WEBVTT 00:00:00.250 --> 00:00:03.050 Compare two meanings of acquiring information. 00:00:04.169 --> 00:00:08.222 For this example, X means dot and Y means stripe. 00:00:08.641 --> 00:00:12.799 A has only a dot, B both, C only a stripe, and 00:00:12.799 --> 00:00:13.736 D neither. 00:00:14.240 --> 00:00:17.504 Their two-bit annotations follow X,Y order. 00:00:17.969 --> 00:00:21.376 These are the same four cards at the start of both branches. 00:00:22.500 --> 00:00:27.654 Both branches begin with all four valuations and only X selected. 00:00:28.125 --> 00:00:31.847 The opens are empty, A and B, and the whole carrier. 00:00:32.336 --> 00:00:37.978 Reset both branches before changing anything; otherwise their effects cannot be compared 00:00:37.988 --> 00:00:38.739 fairly. 00:00:39.602 --> 00:00:43.239 In the first branch we add Y to the observation menu. 00:00:43.837 --> 00:00:47.045 In the second we accept Y as a premise. 00:00:47.464 --> 00:00:51.880 The first changes which events can be certified; the second changes which 00:00:51.880 --> 00:00:54.166 valuations remain possible. 00:00:54.977 --> 00:00:59.687 Adding the available test Y retains A, B, C and D. 00:01:00.456 --> 00:01:04.104 It produces the six-open positive topology. 00:01:04.634 --> 00:01:08.237 The fact that we can now ask for a Y certificate does 00:01:08.237 --> 00:01:12.257 not say the actual valuation satisfies Y. 00:01:13.177 --> 00:01:16.790 Do not remove the Y-false cards in this branch. 00:01:17.444 --> 00:01:23.086 Availability is a feature of the interface, whereas a learned premise asserts 00:01:23.086 --> 00:01:25.592 a condition on possible worlds. 00:01:26.237 --> 00:01:29.894 Confusing the two would make the comparison meaningless. 00:01:31.045 --> 00:01:35.191 Return the other branch to the original one-test menu X. 00:01:35.890 --> 00:01:38.187 Accept Y as true. 00:01:38.606 --> 00:01:43.730 The possible carrier becomes B and C, the intersection of the old 00:01:43.740 --> 00:01:47.269 possibility region with the truth region of Y. 00:01:48.240 --> 00:01:51.529 Restrict every old open to this new carrier. 00:01:52.106 --> 00:01:55.654 The old dot region meets B and C at B alone. 00:01:55.675 --> 00:01:58.595 The whole old carrier restricts to B and C. 00:01:59.339 --> 00:02:04.027 Together with the empty set, these are the induced subspace opens. 00:02:05.172 --> 00:02:10.268 For a surviving point, intersect its old smallest neighborhood with the new 00:02:10.278 --> 00:02:11.765 possibility region. 00:02:12.395 --> 00:02:16.855 Under X alone, B's old neighborhood was A and B; after 00:02:16.876 --> 00:02:18.949 learning Y it becomes B. 00:02:19.773 --> 00:02:23.992 This is restriction, not a newly invented topology. 00:02:25.116 --> 00:02:27.925 Learning a second premise intersects again. 00:02:28.609 --> 00:02:34.692 The final possibility region is unchanged if we regroup the intersections, reverse 00:02:34.692 --> 00:02:38.843 the order of the premises, or repeat one premise. 00:02:39.340 --> 00:02:45.207 These properties are called associativity, commutativity, and idempotence. 00:02:45.701 --> 00:02:51.742 They concern accepted classical premises, not every real-world learning protocol. 00:02:52.605 --> 00:02:56.506 After learning Y, accepting X retains only B. 00:02:57.340 --> 00:03:02.114 On a singleton carrier the induced topology has the empty set and 00:03:02.114 --> 00:03:03.026 that singleton. 00:03:03.631 --> 00:03:07.971 This does not retroactively mean X was learned when it was merely 00:03:07.981 --> 00:03:09.925 added as an available test. 00:03:10.954 --> 00:03:14.668 Inconsistent premises leave no surviving valuation. 00:03:15.200 --> 00:03:19.141 The empty subspace is a legitimate mathematical outcome. 00:03:19.576 --> 00:03:24.438 It does not identify an impossible state card or assign truth values 00:03:24.459 --> 00:03:27.054 to a newly drawn empty-world point. 00:03:28.043 --> 00:03:31.861 Reset to all four states with Y as the sole test. 00:03:32.442 --> 00:03:36.425 Compare adding X as an available test with learning X as a 00:03:36.435 --> 00:03:37.274 premise. 00:03:37.783 --> 00:03:41.613 Give the carrier in each branch and the induced opens in the 00:03:41.613 --> 00:03:42.612 premise branch. 00:04:03.078 --> 00:04:07.288 Adding X keeps the full carrier and gives the positive X,Y 00:04:07.298 --> 00:04:08.199 topology. 00:04:08.822 --> 00:04:13.651 Learning X instead keeps only A and B while retaining the menu 00:04:13.705 --> 00:04:14.125 Y. 00:04:14.662 --> 00:04:19.726 Restricting Y's region gives B, so the premise branch has opens 00:04:19.780 --> 00:04:22.965 empty, B and the carrier A and B. 00:04:24.321 --> 00:04:28.027 State the menu and carrier after every change. 00:04:28.523 --> 00:04:32.603 Refinement changes expressibility on fixed points. 00:04:33.272 --> 00:04:39.419 Premise update changes the current points and retains the induced observation structure. 00:04:39.894 --> 00:04:45.309 Later inference will ask whether all surviving worlds satisfy a conclusion.