00:01:04.917 --> 00:01:07.978 This diagram is our map of the possibilities. 00:01:08.587 --> 00:01:12.613 It does not tell the observer which card is actually inside the 00:01:12.623 --> 00:01:13.349 envelope. 00:01:14.342 --> 00:01:19.458 Now suppose the only available certificate says, “The card has a dot.” 00:01:20.020 --> 00:01:24.247 A certificate is evidence establishing the statement written on it. 00:01:25.203 --> 00:01:27.363 That statement is true for A and B. 00:01:28.062 --> 00:01:29.853 It is false for C and D. 00:01:31.027 --> 00:01:34.396 A and B really are different: B has a stripe. 00:01:34.893 --> 00:01:37.828 But the dot certificate cannot tell them apart. 00:01:38.357 --> 00:01:42.301 Both satisfy exactly the same available test. 00:01:43.258 --> 00:01:48.314 It can distinguish A from D in one direction: certifying a dot 00:01:48.325 --> 00:01:49.226 excludes D. 00:01:49.975 --> 00:01:53.576 It also excludes C, because C has no dot. 00:01:54.090 --> 00:01:58.522 The positive dot certificate leaves A and B possible. 00:01:59.291 --> 00:02:00.976 The reverse needs care. 00:02:01.431 --> 00:02:05.871 We have not supplied a certificate saying “the card has no dot.” 00:02:06.354 --> 00:02:10.675 Without a dot certificate, we cannot simply conclude “no dot.” 00:02:11.173 --> 00:02:15.112 A statement can be false without its negation being available as a 00:02:15.112 --> 00:02:16.205 certificate. 00:02:17.004 --> 00:02:20.432 We can draw a region around the states that a certificate would 00:02:20.432 --> 00:02:21.114 confirm. 00:02:21.575 --> 00:02:25.051 For “has a dot,” that region contains A and B. 00:02:26.216 --> 00:02:31.174 Topology gives precise rules for organizing regions called open sets. 00:02:31.794 --> 00:02:36.471 In this paper, those regions are generated from the observations we choose 00:02:36.471 --> 00:02:37.734 to make available. 00:02:38.644 --> 00:02:42.953 They record which distinctions those observations can express. 00:02:44.102 --> 00:02:45.612 We will learn the rules later. 00:02:46.131 --> 00:02:50.536 For now, the key idea is that topology can describe the structure 00:02:50.547 --> 00:02:55.972 of observation, even when our points are logical states rather than locations 00:02:56.005 --> 00:02:56.689 on a map. 00:02:57.680 --> 00:03:02.227 There are two ways we might improve our information: add an observation, 00:03:02.512 --> 00:03:03.665 or learn a fact. 00:03:04.632 --> 00:03:09.319 First, we could make another certificate available: “The card has a stripe.” 00:03:09.746 --> 00:03:13.104 We would then have a way to distinguish B from A. 00:03:13.657 --> 00:03:18.410 All four cards would still be possible; we have improved our observation 00:03:18.421 --> 00:03:19.001 tools. 00:03:20.143 --> 00:03:23.827 Second, we could learn as a fact that the hidden card has 00:03:23.827 --> 00:03:24.660 a stripe. 00:03:25.078 --> 00:03:28.152 Then only B and C remain possible. 00:03:28.619 --> 00:03:30.311 We have narrowed the candidates. 00:03:31.446 --> 00:03:34.776 Adding an observation changes what we can distinguish. 00:03:35.423 --> 00:03:39.313 Learning a fact changes which states remain possible. 00:03:39.723 --> 00:03:43.629 The paper carefully separates these two operations. 00:03:44.778 --> 00:03:46.418 Let’s check the distinction. 00:03:47.537 --> 00:03:52.847 Return to our original setup, with only the dot certificate available. 00:03:53.781 --> 00:03:58.296 If you obtain that certificate, can you conclude that the card is 00:03:58.349 --> 00:03:58.708 A? 00:03:59.838 --> 00:04:03.570 And if you have no such certificate, can you conclude that the 00:04:03.570 --> 00:04:05.045 card has no dot? 00:04:19.284 --> 00:04:20.825 Both answers are no. 00:04:21.724 --> 00:04:25.030 The dot certificate leaves A and B possible. 00:04:25.330 --> 00:04:30.072 And having no dot certificate does not establish that the card lacks 00:04:30.082 --> 00:04:30.798 a dot. 00:04:31.247 --> 00:04:35.447 A certificate for “no dot” would be a separate observation. 00:04:36.199 --> 00:04:40.409 With only the dot test available, does it distinguish A from D? 00:04:41.193 --> 00:04:42.984 Does it distinguish A from B? 00:04:43.795 --> 00:04:49.274 State which positive certificate could establish the distinction, rather than guessing the 00:04:49.274 --> 00:04:50.175 hidden card. 00:05:04.619 --> 00:05:09.621 The dot test distinguishes A from D in one direction: a positive 00:05:09.621 --> 00:05:11.456 dot certificate excludes D. 00:05:12.195 --> 00:05:16.885 It does not distinguish A from B, because both satisfy the same 00:05:16.885 --> 00:05:18.090 available test. 00:05:18.689 --> 00:05:24.277 The presence of a distinction is different from identifying the actual valuation. 00:05:25.203 --> 00:05:30.987 This is the starting point of propositional logical topology: connect the states 00:05:30.987 --> 00:05:35.831 of a logical system with the distinctions its observations support. 00:05:36.761 --> 00:05:42.349 The paper brings established ideas from logic and topology together around that 00:05:42.349 --> 00:05:43.132 connection. 00:05:44.057 --> 00:05:49.963 We will build the background step by step, including sets, logic, distance, 00:05:49.963 --> 00:05:51.057 and topology. 00:05:52.091 --> 00:05:56.366 Next, we start with sets: a simple language for describing which states 00:05:56.366 --> 00:05:57.886 belong to which regions.