WEBVTT 00:00:00.250 --> 00:00:03.377 In the first two lectures, A, B, C, and D were four 00:00:03.377 --> 00:00:04.387 possible states. 00:00:05.046 --> 00:00:08.388 We can also read each state as a complete assignment of true 00:00:08.388 --> 00:00:09.440 or false values. 00:00:10.595 --> 00:00:14.121 Let X mean “there is a dot,” and Y mean “there is 00:00:14.121 --> 00:00:14.853 a stripe.” 00:00:15.340 --> 00:00:18.505 Card A makes X true and Y false. 00:00:19.016 --> 00:00:20.614 B makes both true. 00:00:21.123 --> 00:00:23.897 C makes X false and Y true. 00:00:24.399 --> 00:00:26.183 D makes both false. 00:00:27.061 --> 00:00:32.476 These four rows exhaust every possible way two true-or-false variables 00:00:32.476 --> 00:00:33.534 can be assigned. 00:00:34.009 --> 00:00:36.956 This is our running two-variable universe. 00:00:37.365 --> 00:00:42.118 Later, whenever we use a different number of variables, we will explicitly 00:00:42.150 --> 00:00:44.353 rebuild and label the carrier. 00:00:45.246 --> 00:00:48.857 A variable such as X is a simple formula. 00:00:49.318 --> 00:00:54.078 At each complete assignment, it has a value: true or false. 00:00:54.878 --> 00:00:58.705 X is true at A and B, the rows with a dot. 00:00:59.208 --> 00:01:03.623 Y is true at B and C, the rows with a stripe. 00:01:04.116 --> 00:01:08.359 A formula can be true in some states and false in others. 00:01:09.014 --> 00:01:12.746 The set of states where it is true is its truth region. 00:01:13.292 --> 00:01:15.853 For X, that region is A and B. 00:01:17.077 --> 00:01:20.187 A formula is the expression we write. 00:01:20.574 --> 00:01:24.678 Its value is true or false at one particular row. 00:01:25.172 --> 00:01:27.040 Keep those roles distinct. 00:01:28.138 --> 00:01:30.025 Now combine two formulas. “ 00:01:30.630 --> 00:01:34.591 X and Y” requires both statements to be true in the same 00:01:34.601 --> 00:01:35.013 row. 00:01:36.137 --> 00:01:38.980 Which card has both a dot and a stripe? 00:01:49.360 --> 00:01:50.000 Only B. 00:01:50.812 --> 00:01:54.968 So X and Y is true at B and false at A, 00:01:55.000 --> 00:01:55.662 C, and D. 00:01:56.926 --> 00:01:59.377 The wedge symbol means “and.” 00:01:59.959 --> 00:02:04.549 Its truth region is the intersection of the truth regions for X 00:02:04.603 --> 00:02:05.492 and for Y. 00:02:06.458 --> 00:02:09.659 X or Y” is true when at least one of the two 00:02:09.671 --> 00:02:10.982 statements is true. 00:02:11.920 --> 00:02:14.278 A qualifies because it has a dot. 00:02:14.823 --> 00:02:17.042 B qualifies because it has both. 00:02:17.650 --> 00:02:20.226 C qualifies because it has a stripe. 00:02:20.746 --> 00:02:24.197 D is the only row where neither statement holds. 00:02:25.330 --> 00:02:28.932 So the truth region of X or Y is A, B, and 00:02:28.943 --> 00:02:29.192 C. 00:02:29.891 --> 00:02:31.968 The vee symbol means “or.” 00:02:32.544 --> 00:02:37.617 Here it is inclusive: B still counts even though both statements hold. 00:02:38.604 --> 00:02:42.500 Negation changes true to false and false to true. “ 00:02:43.059 --> 00:02:47.108 Not X” is true exactly where X is false. 00:02:47.952 --> 00:02:52.367 Since X means dot, not X is true at C and D: 00:02:52.909 --> 00:02:55.018 the two rows with no dot. 00:02:55.788 --> 00:02:58.446 The symbol at the front is read “not.” 00:02:59.045 --> 00:03:02.556 It is a logical operation on a complete assignment. 