-
Title: Disjunction, Tautology and Logical Equivalence
-
Series: Start Learning Logic
-
Parent Series: Start Learning Mathematics
-
YouTube-Title: Start Learning Logic 2 | Disjunction, Tautology and Logical Equivalence
-
Bright video: https://youtu.be/AjdIPOXRgoQ
-
Dark video: https://youtu.be/Jw6KxvMKbys
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: sll02_sub_eng.srt
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Definitions in the video: disjunction
-
Timestamps (n/a)
-
Subtitle in English
1 00:00:00,400 –> 00:00:02,099 Hello and welcome back to
2 00:00:02,109 –> 00:00:03,460 start learning logic.
3 00:00:03,470 –> 00:00:04,900 Now with part two.
4 00:00:05,500 –> 00:00:06,820 And first, I want to thank
5 00:00:06,829 –> 00:00:08,350 all the nice people on Steady
6 00:00:08,399 –> 00:00:10,170 or paypal who support this
7 00:00:10,180 –> 00:00:10,649 channel.
8 00:00:11,229 –> 00:00:12,899 Now to refresh your memory.
9 00:00:12,909 –> 00:00:14,300 Last time we learned the
10 00:00:14,310 –> 00:00:16,100 meaning of logical statements
11 00:00:16,250 –> 00:00:18,180 and we used the names A and
12 00:00:18,190 –> 00:00:19,459 B to denote them.
13 00:00:20,149 –> 00:00:21,409 And then we have seen that
14 00:00:21,420 –> 00:00:23,180 we can formulate new logical
15 00:00:23,190 –> 00:00:24,850 statements with the negation
16 00:00:25,049 –> 00:00:26,090 and the conjunction.
17 00:00:26,979 –> 00:00:28,639 These two new symbols, we
18 00:00:28,649 –> 00:00:30,290 called logical operations.
19 00:00:30,389 –> 00:00:31,920 And this one was the not
20 00:00:31,930 –> 00:00:33,830 operation and this one, the
21 00:00:33,840 –> 00:00:34,720 and operation.
22 00:00:35,470 –> 00:00:37,049 Now, if you have such a formula
23 00:00:37,060 –> 00:00:38,669 with logical operations,
24 00:00:38,680 –> 00:00:40,319 the original statements A
25 00:00:40,330 –> 00:00:42,090 and B are often just
26 00:00:42,099 –> 00:00:43,650 called logical variables
27 00:00:43,659 –> 00:00:44,409 in this case.
28 00:00:44,979 –> 00:00:46,250 So you see we have a lot
29 00:00:46,259 –> 00:00:47,889 of vocabulary which is in
30 00:00:47,900 –> 00:00:49,209 the end, maybe not so
31 00:00:49,220 –> 00:00:49,970 important.
32 00:00:49,979 –> 00:00:51,090 The important thing will
33 00:00:51,099 –> 00:00:52,919 be that you understand all
34 00:00:52,930 –> 00:00:54,130 the symbols we use.
35 00:00:54,840 –> 00:00:55,270 OK.
36 00:00:55,279 –> 00:00:56,939 Then let’s continue introducing
37 00:00:56,950 –> 00:00:58,619 more symbols with logical
38 00:00:58,630 –> 00:00:59,330 operations.
39 00:01:00,009 –> 00:01:01,490 The next one is the so called
40 00:01:01,500 –> 00:01:03,229 disjunction also defined
41 00:01:03,240 –> 00:01:04,730 for two logical statements
42 00:01:04,739 –> 00:01:05,750 A and B.
43 00:01:05,940 –> 00:01:07,379 And it’s similar to the end
44 00:01:07,389 –> 00:01:09,069 operation here because it’s
45 00:01:09,080 –> 00:01:10,349 the or operation,
46 00:01:11,139 –> 00:01:12,470 the symbol we use here is
47 00:01:12,480 –> 00:01:13,339 the flipped one.
48 00:01:13,349 –> 00:01:14,680 So just a V
49 00:01:15,239 –> 00:01:16,540 as in the last video, we
50 00:01:16,550 –> 00:01:18,300 can define the symbol by
51 00:01:18,309 –> 00:01:19,800 writing down the truth table.
52 00:01:20,330 –> 00:01:21,519 The disjunction should be
53 00:01:21,529 –> 00:01:23,019 two when at least one of
54 00:01:23,029 –> 00:01:24,760 the two constituents is two.
55 00:01:25,500 –> 00:01:27,230 So we have two here, two
56 00:01:27,239 –> 00:01:27,779 here.
