WEBVTT 00:00:00.250 --> 00:00:03.599 We now study an explicitly different interpretation. 00:00:04.018 --> 00:00:08.033 Classical formulas use ordinary complement of truth regions. 00:00:08.668 --> 00:00:13.361 In the open-set interpretation, the values of formulas must remain open. 00:00:13.804 --> 00:00:19.535 On a nondiscrete space, a classical complement need not satisfy that requirement. 00:00:20.481 --> 00:00:24.770 For this example there are just two points, zero and one. 00:00:25.092 --> 00:00:29.289 The only opens are the empty set, the singleton one, and the 00:00:29.289 --> 00:00:30.273 whole carrier. 00:00:30.751 --> 00:00:33.301 The singleton zero is not open. 00:00:33.905 --> 00:00:39.166 Interior means the largest open subset of the set being considered. 00:00:39.823 --> 00:00:44.673 Use the labelled two-point space, with X interpreted by the open 00:00:44.705 --> 00:00:46.536 singleton containing one. 00:00:46.997 --> 00:00:50.927 Its classical complement is the singleton containing zero. 00:00:51.425 --> 00:00:55.410 That set is not open in this topology, so it cannot be 00:00:55.410 --> 00:00:57.781 the open-set value of negation. 00:00:58.680 --> 00:01:02.511 Take the largest open subset inside the complement. 00:01:02.532 --> 00:01:04.274 This is Heyting negation. 00:01:04.801 --> 00:01:10.319 It includes every open event disjoint from U, while remaining disjoint from 00:01:10.374 --> 00:01:11.220 U itself. 00:01:11.895 --> 00:01:15.867 It need not preserve every classical falsifying point. 00:01:16.976 --> 00:01:20.992 The complement of the singleton one is the singleton zero. 00:01:21.590 --> 00:01:24.355 Its only open subset is empty. 00:01:24.915 --> 00:01:30.085 Therefore open-set negation of our chosen X is empty, even though 00:01:30.085 --> 00:01:33.029 its classical complement was nonempty. 00:01:34.047 --> 00:01:38.688 Apply the same open-set rule again to the result just obtained. 00:01:39.192 --> 00:01:43.640 The complement of empty is the whole carrier, already open. 00:01:43.986 --> 00:01:48.466 Double open-set negation therefore gives the whole two-point carrier. 00:01:49.423 --> 00:01:51.800 The intermediate empty set matters. 00:01:52.509 --> 00:01:58.010 Replacing this two-step evaluation with a classical simplification would erase it 00:01:58.021 --> 00:02:00.438 and give the wrong open-set result. 00:02:00.860 --> 00:02:06.574 A recursive semantics evaluates each connective using the specified operation. 00:02:07.590 --> 00:02:13.664 Classically, double negation returns the original truth region, the singleton one. 00:02:14.145 --> 00:02:19.257 Taking its interior afterward still gives the singleton one because it is 00:02:19.278 --> 00:02:19.712 open. 00:02:20.326 --> 00:02:24.779 That is different from the recursive open-set result, the whole carrier. 00:02:25.690 --> 00:02:29.401 We have displayed two procedures with different answers. 00:02:30.050 --> 00:02:33.939 This is not an inconsistency within either procedure. 00:02:34.426 --> 00:02:40.503 It shows that classical truth evaluation followed by one interior operation does 00:02:40.503 --> 00:02:44.270 not generally reproduce recursive Heyting evaluation. 00:02:45.351 --> 00:02:50.169 An open event and its Heyting negation are disjoint by construction. 00:02:50.778 --> 00:02:55.531 Every open event disjoint from U is included in this largest open 00:02:55.531 --> 00:02:56.312 exclusion. 00:02:57.016 --> 00:03:02.088 These statements follow from the interior definition, not from assuming that all 00:03:02.088 --> 00:03:05.078 classical complement identities survive. 00:03:05.890 --> 00:03:11.755 In a discrete topology, every complement is already open, so interior removes 00:03:11.776 --> 00:03:12.276 nothing. 00:03:12.875 --> 00:03:16.430 Heyting negation then agrees with classical complement. 00:03:16.955 --> 00:03:22.087 The difference is created by the observability requirement in a nondiscrete topology. 00:03:23.171 --> 00:03:27.403 On the same two-point topology, start with the empty open event 00:03:27.403 --> 00:03:28.135 instead. 00:03:28.632 --> 00:03:31.884 Compute its Heyting negation and double negation. 00:03:32.368 --> 00:03:37.055 Keep the result of the first operation visible before applying the second. 00:03:53.369 --> 00:03:57.905 The first complement is the whole carrier, whose interior is itself. 00:03:58.594 --> 00:04:03.151 Negating that whole carrier gives the interior of empty, so the double 00:04:03.151 --> 00:04:04.497 negation is empty. 00:04:05.096 --> 00:04:09.014 The procedure is the same even though this input happens to return 00:04:09.024 --> 00:04:09.745 to itself. 00:04:10.765 --> 00:04:15.027 Open-set negation means interior of complement at each step. 00:04:15.553 --> 00:04:20.887 It should not be substituted silently into the earlier classical inference videos. 00:04:21.476 --> 00:04:26.099 Next we will find the corresponding open implication by asking for its 00:04:26.099 --> 00:04:27.722 defining universal property.