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Title: Logical Statements, Negations and Conjunction
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Series: Start Learning Logic
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Parent Series: Start Learning Mathematics
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YouTube-Title: Start Learning Logic 1 | Logical Statements, Negations and Conjunction
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Bright video: https://youtu.be/DU4wKBDm2Z4
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Subtitle on GitHub: sll01_sub_eng.srt
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Definitions in the video: proposition, logical statement, negation, conjunction
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Subtitle in English
1 00:00:00,389 –> 00:00:01,970 Hello and welcome to
2 00:00:01,980 –> 00:00:03,569 start learning logic.
3 00:00:03,680 –> 00:00:05,369 And as always, many thanks
4 00:00:05,380 –> 00:00:06,530 to all the nice people that
5 00:00:06,539 –> 00:00:08,090 support this channel on Steady
6 00:00:08,100 –> 00:00:08,890 or paypal.
7 00:00:09,500 –> 00:00:10,800 This is part one where I
8 00:00:10,810 –> 00:00:12,090 want to show you the basic
9 00:00:12,100 –> 00:00:13,500 stuff you need to know about
10 00:00:13,510 –> 00:00:15,050 logic to understand
11 00:00:15,060 –> 00:00:15,770 mathematics.
12 00:00:16,459 –> 00:00:17,319 Therefore, we don’t need
13 00:00:17,329 –> 00:00:18,639 to go into the details of
14 00:00:18,649 –> 00:00:20,370 the wide field of logic and
15 00:00:20,379 –> 00:00:21,309 formal logic.
16 00:00:21,770 –> 00:00:23,120 For us, it would be sufficient
17 00:00:23,129 –> 00:00:24,559 to understand logical
18 00:00:24,569 –> 00:00:26,520 statements and logical operations.
19 00:00:27,360 –> 00:00:28,649 Since a lot of people can
20 00:00:28,659 –> 00:00:30,590 correctly deal with the everyday
21 00:00:30,600 –> 00:00:32,520 logic on an intuitive level,
22 00:00:32,630 –> 00:00:34,310 this is often skipped when
23 00:00:34,319 –> 00:00:35,700 you start with mathematics.
24 00:00:36,189 –> 00:00:37,509 However, I think it’s important
25 00:00:37,520 –> 00:00:39,130 to see that we have the logic
26 00:00:39,139 –> 00:00:40,830 as the foundation because
27 00:00:40,840 –> 00:00:42,619 we need to define everything
28 00:00:42,630 –> 00:00:44,110 in an unambiguous way in
29 00:00:44,119 –> 00:00:44,869 mathematics.
30 00:00:45,580 –> 00:00:46,919 Hence, for us, our starting
31 00:00:46,930 –> 00:00:48,319 point would be logical
32 00:00:48,330 –> 00:00:49,950 statements also called
33 00:00:49,959 –> 00:00:50,759 propositions.
34 00:00:51,509 –> 00:00:53,000 Now, such a proposition is
35 00:00:53,009 –> 00:00:54,700 just a statement which means
36 00:00:54,709 –> 00:00:56,159 a meaningful declarative
37 00:00:56,169 –> 00:00:57,400 sentence that has a
38 00:00:57,409 –> 00:00:59,000 well-defined truth value.
39 00:00:59,650 –> 00:01:01,560 So it can be either a true
40 00:01:01,569 –> 00:01:02,819 or a false statement.
41 00:01:03,540 –> 00:01:04,550 Therefore, when you take
42 00:01:04,559 –> 00:01:06,029 examples outside the
43 00:01:06,040 –> 00:01:07,410 mathematical world, you have
44 00:01:07,419 –> 00:01:09,029 to be careful what really
45 00:01:09,040 –> 00:01:10,550 satisfies this condition
46 00:01:10,559 –> 00:01:10,900 here.
47 00:01:11,480 –> 00:01:13,349 So you can’t take any opinions
48 00:01:13,360 –> 00:01:14,860 or questions because a
49 00:01:14,870 –> 00:01:16,209 question does not have a
50 00:01:16,220 –> 00:01:17,750 well-defined truth value.
51 00:01:18,470 –> 00:01:18,860 OK.
52 00:01:18,870 –> 00:01:20,529 Then let’s look at some examples
53 00:01:20,540 –> 00:01:21,690 where you should always keep
54 00:01:21,699 –> 00:01:23,069 in mind that we first have
55 00:01:23,080 –> 00:01:24,809 to agree on some definitions
56 00:01:24,819 –> 00:01:26,309 such that the examples make
57 00:01:26,319 –> 00:01:26,690 sense.
