00:00 Good day, my name is Afeez Adedokun, I will be taking you furthermaths and the topic is Logical Reasoning.

00:21 At the end of this topic, students should be able to know understanding basic issues

00:28 in intelligence system, give basic definition of intelligence system, understand modeling

00:36 of the watch, learn about the propositional and predicate logical resolution, give introduction

00:45 to proving of theory.

00:50 What is intelligence system?

00:53 An intelligence system is a system that senses its environments and learns each situation

01:01 which action permits is to reach its objectives.

01:05 An intelligence system reaches its objective by continually acting mentally and externally.

01:14 It uses energy for its external processes in order to act.

01:18 We need that and for us to have an intelligence system, we must have an environment to

01:25 learn.

01:26 For students to be an intelligence student, you or she must have an environment in which

01:33 it will be learning one or two things up for a particular situation.

01:40 So and that is what the intelligence system is trying to tell us, that an intelligence

01:45 system reaches objective by continually acting mentally and externally.

01:52 And it uses energy for its external process in order to act.

01:58 From the definition of intelligence system, the following basic issues arise.

02:05 And the first one, a system of the system.

02:08 By student to learn, there must be a system as a learning, there's a system in which you

02:14 are learning.

02:15 A student's teacher approach is a system in which you are used to learn.

02:21 Also an environment in which the system can be interacted.

02:26 So we must have an environment and an environment must exist when we want to have an intelligent

02:34 system.

02:36 We must have an environment.

02:38 The first one, we must have a system.

02:41 Having had a system, we must have an environment.

02:45 Then the third one is intelligent system must be able to communicate with its environment.

02:51 So we must be able as a student, you must be able to communicate with your environment.

02:59 And the fourth one is the intelligent system must have an objective.

03:05 Why do we learn?

03:06 You must have an objective, why you are learning.

03:09 So there must be an objective if you want to learn.

03:16 Then the fifth one, the intelligent system must be able to select its response in order

03:21 to reach its objective.

03:23 So you must be able to know that this is what we have learned so far.

03:26 And then in order to achieve the objective of why you are learning.

03:32 Because this is what we have learned so far.

03:34 And that will be able to make you to know what you have learned so far.

03:39 And the objective of why you are even learning.

03:41 The intelligent system must be able to learn, must be able to work, to learn as well.

03:47 And the sixth one is, the seventh one is it must be able to act in order to achieve the

03:53 selected response.

03:55 So you must be able to act of what you have learned, must be able to use what you have learned

03:59 in order to achieve the selected response.

04:02 So these are the basic issues when from the definition of intelligent system.

04:10 Now, propositions, a propositions or statements.

04:15 We talk about propositions in logicalizing in SS1.

04:22 And the way we define that in SS1 is also a little bit different from that.

04:27 It's just the same thing about a little bit different.

04:30 A propositions or statements is the verb or written declarative sentence.

04:36 That is either true or false, but they cannot be both.

04:40 That is what we define in SS1.

04:42 Then yet, we say it is a verb or written declarations system.

04:47 And that we say it.

04:49 And that we say it or we write it.

04:51 They are propositions or they are called system or assessment.

04:56 Either true or false.

04:58 When we talk sometimes, it's possible for us to be what?

05:01 So not to be true.

05:03 And when we also talk, it's possible for us to be false.

05:06 But we cannot be false and we cannot be just like we say that what we have said so far

05:12 now is true and it's also false.

05:14 It's not possible for us to have that.

05:16 So is that you, it is true or false?

05:18 So when you want to propose something, is that that thing?

05:22 What's your propositions?

05:23 It's what?

05:24 It's true or false.

05:26 For example, 4 plus 5 equals 9.

05:30 4 plus 5 is a mathematical expression.

05:34 And mathematical expression can also be false.

05:36 It can also be a proposition or can also be our statements.

05:41 And that statement as it is is what is true.

05:44 Because 4 plus 5 will give you 9.

05:47 The second statement is 2 plus 2 equals 5.

05:50 It's a mathematical expression.

05:51 And it is not true or false.

05:53 And if you look at 2 plus 2, no 2 plus 2 is 4.

05:56 So it means that this statement is what is not true.

05:59 So the 2 plus 2 is a statement, it's proposition.

06:02 But it's not true.

06:03 It's a force.

06:05 So a proposition is a verb or a written declarative sentence that is either true or false.

06:15 The true or false still of a statement or proposition is called its true value.

06:20 That is when you have, what are you saying that a statement is false or true?

06:25 What we call that is a true value.

06:28 For example, like we said, 4 plus 5 is 9.

06:31 That 4 plus 5 is 9 is what?

06:34 Is it true statement?

06:35 Is it statement?

06:36 Is proposition?

06:38 And is also what?

06:39 Is it true value?