57 00:01:27,930 –> 00:01:29,919 And also the first line because
58 00:01:29,930 –> 00:01:31,489 both of them are two and
59 00:01:31,500 –> 00:01:32,940 we don’t have an exclusive
60 00:01:32,949 –> 00:01:33,519 or here.
61 00:01:34,050 –> 00:01:35,510 However, when both of them
62 00:01:35,519 –> 00:01:37,319 are false, we have a false
63 00:01:38,029 –> 00:01:39,419 similar to the conjunction.
64 00:01:39,430 –> 00:01:41,059 You can also visualize this
65 00:01:41,069 –> 00:01:43,019 with a circuit and a lamp.
66 00:01:43,690 –> 00:01:45,260 In this case, the two switches
67 00:01:45,269 –> 00:01:46,910 just have to be parallel.
68 00:01:47,470 –> 00:01:48,879 Now having these logical
69 00:01:48,889 –> 00:01:50,790 operations, we can combine
70 00:01:50,800 –> 00:01:52,349 them in a lot of ways to
71 00:01:52,360 –> 00:01:54,190 get out new logical statements.
72 00:01:54,910 –> 00:01:55,989 Therefore, the natural thing
73 00:01:56,000 –> 00:01:57,900 to do is writing down a truth
74 00:01:57,910 –> 00:01:59,349 table for such a new
75 00:01:59,360 –> 00:01:59,970 combination.
76 00:02:00,970 –> 00:02:02,449 So let’s consider a simple
77 00:02:02,459 –> 00:02:03,269 example here.
78 00:02:03,919 –> 00:02:05,650 What about not A
79 00:02:05,680 –> 00:02:06,500 or a?
80 00:02:07,309 –> 00:02:09,089 You see what we usually assume
81 00:02:09,100 –> 00:02:11,000 is that the NOT operator
82 00:02:11,008 –> 00:02:12,300 binds closest.
83 00:02:12,309 –> 00:02:13,389 Therefore, we don’t need
84 00:02:13,399 –> 00:02:14,699 any parentheses here.
85 00:02:15,399 –> 00:02:16,759 Now, in this example, our
86 00:02:16,770 –> 00:02:18,479 combination just has one
87 00:02:18,490 –> 00:02:20,190 logic variable which means
88 00:02:20,199 –> 00:02:21,860 our truth table just needs
89 00:02:21,869 –> 00:02:22,839 two rows.
90 00:02:23,369 –> 00:02:23,720 OK?
91 00:02:23,729 –> 00:02:24,779 So let’s fill that in.
92 00:02:24,789 –> 00:02:26,600 We know that not just flips
93 00:02:26,610 –> 00:02:28,490 the truth values and
94 00:02:28,500 –> 00:02:30,419 now the or just needs one
95 00:02:30,429 –> 00:02:30,949 true.
96 00:02:30,960 –> 00:02:32,809 So we have the true here and
97 00:02:32,820 –> 00:02:33,580 the true here.
98 00:02:34,199 –> 00:02:35,529 Now you see this
99 00:02:35,539 –> 00:02:37,339 combination always gets you
100 00:02:37,350 –> 00:02:39,119 a true statement out no
101 00:02:39,130 –> 00:02:40,500 matter what the truth value
102 00:02:40,509 –> 00:02:41,809 of the input A was.
103 00:02:42,550 –> 00:02:43,880 In logic, we call such a
104 00:02:43,889 –> 00:02:45,610 thing a tautology.
105 00:02:46,270 –> 00:02:47,619 So it’s a combination of
106 00:02:47,630 –> 00:02:49,509 logical operations and logical
107 00:02:49,520 –> 00:02:51,460 variables that is always
108 00:02:51,470 –> 00:02:53,240 true no matter what the truth
109 00:02:53,250 –> 00:02:54,910 values are of the logical
110 00:02:54,919 –> 00:02:56,529 variables that are contained
111 00:02:56,539 –> 00:02:57,369 in the formula.
112 00:02:57,869 –> 00:02:59,330 Of course, this was a simple
113 00:02:59,339 –> 00:03:00,050 example.
114 00:03:00,059 –> 00:03:01,770 You could have many different
115 00:03:01,779 –> 00:03:03,169 logical variables in the
116 00:03:03,179 –> 00:03:03,770 formula.
117 00:03:03,919 –> 00:03:05,119 But then the truth table
118 00:03:05,130 –> 00:03:06,270 would be much bigger.