58 00:01:27,480 –> 00:01:29,459 So here Mars is a planet
59 00:01:29,470 –> 00:01:30,900 is a logical statement.
60 00:01:31,010 –> 00:01:32,699 When we know what object
61 00:01:32,709 –> 00:01:34,440 we call Mars and what a
62 00:01:34,449 –> 00:01:35,779 planet is by definition.
63 00:01:36,419 –> 00:01:38,110 Then this logical statement
64 00:01:38,120 –> 00:01:39,379 is indeed true.
65 00:01:40,169 –> 00:01:41,709 In this way, we can also
66 00:01:41,720 –> 00:01:43,489 formulate the sentence Pluto
67 00:01:43,500 –> 00:01:44,879 is a planet which is by the
68 00:01:44,889 –> 00:01:46,830 same agreement also a logical
69 00:01:46,839 –> 00:01:47,430 statement.
70 00:01:47,900 –> 00:01:49,489 However, this one now has
71 00:01:49,500 –> 00:01:51,069 the truth value false.
72 00:01:51,790 –> 00:01:53,440 So you see such real world
73 00:01:53,449 –> 00:01:54,849 examples can really help
74 00:01:54,860 –> 00:01:56,290 to understand the concepts.
75 00:01:56,449 –> 00:01:57,940 But of course, they can lead
76 00:01:57,949 –> 00:01:59,169 to some discussions.
77 00:02:00,010 –> 00:02:01,750 Therefore, I prefer examples
78 00:02:01,760 –> 00:02:03,419 in mathematics because they
79 00:02:03,430 –> 00:02:05,069 are all definitions are given.
80 00:02:05,080 –> 00:02:06,510 For example, we could choose
81 00:02:06,519 –> 00:02:08,360 the sentence one plus
82 00:02:08,369 –> 00:02:10,270 one is equal to two.
83 00:02:11,250 –> 00:02:12,589 Of course, here you also
84 00:02:12,600 –> 00:02:13,979 have to know what the numbers
85 00:02:13,990 –> 00:02:15,250 mean, what the plus sign
86 00:02:15,259 –> 00:02:16,500 means and what the equality
87 00:02:16,509 –> 00:02:17,119 sign means.
88 00:02:17,880 –> 00:02:19,350 But if you know these things,
89 00:02:19,360 –> 00:02:20,740 you know, this is a true
90 00:02:20,750 –> 00:02:22,020 logical statement.
91 00:02:22,750 –> 00:02:24,490 Of course, such logical statements
92 00:02:24,500 –> 00:02:26,020 coming from mathematics can
93 00:02:26,029 –> 00:02:27,940 also be formulated with English
94 00:02:27,949 –> 00:02:28,440 words.
95 00:02:28,520 –> 00:02:30,300 For example, we could say
96 00:02:30,320 –> 00:02:31,919 the number five is smaller
97 00:02:31,929 –> 00:02:32,979 than the number two.
98 00:02:33,500 –> 00:02:34,940 So here we need only knowledge
99 00:02:34,949 –> 00:02:36,309 of the natural numbers and
100 00:02:36,320 –> 00:02:37,250 how they are ordered.
101 00:02:37,960 –> 00:02:39,710 Then you can say yes, this
102 00:02:39,720 –> 00:02:41,220 is also a logical statement,
103 00:02:41,320 –> 00:02:42,380 but it is false.
104 00:02:43,119 –> 00:02:43,449 OK.
105 00:02:43,460 –> 00:02:45,289 Then let’s go to a counterexample.
106 00:02:45,300 –> 00:02:47,160 What about good morning?
107 00:02:47,600 –> 00:02:49,130 Of course, this seems very
108 00:02:49,139 –> 00:02:51,089 nice, but we can’t associate
109 00:02:51,100 –> 00:02:52,419 a well-defined truth value
110 00:02:52,429 –> 00:02:52,809 to it.
111 00:02:53,660 –> 00:02:55,000 It simply does not make sense
112 00:02:55,009 –> 00:02:56,350 to say that this one here
113 00:02:56,360 –> 00:02:58,240 is true or that it is false.
114 00:02:58,789 –> 00:03:00,520 So it’s clearly not a logical
115 00:03:00,529 –> 00:03:01,080 statement.
