WEBVTT 00:00:00.250 --> 00:00:03.719 A truth region collects the worlds where a formula is true. 00:00:04.138 --> 00:00:09.054 AND corresponds to intersection, inclusive OR to union, and classical NOT to 00:00:09.075 --> 00:00:11.313 complement inside the current carrier. 00:00:11.789 --> 00:00:15.727 Our cards use X for dot and Y for stripe: A has 00:00:15.748 --> 00:00:20.663 only a dot, B both, C only a stripe, and D neither. 00:00:21.147 --> 00:00:24.699 Premises must be evaluated together at one world. 00:00:25.523 --> 00:00:28.898 Collect the valuations satisfying all premises. 00:00:29.507 --> 00:00:35.182 A conclusion follows classically when every one of those surviving valuations also 00:00:35.215 --> 00:00:36.963 satisfies the conclusion. 00:00:37.453 --> 00:00:42.011 This is a universal claim, so one failing survivor is enough to 00:00:42.032 --> 00:00:42.945 disprove it. 00:00:43.870 --> 00:00:48.319 In set language, the common premise region must be included in the 00:00:48.319 --> 00:00:50.012 conclusion's truth region. 00:00:50.641 --> 00:00:54.287 Inclusion permits equality but does not require it. 00:00:54.778 --> 00:01:00.484 A stronger group of premises may select fewer valuations than its consequence 00:01:00.484 --> 00:01:01.180 selects. 00:01:02.314 --> 00:01:05.103 Take X and Y together as premises. 00:01:05.727 --> 00:01:08.931 Their common truth region is B alone. 00:01:09.414 --> 00:01:12.528 The conclusion X is true at A and B. 00:01:13.258 --> 00:01:17.414 Every premise survivor is therefore in the conclusion region. 00:01:18.538 --> 00:01:22.675 The inclusion is proper, yet the inference is valid. 00:01:23.142 --> 00:01:28.286 The conclusion allows an additional world A, which does not satisfy both 00:01:28.286 --> 00:01:29.058 premises. 00:01:29.692 --> 00:01:34.445 A consequence can be weaker than the information supplied by the premises. 00:01:35.594 --> 00:01:38.827 Now ask whether X alone entails Y. 00:01:39.298 --> 00:01:41.837 The premise region contains A and B. 00:01:42.562 --> 00:01:48.964 A satisfies X and falsifies Y, so A is a surviving counterexample. 00:01:49.348 --> 00:01:55.230 B's success cannot rescue a universally quantified claim that fails at 00:01:55.240 --> 00:01:55.560 A. 00:01:56.607 --> 00:02:02.120 In general, compute the premise region with the conclusion region removed. 00:02:02.570 --> 00:02:06.324 A nonempty remainder consists of counterexamples. 00:02:06.978 --> 00:02:13.044 An empty remainder is exactly inclusion, so the same finite procedure tests 00:02:13.076 --> 00:02:17.015 validity and identifies where invalidity occurs. 00:02:18.140 --> 00:02:23.226 If each formula entails the other, the two truth regions are included 00:02:23.226 --> 00:02:25.392 in one another and therefore equal. 00:02:25.871 --> 00:02:28.073 This is semantic equivalence. 00:02:28.493 --> 00:02:33.088 It is stronger than sharing some true rows or one-way entailment. 00:02:34.063 --> 00:02:38.611 If the premises are inconsistent, their common region is empty. 00:02:39.030 --> 00:02:44.325 There is no surviving valuation that could satisfy all premises and falsify 00:02:44.346 --> 00:02:45.247 the conclusion. 00:02:45.882 --> 00:02:53.591 Classical semantic consequence therefore holds vacuously: the universal claim has no counterexample. 00:02:54.001 --> 00:02:58.797 This does not make the premises satisfiable or provide an actual world 00:02:58.797 --> 00:03:00.413 witnessing the conclusion. 00:03:01.558 --> 00:03:07.390 Choosing fewer or more observations changes which events are certifiable. 00:03:07.800 --> 00:03:12.194 It does not change the classical truth regions of the fixed formulas. 00:03:12.823 --> 00:03:18.336 A valid inclusion remains valid even when the available interface cannot certify 00:03:18.336 --> 00:03:21.315 its conclusion at every satisfying point. 00:03:22.366 --> 00:03:27.097 Conjoining premises intersects subsets of valuations. 00:03:27.707 --> 00:03:34.532 Intersecting two topologies instead intersects collections of subsets on one carrier. 00:03:35.067 --> 00:03:39.483 Those are different types of operation, and the latter is not how 00:03:39.504 --> 00:03:42.012 we compute a common premise region. 00:03:43.121 --> 00:03:47.235 Do X or Y together with not X entail Y? 00:03:47.861 --> 00:03:52.024 Find the common premise region, compare it with Y's region, and 00:03:52.024 --> 00:03:54.856 explain why no counterexample remains. 00:04:13.592 --> 00:04:16.555 X or Y allows A, B and C. 00:04:17.311 --> 00:04:19.416 Not X allows C and D. 00:04:20.245 --> 00:04:24.520 Their intersection is C, which lies inside Y's region B and 00:04:24.520 --> 00:04:24.803 C. 00:04:25.557 --> 00:04:30.063 The inference is valid by inclusion, whether or not we attach a 00:04:30.063 --> 00:04:31.557 traditional rule name. 00:04:32.495 --> 00:04:39.846 The method is stable: evaluate formulas, intersect premise regions, then test inclusion. 00:04:40.490 --> 00:04:46.013 Observational topology adds structure to these regions and their maps; it does 00:04:46.035 --> 00:04:50.180 not introduce a new criterion of classical logical validity.