00:03:03.555 --> 00:03:08.905 One important reminder: this drawing describes the truth region of not X. 00:03:09.549 --> 00:03:13.174 It does not say that an observer has obtained a certificate of 00:03:13.228 --> 00:03:13.980 no dot. 00:03:14.493 --> 00:03:17.632 Description and acquired evidence are different jobs. 00:03:18.729 --> 00:03:23.947 The connective that most often surprises people is implication: “if X, then 00:03:23.969 --> 00:03:24.340 Y.” 00:03:24.912 --> 00:03:29.763 It is false in exactly one situation: X is true but Y 00:03:29.774 --> 00:03:30.744 is false. 00:03:31.154 --> 00:03:33.687 Card A is that counterexample. 00:03:33.924 --> 00:03:38.048 It has the dot required by X, but lacks the stripe required 00:03:38.048 --> 00:03:38.816 by Y. 00:03:39.759 --> 00:03:45.212 B makes the implication true: whenever X holds there, Y holds too. 00:03:46.161 --> 00:03:49.216 C and D also make the implication true. 00:03:49.658 --> 00:03:55.084 In those rows X is false, so neither supplies the forbidden combination, 00:03:55.374 --> 00:03:57.690 X true and Y false. 00:03:58.187 --> 00:04:02.722 This does not establish Y: Y is true at C and false 00:04:02.722 --> 00:04:03.102 at D. 00:04:04.336 --> 00:04:08.399 This is a rule about truth values, not a claim that a 00:04:08.409 --> 00:04:10.086 dot causes a stripe. 00:04:10.593 --> 00:04:16.906 The implication has one forbidden pattern: true X together with false Y. 00:04:17.849 --> 00:04:20.603 A truth table is simply a complete map. 00:04:21.130 --> 00:04:26.240 Each row is a possible assignment, and each formula column records its 00:04:26.240 --> 00:04:27.845 value at that row. 00:04:28.864 --> 00:04:32.312 Before we move on, look at the implication column. 00:04:32.823 --> 00:04:35.309 Which row is false, and why? 00:04:47.775 --> 00:04:52.346 A is the only false row, because it alone has X true 00:04:52.346 --> 00:04:53.625 and Y false. 00:04:54.086 --> 00:04:58.718 The table evaluates a formula at every complete possibility. 00:04:59.188 --> 00:05:03.851 It does not select the actual card or report an observation. 00:05:04.774 --> 00:05:09.086 Some formulas have a special pattern across the whole table. 00:05:09.996 --> 00:05:12.998 X or not X” is true in every row. 00:05:13.517 --> 00:05:18.362 A formula true at every complete assignment is called a tautology. 00:05:19.254 --> 00:05:22.392 X and not X” is false in every row. 00:05:22.939 --> 00:05:27.762 A formula false at every complete assignment is called a contradiction. 00:05:28.585 --> 00:05:32.438 Their truth regions are the whole universe and the empty set. 00:05:33.057 --> 00:05:37.962 These are logical facts about the table, before we add any observations 00:05:37.962 --> 00:05:38.896 or premises. 00:05:40.035 --> 00:05:43.475 For a fresh check, evaluate Y and not X. 00:05:44.160 --> 00:05:47.249 Which states make this compound formula true? 00:05:47.754 --> 00:05:52.445 Use both requirements at the same valuation, and keep the outer carrier 00:05:52.499 --> 00:05:53.108 fixed. 00:06:07.379 --> 00:06:09.538 Not X is true at C and D. 00:06:10.297 --> 00:06:12.358 Y is true at B and C. 00:06:13.083 --> 00:06:18.078 To satisfy their conjunction a valuation must belong to both regions. 00:06:19.142 --> 00:06:21.112 Only C belongs to both. 00:06:21.630 --> 00:06:24.813 At C, Y is true and X is false. 00:06:25.268 --> 00:06:29.391 At D, not X is true but Y is false, so D 00:06:29.587 --> 00:06:31.780 does not satisfy the conjunction. 00:06:32.878 --> 00:06:37.963 This new calculation uses the same truth-table rules, rather than repeating 00:06:37.963 --> 00:06:40.104 the implication counterexample. 00:06:40.570 --> 00:06:43.284 Every formula gives a truth region. 00:06:43.906 --> 00:06:49.137 Next we will translate those regions into set operations and compare formulas 00:06:49.147 --> 00:06:51.035 across all valuations.