116 00:03:01,880 –> 00:03:03,419 Another example now in the
117 00:03:03,429 –> 00:03:04,679 mathematical world would
118 00:03:04,690 –> 00:03:06,160 be to say I have a variable
119 00:03:06,169 –> 00:03:08,080 X plus one and
120 00:03:08,089 –> 00:03:09,699 this is equal to one.
121 00:03:10,169 –> 00:03:11,419 So this might be a tricky
122 00:03:11,429 –> 00:03:13,300 one because it’s not a logical
123 00:03:13,309 –> 00:03:13,860 statement.
124 00:03:13,869 –> 00:03:15,460 Despite looking like one.
125 00:03:16,149 –> 00:03:17,410 The reason for this is that
126 00:03:17,419 –> 00:03:19,250 the variable X here is not
127 00:03:19,259 –> 00:03:19,850 fixed.
128 00:03:19,860 –> 00:03:21,559 So we can’t give the statement
129 00:03:21,570 –> 00:03:23,250 a well defined truth value
130 00:03:23,259 –> 00:03:24,869 because it would depend what
131 00:03:24,880 –> 00:03:25,889 X really is.
132 00:03:26,779 –> 00:03:28,490 This means that in the moment
133 00:03:28,500 –> 00:03:30,039 that X gets a value,
134 00:03:30,050 –> 00:03:31,509 this statement becomes a
135 00:03:31,520 –> 00:03:32,600 logical statement.
136 00:03:33,059 –> 00:03:34,669 Of course, such constructions
137 00:03:34,679 –> 00:03:36,440 are all over mathematics.
138 00:03:36,449 –> 00:03:37,889 Therefore, they will be important
139 00:03:37,899 –> 00:03:39,440 later and we call such a
140 00:03:39,449 –> 00:03:40,669 thing a pretty kit.
141 00:03:41,460 –> 00:03:42,820 Now, the next step is when
142 00:03:42,830 –> 00:03:44,600 we have such logical statements,
143 00:03:44,610 –> 00:03:46,399 we also want to combine them,
144 00:03:47,080 –> 00:03:48,389 they are just called logical
145 00:03:48,399 –> 00:03:49,639 operations and they give
146 00:03:49,649 –> 00:03:51,440 us new logical statements.
147 00:03:52,029 –> 00:03:53,500 For example, the negation
148 00:03:53,509 –> 00:03:55,059 is defined for given
149 00:03:55,070 –> 00:03:56,899 logical statement A which
150 00:03:56,910 –> 00:03:58,479 could be, for example, one
151 00:03:58,490 –> 00:03:59,970 plus one is equal to two.
152 00:04:00,800 –> 00:04:02,440 Then the new logical statement,
153 00:04:02,449 –> 00:04:04,210 the negation of A is denoted
154 00:04:04,220 –> 00:04:05,110 by this line.
155 00:04:05,119 –> 00:04:07,100 In front of A, you should
156 00:04:07,110 –> 00:04:08,740 read it as not A and it
157 00:04:08,750 –> 00:04:10,160 simply switches the truth
158 00:04:10,169 –> 00:04:10,740 value.
159 00:04:11,289 –> 00:04:12,240 Therefore, the best thing
160 00:04:12,250 –> 00:04:14,100 to define such a logic operation
161 00:04:14,110 –> 00:04:15,699 is to use a so called truth
162 00:04:15,710 –> 00:04:17,220 table where we put in all
163 00:04:17,230 –> 00:04:18,820 possible inputs and the
164 00:04:18,829 –> 00:04:20,059 corresponding outcomes.
165 00:04:20,908 –> 00:04:22,158 Of course, here we only have
166 00:04:22,169 –> 00:04:22,998 two inputs.
167 00:04:23,009 –> 00:04:24,838 A could be either true or
168 00:04:24,848 –> 00:04:25,378 false.
169 00:04:25,769 –> 00:04:26,919 So the truth value should
170 00:04:26,928 –> 00:04:27,618 be switched.
171 00:04:27,628 –> 00:04:29,378 So not a should be false
172 00:04:29,389 –> 00:04:30,989 when A is true and the other
173 00:04:30,998 –> 00:04:31,579 way around.
174 00:04:32,239 –> 00:04:32,619 OK.
175 00:04:32,630 –> 00:04:33,959 Also here let’s look at some
176 00:04:33,970 –> 00:04:35,660 examples to get an idea what
177 00:04:35,670 –> 00:04:36,739 we are talking about.