119 00:03:06,919 –> 00:03:08,119 Of course, knowing which
120 00:03:08,130 –> 00:03:09,850 combinations are tautologies is
121 00:03:09,860 –> 00:03:11,419 very helpful because you
122 00:03:11,429 –> 00:03:13,020 always can substitute this
123 00:03:13,029 –> 00:03:13,630 combination.
124 00:03:13,639 –> 00:03:15,199 Then with just a true
125 00:03:15,929 –> 00:03:17,419 now related to that is the
126 00:03:17,429 –> 00:03:18,960 notion of logical
127 00:03:18,970 –> 00:03:19,820 equivalence.
128 00:03:20,720 –> 00:03:22,179 It also tells you when you
129 00:03:22,190 –> 00:03:24,050 can substitute one formula
130 00:03:24,070 –> 00:03:25,979 with another, we
131 00:03:25,990 –> 00:03:27,699 call two logical statements.
132 00:03:27,710 –> 00:03:29,300 So combinations as we had
133 00:03:29,309 –> 00:03:30,830 before, logically
134 00:03:30,839 –> 00:03:32,410 equivalent, if the truth
135 00:03:32,419 –> 00:03:33,830 table looks the same for
136 00:03:33,839 –> 00:03:35,250 both more
137 00:03:35,259 –> 00:03:36,529 concretely, you have to look
138 00:03:36,539 –> 00:03:38,139 at all possible assignments
139 00:03:38,149 –> 00:03:40,119 of truth values for the logical
140 00:03:40,130 –> 00:03:41,839 variables that are included
141 00:03:41,850 –> 00:03:43,020 in both formulas.
142 00:03:43,720 –> 00:03:45,360 And if you get the same output
143 00:03:45,369 –> 00:03:47,130 for both formulas, we call
144 00:03:47,139 –> 00:03:48,820 them logically equivalent.
145 00:03:49,500 –> 00:03:49,880 OK?
146 00:03:49,889 –> 00:03:51,270 So it looks like a reasonable
147 00:03:51,279 –> 00:03:51,800 term.
148 00:03:51,809 –> 00:03:53,520 But to understand it, I think
149 00:03:53,529 –> 00:03:54,960 we should look at an example
150 00:03:55,809 –> 00:03:56,169 here.
151 00:03:56,179 –> 00:03:58,020 I want to consider A or
152 00:03:58,029 –> 00:03:59,619 B with a NOT in
153 00:03:59,630 –> 00:04:01,440 front and
154 00:04:01,449 –> 00:04:03,309 also not A
155 00:04:03,350 –> 00:04:04,779 and not B.
156 00:04:05,580 –> 00:04:07,029 So please keep in mind we
157 00:04:07,039 –> 00:04:08,910 need parentheses here, but
158 00:04:08,919 –> 00:04:09,550 not here.
159 00:04:10,259 –> 00:04:11,850 But of course, if you want,
160 00:04:11,910 –> 00:04:13,610 you are allowed to use them
161 00:04:13,619 –> 00:04:13,960 here.
162 00:04:14,899 –> 00:04:16,600 OK, then let’s do a large
163 00:04:16,608 –> 00:04:17,480 truth table.
164 00:04:18,200 –> 00:04:19,730 Of course, we need here our
165 00:04:19,738 –> 00:04:21,548 inputs A and B
166 00:04:21,738 –> 00:04:23,170 and the outputs on the right
167 00:04:23,179 –> 00:04:23,779 hand side.
168 00:04:24,529 –> 00:04:25,559 Then in the next step, we
169 00:04:25,570 –> 00:04:27,209 can fill in all possible
170 00:04:27,220 –> 00:04:28,619 combinations of the truth
171 00:04:28,630 –> 00:04:30,029 values of the logical
172 00:04:30,040 –> 00:04:30,750 variables.
173 00:04:31,390 –> 00:04:32,809 Afterwards, we have to think
174 00:04:32,820 –> 00:04:34,350 what are good intermediate
175 00:04:34,359 –> 00:04:36,010 steps to reach our result.
176 00:04:36,019 –> 00:04:37,890 On the right hand side here.
177 00:04:37,899 –> 00:04:39,170 For example, I would say
178 00:04:39,179 –> 00:04:41,010 we calculate first A or
179 00:04:41,019 –> 00:04:42,769 B for this, we
180 00:04:42,779 –> 00:04:44,239 already know the truth table.
181 00:04:44,250 –> 00:04:46,089 We have true true true
182 00:04:46,100 –> 00:04:46,760 false.
