WEBVTT 00:00:00.250 --> 00:00:04.153 Our four-card table uses X for dot and Y for stripe, 00:00:04.287 --> 00:00:05.352 in that bit order. 00:00:05.835 --> 00:00:10.290 A is one-zero, B one-one, C zero-one and D 00:00:10.408 --> 00:00:11.309 zero-zero. 00:00:11.826 --> 00:00:16.557 Classical X implies Y fails only when X is true and Y 00:00:16.622 --> 00:00:17.377 false. 00:00:17.786 --> 00:00:21.150 Its truth region collects the satisfying worlds. 00:00:21.770 --> 00:00:27.592 The positive observation opens describe what the chosen interface can certify. 00:00:28.165 --> 00:00:31.158 One means true and zero means false. 00:00:32.168 --> 00:00:38.672 Under positive observations X and Y, implication is true at B, C 00:00:38.790 --> 00:00:39.203 and D. 00:00:40.022 --> 00:00:43.994 Being inside this truth region answers a semantic question. 00:00:44.604 --> 00:00:50.810 Whether available positive certificates can establish membership is a different question, governed 00:00:50.810 --> 00:00:52.147 by the topology. 00:00:53.093 --> 00:00:58.215 Recall the four smallest neighborhoods: A has A and B; B has 00:00:58.247 --> 00:01:02.801 B alone; C has B and C; D has the whole carrier. 00:01:03.353 --> 00:01:08.865 Keep these sets beside E, the implication region B, C and D. 00:01:09.714 --> 00:01:14.571 We will test containment and intersection, rather than infer them from the 00:01:14.571 --> 00:01:16.156 drawing's geometry. 00:01:17.154 --> 00:01:22.867 A point is interior when some open neighborhood lies entirely inside E. 00:01:23.403 --> 00:01:28.127 In our finite space it is enough to test the smallest neighborhood. 00:01:28.582 --> 00:01:32.858 If it fits, it is a sufficient positive certificate; if it does 00:01:32.868 --> 00:01:35.669 not, no larger neighborhood will fit. 00:01:36.747 --> 00:01:41.195 B's singleton neighborhood fits inside the implication region. 00:01:41.849 --> 00:01:44.790 C's neighborhood, B and C, also fits. 00:01:45.424 --> 00:01:48.495 A's neighborhood contains A outside E. 00:01:48.495 --> 00:01:52.999 D's whole-carrier neighborhood also contains A, so those two points 00:01:53.031 --> 00:01:54.790 fail the interior test. 00:01:55.830 --> 00:01:57.708 The interior is B and C. 00:01:58.442 --> 00:02:03.064 In particular D satisfies implication but is not interior. 00:02:03.578 --> 00:02:07.864 That is a concrete instance of a true fact that this positive 00:02:07.864 --> 00:02:10.718 interface cannot certify at that point. 00:02:11.611 --> 00:02:16.355 A point belongs to the closure when every open neighborhood meets E. 00:02:16.985 --> 00:02:21.901 Since the smallest neighborhood is contained in all others, it suffices here 00:02:21.901 --> 00:02:25.861 to ask whether that smallest neighborhood meets E at all. 00:02:26.849 --> 00:02:29.421 Even A's neighborhood meets E through B. 00:02:30.180 --> 00:02:34.822 The other three points already lie in E, and each neighborhood contains 00:02:34.843 --> 00:02:35.743 its centre. 00:02:36.116 --> 00:02:41.451 Every point therefore passes the closure test, giving the whole carrier as 00:02:41.473 --> 00:02:42.508 the closure. 00:02:43.343 --> 00:02:49.843 If any open neighborhood witnesses interior membership, the smallest neighborhood is contained 00:02:49.843 --> 00:02:52.555 in it and also fits inside the event. 00:02:53.130 --> 00:02:56.792 Conversely the smallest neighborhood itself is open. 00:02:57.289 --> 00:03:03.377 That proves the interior criterion, rather than merely verifying our four examples. 00:03:04.491 --> 00:03:09.862 If the smallest neighborhood meets E, every containing neighborhood meets E. 00:03:10.434 --> 00:03:15.238 If it misses E, it is an explicit neighborhood proving failure of 00:03:15.259 --> 00:03:16.505 closure membership. 00:03:17.031 --> 00:03:22.632 These arguments depend on having smallest open neighborhoods, as our finite setting 00:03:22.632 --> 00:03:23.522 guarantees. 00:03:24.671 --> 00:03:29.704 Closure does not assign a likelihood, confirm that E is true, or 00:03:29.737 --> 00:03:31.449 model a delayed report. 00:03:31.932 --> 00:03:35.735 A can lie in the closure while falsifying implication. 00:03:36.232 --> 00:03:40.855 The claim is about the available neighborhoods and the inability of those 00:03:40.855 --> 00:03:42.678 neighborhoods to avoid E. 00:03:43.787 --> 00:03:48.259 The interior is also the union of all opens contained in E. 00:03:48.782 --> 00:03:52.255 This description works beyond our finite example. 00:03:52.691 --> 00:03:57.498 The neighborhood formula makes it especially easy to compute here, provided we 00:03:57.509 --> 00:04:00.581 preserve the correct information-order direction. 00:04:01.679 --> 00:04:06.291 Under positive X and Y, compute the interior and closure of the 00:04:06.291 --> 00:04:08.065 singleton containing A. 00:04:08.634 --> 00:04:14.367 Which smallest neighborhoods fit entirely inside it, and which merely meet it? 00:04:34.897 --> 00:04:41.018 No smallest neighborhood fits inside A alone, because even A's neighborhood 00:04:41.050 --> 00:04:42.461 also contains B. 00:04:43.155 --> 00:04:44.948 The interior is empty. 00:04:45.504 --> 00:04:49.270 A's and D's neighborhoods meet the singleton, while B's 00:04:49.280 --> 00:04:52.744 and C's do not, so its closure is A and D. 00:04:53.923 --> 00:04:58.294 Interior asks whether a whole neighborhood fits inside the event. 00:04:58.855 --> 00:05:02.784 Closure asks whether the neighborhood has any contact with it. 00:05:03.285 --> 00:05:07.419 The two tests have different quantifiers and can give very different answers 00:05:07.430 --> 00:05:09.448 on the same fixed truth region.