178 00:04:37,290 –> 00:04:38,380 So let’s take the logical
179 00:04:38,390 –> 00:04:40,019 statement A given by the
180 00:04:40,029 –> 00:04:41,380 English sentence, the wine
181 00:04:41,390 –> 00:04:42,290 bottle is full.
182 00:04:42,959 –> 00:04:44,579 Then the negation of A is
183 00:04:44,589 –> 00:04:46,140 simply given by putting in
184 00:04:46,149 –> 00:04:47,660 the word not into the
185 00:04:47,670 –> 00:04:48,190 sentence.
186 00:04:49,000 –> 00:04:50,209 Of course, here we assume
187 00:04:50,220 –> 00:04:51,299 that you know which wine
188 00:04:51,309 –> 00:04:52,809 bottle we talk about and
189 00:04:52,820 –> 00:04:53,910 what full means.
190 00:04:53,920 –> 00:04:55,549 But then you see negation
191 00:04:55,559 –> 00:04:57,220 is very easy just putting
192 00:04:57,230 –> 00:04:58,989 in not at the correct position.
193 00:04:59,679 –> 00:05:01,019 An important thing to note
194 00:05:01,029 –> 00:05:02,390 here is that this is not
195 00:05:02,399 –> 00:05:04,140 the same as saying the wine
196 00:05:04,149 –> 00:05:05,420 bottle is empty.
197 00:05:06,119 –> 00:05:07,320 In other words, negation
198 00:05:07,329 –> 00:05:08,630 does not mean the complete
199 00:05:08,640 –> 00:05:09,269 opposite.
200 00:05:09,279 –> 00:05:10,899 It just means switching it
201 00:05:10,910 –> 00:05:11,670 to values.
202 00:05:12,390 –> 00:05:13,519 As another example, let’s
203 00:05:13,529 –> 00:05:14,929 consider the logical statement
204 00:05:14,940 –> 00:05:16,890 A S two plus two
205 00:05:16,899 –> 00:05:18,209 is equal to five,
206 00:05:18,959 –> 00:05:20,709 then not A is of course two
207 00:05:20,720 –> 00:05:22,589 plus two is not equal
208 00:05:22,600 –> 00:05:23,279 to five.
209 00:05:23,829 –> 00:05:25,059 Now, what you might think
210 00:05:25,070 –> 00:05:26,420 is we just have to look at
211 00:05:26,429 –> 00:05:28,140 the second line here because
212 00:05:28,149 –> 00:05:29,700 we already know A is
213 00:05:29,709 –> 00:05:30,380 false.
214 00:05:30,390 –> 00:05:32,269 So not A should be just a
215 00:05:32,279 –> 00:05:33,059 true statement.
216 00:05:33,609 –> 00:05:34,600 However, that’s not what
217 00:05:34,609 –> 00:05:35,700 the negation does.
218 00:05:35,769 –> 00:05:37,670 It considers both cases at
219 00:05:37,679 –> 00:05:38,470 the same time.
220 00:05:38,910 –> 00:05:40,260 This means that if we live
221 00:05:40,269 –> 00:05:41,670 in a strange world where
222 00:05:41,679 –> 00:05:43,059 this logical statement is
223 00:05:43,070 –> 00:05:44,739 true, then the negation
224 00:05:44,750 –> 00:05:46,260 here should also be
225 00:05:46,269 –> 00:05:47,019 false.
226 00:05:47,720 –> 00:05:48,869 Therefore, please always
227 00:05:48,880 –> 00:05:50,489 consider these two possible
228 00:05:50,500 –> 00:05:51,869 worlds, which means these
229 00:05:51,880 –> 00:05:52,769 two lines here.
230 00:05:53,760 –> 00:05:55,010 Then if you get the correct
231 00:05:55,019 –> 00:05:56,529 truth table, you’ve found
232 00:05:56,540 –> 00:05:57,609 the correct negation.
233 00:05:58,320 –> 00:05:59,540 Besides the negation, we
234 00:05:59,549 –> 00:06:00,880 have another fundamental
235 00:06:00,890 –> 00:06:02,329 logic operation, which is
236 00:06:02,339 –> 00:06:03,619 called the conjunction.
237 00:06:04,410 –> 00:06:06,339 It combines two logical statements,
238 00:06:06,350 –> 00:06:07,989 let’s call them A and
239 00:06:08,000 –> 00:06:08,470 B.
