WEBVTT 00:00:00.250 --> 00:00:05.046 A complete valuation answers every variable question in our chosen language. 00:00:05.057 --> 00:00:09.410 In the card example, X means dot and Y means stripe. 00:00:09.491 --> 00:00:13.411 Where a state has a two-bit annotation, those bits give its 00:00:13.465 --> 00:00:18.388 fixed X,Y values: one for true, zero for false. 00:00:18.908 --> 00:00:22.869 A selected formula list will have its own ordered headers. 00:00:24.008 --> 00:00:27.258 A makes X true and Y false. 00:00:27.733 --> 00:00:33.197 That state remains A whether we observe many formulas or only one. 00:00:34.096 --> 00:00:38.783 Now choose the ordered list X, Y as our observations. 00:00:39.412 --> 00:00:41.091 We call the list Phi. 00:00:41.575 --> 00:00:46.414 Ordered means that each position has a fixed meaning: position one is 00:00:46.468 --> 00:00:49.058 X, and position two is Y. 00:00:49.992 --> 00:00:52.477 An index is just a position label. 00:00:52.913 --> 00:00:57.449 We can write formula one and formula two to name those entries. 00:00:58.068 --> 00:01:03.158 An indexed family can include repetitions, so it is more informative than 00:01:03.158 --> 00:01:05.588 an ordinary set of formula names. 00:01:06.732 --> 00:01:08.820 A state supplies the input. 00:01:09.352 --> 00:01:13.433 The formula list specifies what to evaluate there. 00:01:13.944 --> 00:01:16.853 Neither is the resulting list of answers. 00:01:17.982 --> 00:01:20.112 Evaluate the list at A. 00:01:20.720 --> 00:01:24.081 X is true, so the first entry is one. 00:01:24.588 --> 00:01:28.296 Y is false, so the second entry is zero. 00:01:28.835 --> 00:01:32.551 The signature of A for this list is one, zero. 00:01:33.524 --> 00:01:35.927 At B both entries are one. 00:01:36.370 --> 00:01:38.551 At C they are zero, one. 00:01:38.987 --> 00:01:41.424 At D they are zero, zero. 00:01:41.891 --> 00:01:45.716 The formula headers tell us how to read every vector. 00:01:46.550 --> 00:01:51.184 A signature is the complete truth vector of the selected tuple at 00:01:51.205 --> 00:01:52.432 one valuation. 00:01:53.052 --> 00:01:58.328 It may have fewer entries than the valuation, or more, because we 00:01:58.339 --> 00:02:02.321 choose the formulas independently of the variable list. 00:02:03.435 --> 00:02:07.494 The zero at A says that Y is false at A. 00:02:08.057 --> 00:02:11.399 It does not mean that a reply has not arrived. 00:02:12.009 --> 00:02:17.311 Every entry of this semantic profile already has a definite value. 00:02:18.079 --> 00:02:23.895 The observation interface answers a different question: which affirmative events are available 00:02:23.895 --> 00:02:25.056 as certificates? 00:02:25.676 --> 00:02:30.580 With X and Y selected, an X certificate establishes X, and a 00:02:30.635 --> 00:02:33.204 Y certificate establishes Y. 00:02:34.067 --> 00:02:38.720 Absence of a Y certificate does not itself establish not Y. 00:02:39.192 --> 00:02:43.867 The topology gives no waiting time or certificate-delivery rule that would 00:02:43.867 --> 00:02:45.588 license that conclusion. 00:02:46.487 --> 00:02:51.239 Knowing the complete semantic profile would tell us that Y is false 00:02:51.239 --> 00:02:51.807 at A. 00:02:52.311 --> 00:02:55.794 Having no Y certificate is a different situation. 00:02:56.413 --> 00:03:01.763 Keep the instructor's mathematical table separate from the evidence actually supplied 00:03:01.763 --> 00:03:02.793 to an observer. 00:03:03.815 --> 00:03:09.053 A viewer says, “The second bit is zero because no stripe certificate 00:03:09.053 --> 00:03:09.731 arrived.” 00:03:10.276 --> 00:03:14.747 Does that correctly explain the zero in A's complete signature? 00:03:26.964 --> 00:03:27.380 No. 00:03:27.880 --> 00:03:30.928 The bit is zero because Y is false at A. 00:03:31.508 --> 00:03:36.261 Missing delivery is not what the signature records, and our topology does 00:03:36.282 --> 00:03:37.941 not model delivery. 00:03:38.804 --> 00:03:43.241 Choose a different list: the single conjunction formula X and Y. 00:03:43.690 --> 00:03:45.653 There is just one coordinate. 00:03:45.951 --> 00:03:50.248 B has signature one, while A, C and D all have signature 00:03:50.258 --> 00:03:53.823 zero, because each makes that conjunction false. 00:03:54.640 --> 00:03:57.721 Those three states remain different valuations. 00:03:58.370 --> 00:04:01.551 This observation simply does not distinguish them. 00:04:02.129 --> 00:04:05.731 A zero signature is not an extra state and is not an 00:04:05.785 --> 00:04:07.650 unknown truth value. 00:04:08.448 --> 00:04:12.331 If the list is only X instead, A and B both give 00:04:12.342 --> 00:04:14.985 one, while C and D both give zero. 00:04:15.548 --> 00:04:19.748 The same carrier now has a different grouping of equal signatures. 