WEBVTT 00:00:00.250 --> 00:00:04.753 Our original X,Y values are A: one-zero, B: one-one, 00:00:04.774 --> 00:00:07.868 C: zero-one and D: zero-zero. 00:00:08.012 --> 00:00:12.179 A selected signature lists answers under the chosen formula headers. 00:00:12.788 --> 00:00:15.241 Its support lists the true slots. 00:00:15.327 --> 00:00:20.297 A point’s smallest neighborhood contains exactly the worlds preserving those selected 00:00:20.307 --> 00:00:21.045 positives. 00:00:21.655 --> 00:00:26.535 Keep the current tuple, weights and possible worlds as separate inputs. 00:00:27.160 --> 00:00:30.596 One means true and zero means false. 00:00:31.395 --> 00:00:35.686 Begin with the full four-state carrier and the menu X alone. 00:00:36.151 --> 00:00:39.299 Keep presentation and weights as separate labels. 00:00:39.918 --> 00:00:44.638 We will reconstruct several structures and then change one ingredient at a 00:00:44.638 --> 00:00:47.919 time so that their dependencies remain visible. 00:00:48.853 --> 00:00:54.387 The selected X column gives two realized signatures and two equal-signature 00:00:54.387 --> 00:00:55.081 classes. 00:00:55.695 --> 00:00:59.321 The smallest neighborhood of A or B is A and B. 00:01:00.000 --> 00:01:03.754 The smallest neighborhood of C or D is the whole carrier, because 00:01:03.754 --> 00:01:06.427 no positive requirement is imposed there. 00:01:07.197 --> 00:01:09.899 Our order preserves selected positives. 00:01:10.595 --> 00:01:14.263 Symmetric distance counts all disagreements with their weights. 00:01:14.912 --> 00:01:20.138 Directed loss counts only true-to-false changes, keeping the source first. 00:01:21.252 --> 00:01:25.300 The false-signature class precedes the true-signature class. 00:01:25.934 --> 00:01:31.100 With unit weight the symmetric distance across classes is one, while directed 00:01:31.111 --> 00:01:34.800 loss is zero from false to true and one in the reverse 00:01:34.800 --> 00:01:35.489 direction. 00:01:35.996 --> 00:01:40.775 The direction identifies loss, not a temporal learning process. 00:01:41.909 --> 00:01:46.564 Positive X generates three opens on the four valuations. 00:01:47.273 --> 00:01:52.092 Symmetric signature balls give the four-open partition topology. 00:01:52.603 --> 00:01:57.844 The two constructions use the same signatures but answer different questions about 00:01:57.865 --> 00:02:00.752 one-sided versus two-sided distinctions. 00:02:01.856 --> 00:02:04.420 Add Y as an available test. 00:02:04.986 --> 00:02:07.296 All four signatures now differ. 00:02:07.826 --> 00:02:12.796 The six positive opens and the D–A/C–B information diagram 00:02:12.850 --> 00:02:14.820 follow from those signatures. 00:02:15.403 --> 00:02:19.541 The carrier remains fixed because no premise has been learned. 00:02:20.516 --> 00:02:25.433 Interior asks whether a whole smallest neighborhood fits in the event. 00:02:26.032 --> 00:02:28.864 Closure asks whether it meets the event. 00:02:29.463 --> 00:02:32.720 Boundary is closure with interior removed. 00:02:33.251 --> 00:02:38.701 Here the event is classical X implies Y, false only at true 00:02:38.723 --> 00:02:40.433 X and false Y. 00:02:41.443 --> 00:02:46.411 Implication's interior is B and C, while its closure is the 00:02:46.411 --> 00:02:49.352 whole carrier and its boundary A and D. 00:02:50.118 --> 00:02:56.650 D satisfies the formula without a sufficient positive certificate under this interface. 00:02:57.130 --> 00:03:01.643 Truth evaluation and certifiability remain distinct. 00:03:02.589 --> 00:03:06.918 Now accept two premises: X implies Y, and X. 00:03:07.592 --> 00:03:10.067 Their common truth region is B. 00:03:10.819 --> 00:03:16.592 Restrict the ambient topology to that surviving carrier rather than declaring a 00:03:16.645 --> 00:03:18.833 new unrelated topology. 00:03:19.333 --> 00:03:25.230 Every surviving world satisfies Y, so the conclusion follows by inclusion. 00:03:26.160 --> 00:03:31.052 The menu determines signatures and the qualitative topology and order. 00:03:31.611 --> 00:03:35.050 Weights additionally determine numerical distances. 00:03:35.670 --> 00:03:38.882 Premises determine a possibility subregion. 00:03:39.405 --> 00:03:44.625 Keeping these arrows separate prevents treating a topology as a complete record 00:03:44.635 --> 00:03:46.551 of every modelling choice. 00:03:47.396 --> 00:03:52.847 For this two-slot exercise, use unit weight on each selected coordinate. 00:03:53.302 --> 00:03:58.174 Read the new formula headers when building the signatures; keep the original 00:03:58.207 --> 00:04:01.299 X,Y valuation annotations fixed. 00:04:02.424 --> 00:04:07.036 Return to the full carrier and select X as observation one and 00:04:07.067 --> 00:04:09.585 X or Y as observation two. 00:04:10.205 --> 00:04:16.814 Find the signatures, equal-signature classes, smallest neighborhoods and the directed losses 00:04:16.836 --> 00:04:18.512 between C and A. 00:04:19.081 --> 00:04:21.045 Pause and work from the table. 00:04:51.121 --> 00:04:54.304 A and B have both selected formulas true. 00:04:54.709 --> 00:04:58.185 C has only the second true, and D has neither. 00:04:58.633 --> 00:05:03.648 The classes are A and B together, C alone and D alone. 00:05:04.058 --> 00:05:08.089 Their quotient supports form a three-level inclusion chain. 00:05:08.912 --> 00:05:12.656 Above the class A and B there are no additional signatures. 00:05:13.301 --> 00:05:17.749 Above C lies that class too; above D lies everything. 00:05:18.398 --> 00:05:24.901 These upward neighborhoods reconstruct the topology with four opens: empty, A and 00:05:24.933 --> 00:05:28.728 B, A and B and C, and the carrier. 00:05:29.812 --> 00:05:33.392 C loses no selected positive when moving to A. 00:05:34.002 --> 00:05:38.678 In the reverse direction A loses the first coordinate but keeps the 00:05:38.689 --> 00:05:40.481 second, costing one. 00:05:40.916 --> 00:05:46.515 Distinct complete valuations A and B still have identical signatures under this 00:05:46.569 --> 00:05:47.712 new tuple. 00:05:48.462 --> 00:05:53.757 Making both signs available for this tuple opens its three original classes, 00:05:54.038 --> 00:05:57.119 yielding eight opens on the four valuations. 00:05:57.763 --> 00:05:59.651 It does not split A from B. 00:06:00.395 --> 00:06:04.670 This is another reason both signs need not mean discreteness on the 00:06:04.670 --> 00:06:05.867 original carrier. 00:06:06.948 --> 00:06:13.295 A successful reconstruction names each input, computes the resulting objects and explains 00:06:13.295 --> 00:06:14.499 their dependencies. 00:06:15.109 --> 00:06:20.144 If a remembered answer disagrees with the displayed formula table, return to 00:06:20.154 --> 00:06:20.879 the table. 00:06:21.340 --> 00:06:26.006 The framework is a chain of constructions, not a single diagram to 00:06:26.027 --> 00:06:26.913 recite.