240 00:06:09,320 –> 00:06:10,480 And then the conjunction
241 00:06:10,489 –> 00:06:12,329 is defined by this wedge
242 00:06:12,339 –> 00:06:13,739 symbol in between.
243 00:06:14,839 –> 00:06:16,589 You should read it as A and
244 00:06:16,600 –> 00:06:18,279 B because it’s the end
245 00:06:18,290 –> 00:06:19,989 operation, which means this
246 00:06:20,000 –> 00:06:21,510 one is only true.
247 00:06:21,519 –> 00:06:23,179 If A and B are
248 00:06:23,190 –> 00:06:24,549 true simultaneously
249 00:06:25,049 –> 00:06:26,269 with this information, you
250 00:06:26,279 –> 00:06:27,549 are already able to write
251 00:06:27,559 –> 00:06:29,190 down the whole truth table
252 00:06:29,200 –> 00:06:30,470 which you would see as the
253 00:06:30,480 –> 00:06:31,450 correct definition.
254 00:06:32,250 –> 00:06:33,649 Now, instead of two rows,
255 00:06:33,660 –> 00:06:35,220 we have four rows because
256 00:06:35,230 –> 00:06:37,200 we have four possible combinations
257 00:06:37,209 –> 00:06:38,989 of the two values of A and
258 00:06:39,000 –> 00:06:39,380 B.
259 00:06:39,950 –> 00:06:41,399 Now from bottom to top, you
260 00:06:41,410 –> 00:06:43,109 know, this one is false because
261 00:06:43,119 –> 00:06:44,589 we have at least one false
262 00:06:44,600 –> 00:06:46,149 here, the same here
263 00:06:46,359 –> 00:06:47,429 and the same here.
264 00:06:47,440 –> 00:06:49,109 Everything’s false except
265 00:06:49,119 –> 00:06:50,029 the first row.
266 00:06:50,040 –> 00:06:51,709 This one is true because
267 00:06:51,720 –> 00:06:52,750 both are true.
268 00:06:53,500 –> 00:06:54,880 A good visualization for
269 00:06:54,890 –> 00:06:56,440 the and operation would be
270 00:06:56,450 –> 00:06:58,279 to consider a circuit
271 00:06:58,290 –> 00:06:59,880 with a lamp and a battery.
272 00:07:00,579 –> 00:07:01,760 Now, what we can do is to
273 00:07:01,769 –> 00:07:03,329 put A and B as
274 00:07:03,359 –> 00:07:05,200 switches into the circuit.
275 00:07:05,959 –> 00:07:07,540 In this picture, false should
276 00:07:07,549 –> 00:07:09,279 mean off and true should
277 00:07:09,290 –> 00:07:11,179 mean on and the
278 00:07:11,190 –> 00:07:12,940 lamp we see is our output.
279 00:07:13,630 –> 00:07:14,890 So here you see, we need
280 00:07:14,899 –> 00:07:16,820 both switches to be on to
281 00:07:16,829 –> 00:07:18,369 get an on output.
282 00:07:18,380 –> 00:07:20,329 So true means true,
283 00:07:20,369 –> 00:07:22,209 otherwise we are false.
284 00:07:23,019 –> 00:07:23,450 OK.
285 00:07:23,459 –> 00:07:24,559 In the next video I show
286 00:07:24,570 –> 00:07:26,279 you other logical operations
287 00:07:26,290 –> 00:07:27,880 that are also very important
288 00:07:27,890 –> 00:07:28,820 in mathematics.
289 00:07:29,320 –> 00:07:30,570 So I hope I see you there
290 00:07:30,579 –> 00:07:31,839 and have a nice day.
291 00:07:31,850 –> 00:07:32,630 Bye.
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Quiz Content
Q1: Which of the following is not a logical statement?
A1: $1 < 0$
A2: Earth is a planet.
A3: $1 > 0$
A4: Run for your life!
Q2: Which of the following is a false logical statement?
A1: $1 = 1$
A2: Earth is a planet.
A3: $1 + 1 = 2$
A4: Pluto is a planet.
Q3: Let $A$ be a logical statement. How do we denote the negation?
A1: $A-$
A2: $\bullet A$
A3: $\circ A$
A4: $\neg A$
Q4: Let $A, B$ be logical statements, where $A$ is true and $B$ is false. What is the truth value of $A \wedge B$?
A1: True.
A2: False.
A3: One cannot say.
A4: True and false at the same time.
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Last update: 2024-10