183 00:04:47,459 –> 00:04:48,579 And then we just have to
184 00:04:48,589 –> 00:04:50,200 use the negation which
185 00:04:50,209 –> 00:04:51,950 switches all the truth values.
186 00:04:51,959 –> 00:04:53,750 So we have False False False
187 00:04:53,760 –> 00:04:54,279 true.
188 00:04:54,959 –> 00:04:56,059 So now we finish with the
189 00:04:56,070 –> 00:04:56,940 first one.
190 00:04:56,950 –> 00:04:58,070 And for the second one, I
191 00:04:58,079 –> 00:04:59,440 would say we first write
192 00:04:59,450 –> 00:05:01,399 down not A and afterwards
193 00:05:01,410 –> 00:05:02,679 we write down not B
194 00:05:03,380 –> 00:05:03,820 OK.
195 00:05:03,829 –> 00:05:05,040 Filling this in was not so
196 00:05:05,049 –> 00:05:06,559 hard because we already know
197 00:05:06,570 –> 00:05:07,519 how to do this.
198 00:05:07,720 –> 00:05:09,119 And now for the last step,
199 00:05:09,130 –> 00:05:11,119 we have to use the and operator
200 00:05:11,850 –> 00:05:12,839 in the first line, we have
201 00:05:12,850 –> 00:05:13,899 false and false.
202 00:05:13,910 –> 00:05:15,820 So this is false, then we
203 00:05:15,829 –> 00:05:17,420 have false and two still
204 00:05:17,429 –> 00:05:18,010 false.
205 00:05:18,149 –> 00:05:19,790 Then we have two and false
206 00:05:19,799 –> 00:05:20,929 still false.
207 00:05:21,209 –> 00:05:22,170 And in the last line, we
208 00:05:22,179 –> 00:05:23,829 have two and two, which is
209 00:05:23,839 –> 00:05:24,239 two.
210 00:05:24,950 –> 00:05:26,529 And there we have our result,
211 00:05:26,540 –> 00:05:28,010 the truth values for both
212 00:05:28,019 –> 00:05:29,390 combinations match
213 00:05:29,399 –> 00:05:30,089 completely.
214 00:05:31,010 –> 00:05:32,500 In other words, they are
215 00:05:32,510 –> 00:05:34,100 logically equivalent.
216 00:05:34,869 –> 00:05:36,130 And the symbol we use for
217 00:05:36,140 –> 00:05:37,660 this is this double
218 00:05:37,679 –> 00:05:39,529 arrow for
219 00:05:39,540 –> 00:05:39,690 you.
220 00:05:39,700 –> 00:05:40,920 This means if you have a
221 00:05:40,929 –> 00:05:42,670 large formula as a logical
222 00:05:42,679 –> 00:05:43,929 statement where you have
223 00:05:43,940 –> 00:05:45,269 this part inside,
224 00:05:45,980 –> 00:05:47,290 you can easily substitute
225 00:05:47,299 –> 00:05:49,010 this with this one without
226 00:05:49,019 –> 00:05:50,489 changing the truth value
227 00:05:50,709 –> 00:05:52,359 and also the other way around.
228 00:05:53,049 –> 00:05:54,399 So the logical equivalence
229 00:05:54,410 –> 00:05:55,630 is what you can use to
230 00:05:55,640 –> 00:05:57,239 simplify a complicated
231 00:05:57,250 –> 00:05:59,029 expression step by step.
232 00:06:00,010 –> 00:06:01,589 Of course, we will see examples
233 00:06:01,600 –> 00:06:02,070 of this.
234 00:06:02,160 –> 00:06:03,529 But first, I want to use
235 00:06:03,540 –> 00:06:05,160 the next video to show you
236 00:06:05,170 –> 00:06:06,700 more logical operations.
237 00:06:07,549 –> 00:06:08,760 So I hope I see you there
238 00:06:08,769 –> 00:06:10,089 and have a nice day.
239 00:06:10,179 –> 00:06:10,850 Bye.
-
Quiz Content
Q1: Let $A,B$ be two logical statements which are both false. What is the truth value of the disjunction $A \vee B$?
A1: False
A2: True
A3: One cannot say.
A4: True and false at he same time.
Q2: What can one say about $\neg A \vee A$?
A1: Always true.
A2: Always false.
A3: One cannot say.
Q3: Is $A$ logically equivalent to $(\neg A \vee A) \wedge A$?
A1: Yes!
A2: No!
A3: One cannot say.
-
Last update: 2024-10