00:04:20.898 --> 00:04:24.174 Selecting a test changes the presentation. 00:04:24.645 --> 00:04:29.203 It does not remove any possible valuation or assert that the selected 00:04:29.213 --> 00:04:30.564 formula is true. 00:04:31.039 --> 00:04:35.186 Learning a premise is a separate operation later in the course. 00:04:36.057 --> 00:04:40.290 We can also describe a signature by listing the positions where its 00:04:40.300 --> 00:04:41.708 entries are one. 00:04:42.169 --> 00:04:45.313 This set of indices is called its support. 00:04:45.799 --> 00:04:50.779 An index names a position, not merely a formula: two positions can 00:04:50.789 --> 00:04:54.554 carry the same formula and still be two separate members of the 00:04:54.554 --> 00:04:55.313 support. 00:04:56.213 --> 00:05:00.130 For the list X, Y, A's support is the singleton containing 00:05:00.140 --> 00:05:01.174 index one. 00:05:01.630 --> 00:05:05.178 B's support contains indices one and two. 00:05:05.801 --> 00:05:08.437 C's support contains index two. 00:05:09.117 --> 00:05:14.421 D's support is empty, because neither selected formula is true there. 00:05:15.266 --> 00:05:20.131 An empty support is still the profile of a perfectly definite state. 00:05:20.574 --> 00:05:23.156 D has not disappeared from the carrier. 00:05:24.266 --> 00:05:26.731 Order matters when reading the vector. 00:05:27.217 --> 00:05:32.325 If we list Y before X, A's vector becomes zero, one. 00:05:32.724 --> 00:05:36.503 The state did not change; the coordinate meanings did. 00:05:37.022 --> 00:05:41.569 Never compare bit strings after silently changing their headers. 00:05:42.708 --> 00:05:46.956 A selected observation may itself contain NOT. 00:05:47.472 --> 00:05:50.310 Consider the tuple X, not X. 00:05:51.029 --> 00:05:54.025 Four two-bit patterns can be written down. 00:05:54.629 --> 00:05:57.776 Which of them can actually occur at a valuation? 00:06:14.025 --> 00:06:17.254 Only one, zero and zero, one. 00:06:17.720 --> 00:06:21.645 A and B make X true and not X false. 00:06:22.170 --> 00:06:25.941 C and D make X false and not X true. 00:06:26.796 --> 00:06:31.592 Zero, zero is impossible because one of a statement and its negation 00:06:31.689 --> 00:06:33.918 must be true in classical logic. 00:06:34.471 --> 00:06:38.446 One, one is impossible because they cannot both be true. 00:06:38.988 --> 00:06:43.860 Written bit patterns need not all be realized by the selected formulas. 00:06:44.969 --> 00:06:49.917 A certificate for not X is affirmative evidence of that selected formula. “ 00:06:50.420 --> 00:06:56.137 Positive observation” means affirming an available formula, not banning negation from its 00:06:56.137 --> 00:06:56.854 syntax. 00:06:57.979 --> 00:07:01.307 An indexed list may repeat a formula. 00:07:01.389 --> 00:07:05.138 For X, X, Y, A has signature one, one, zero. 00:07:05.635 --> 00:07:10.536 Its support contains indices one and two, even though those positions carry 00:07:10.536 --> 00:07:11.856 the same formula. 00:07:12.716 --> 00:07:16.986 The repeated X contributes the same truth region twice. 00:07:17.483 --> 00:07:22.908 The topology generated from these regions is therefore unchanged from the list 00:07:23.006 --> 00:07:27.397 X, Y: no new generating subset was added. 00:07:28.196 --> 00:07:32.341 But the presentation still remembers two separate coordinates. 00:07:32.991 --> 00:07:38.403 Later, a distance can count contributions coordinate by coordinate. 00:07:38.910 --> 00:07:43.834 Topology alone does not retain all of that presentation data. 00:07:44.806 --> 00:07:46.554 Use two observations. 00:07:47.190 --> 00:07:48.930 Observation one is Y. 00:07:49.461 --> 00:07:53.417 Observation two is the conjunction X and Y. 00:07:53.968 --> 00:07:58.014 At C, write the two-entry signature and its support. 00:07:58.600 --> 00:08:00.565 Then find D’s signature. 00:08:01.067 --> 00:08:05.071 Does this two-observation tuple distinguish A from D? 00:08:25.646 --> 00:08:29.511 At C, Y is true but X and Y is false. 00:08:29.946 --> 00:08:35.727 The signature is one, zero and its support contains only index one. 00:08:36.137 --> 00:08:41.045 At D both formulas are false, so the signature is zero, zero. 00:08:42.001 --> 00:08:47.242 A also gives zero, zero: Y is false and the conjunction is 00:08:47.274 --> 00:08:47.879 false. 00:08:48.362 --> 00:08:53.815 This tuple cannot distinguish A from D, despite their different X values. 00:08:54.890 --> 00:09:00.609 A valuation, a selected formula tuple and its complete signature are three 00:09:00.609 --> 00:09:01.824 different objects. 00:09:02.479 --> 00:09:08.240 Equal signatures identify the distinctions the chosen observations cannot make. 00:09:08.820 --> 00:09:13.866 Next we will use their truth regions to construct the observation space 00:09:13.866 --> 00:09:15.009 explicitly.