06:42 When you say 2 plus 2 is 5, is it statement?

06:45 But it is not what.

06:46 It is a force.

06:47 It is not true.

06:49 So the true or falseity of a statement is called its true values.

06:56 A true statement asks a true value t.

06:59 Why a false statement asks a true value f?

07:03 When a statement, for example, let's say we have p, which is statement, 4 plus 5, you

07:09 call us name.

07:19 Let's say we also have q, which is a statement of 2 plus 2 equals 5.

07:26 Normally, 2 plus 4, 2 plus 2, we give us what?

07:29 We give us 4.

07:30 So this is not true.

07:31 So it means q.

07:33 For logical reason, if we want to represent it, we are going to have what?

07:36 We are going to have not true.

07:38 So we are going to have 4 here.

07:41 So 2 plus 2 is not equals to 5.

07:47 Now, let's consider the following declarative statement.

07:50 The heart is the planet.

07:52 3 plus 4 is 8.

07:53 Are you in Nigerian?

07:55 X plus 5 equals 9.

07:57 Come out.

07:58 What a great day.

08:00 All these are likely statements.

08:03 Both there are some statements in all these that are no words.

08:07 They are not a proposition or a statement.

08:11 The heart is the planet is the statement and is what is a true statement.

08:15 3 plus 4 is what is equal to 8.

08:18 Is a statement but is not true.

08:21 Are you in Nigerian?

08:22 This is not a statement.

08:25 A statement of position must not be a question declarative.

08:30 It must not be in form of a question.

08:32 X plus 5 equals 9.

08:34 Depending on the value of our X that will make us to know whether it is true or false.

08:38 But this is what.

08:39 This is the statement.

08:41 Come out is not a statement and it's not a proposition.

08:47 What a great day.

08:48 We have talked about all these in this as well.

08:50 So you should know what the type of statement is or these come out.

08:54 What it is.

08:55 It's like a command, a command.

08:58 So it's not a statement.

08:59 What a great day.

09:00 An exclamatory statement is not what.

09:04 What a great success.

09:05 You know, it's all these are words, they are not statement or proposition.

09:09 So we have a simple statement or proposition are represented by the letters PQR.

09:15 So in this course, in this topic, we want to represent all our sentence, all our position

09:21 in form of a PQR and so on, which are called, which are called proposition.

09:29 Now, proposition variables.

09:33 The proposition variables can be assigned to values, namely true or false.

09:39 These are two propositions are constant, namely true or false.

09:43 You can only have a true statement or a false statement.

09:47 You cannot have boats, you cannot say that statement is what is boats, true and false.

09:57 But if, for example, pure percentage proposition, the moon is a planet, to say that the proposition,

10:05 the moon is a planet is what is a force.

10:08 So we can simply say that the value of P is what is F, just like we said here.

10:14 So if P, for example, if P is a statement, the moon, the moon is a planet, the moon is

10:24 a planet.

10:25 This is statements and this is not true, it's not true, it's a force.

10:33 So we have F, the value of P here is going to be what is going to be F.

10:39 The compound proposition, a proposition can be obtained from the combination of two

10:45 or more same two sentence, true or more propositions by means of logical predictors, called

10:53 conductives, that is, when you have two statements, you can connect two statements or more than

10:59 that by what we call the conductives.

11:03 So when you have two or more statements, then we have what we call compound proposition.

11:08 You can also call it compound proposition or we call it molecular or composite proposition.

11:18 But the connective and the connective and the world are frices or symbols that are used

11:26 to form compound proposition.

11:28 If you have two statements, for example, and you want to join these two statements together,

11:33 the moon is a planet is a statement, 4 plus 5 is a statement as well.

11:39 So if you want to join the moon is a planet and you want to join 4 plus 5 together, there

11:47 should be a connectives and that connective is the world of frices or symbols that are

11:52 used to form compound proposition.

11:55 The basic connective and negation, the basic connective are what and negation and the

12:02 conjunction, the junction, disjunction, implication and equivalency, all these are what they

12:08 are connectives.

12:09 So the symbols used to represent the connectives are shown in the table below.

12:14 You have a connective word, knots, the symbol is this, the symbol is this and the composite

12:22 main is negation.

12:24 Just you want to come out, you want to, you want to, you have the two statements and you

12:29 want to come back, you want to compound them, you use and probably one is one is true and

12:36 the other is false, you use these negations towards to connect them and the symbol is

12:42 not P, the symbol form, the symbolic form is not P, the AND, the symbol is this, the AND,

12:52 we have the AND, you know the negation, the first one, we said we use this, then we have

12:59 AND, we can also use AND to connect two statements and that AND is, the symbol A is this and

13:09 the last composite main is what is conjunction and when you have, how can we represent that,

13:15 you represent that by this, the symbolic form of that, all is disjunction, we use this

13:22 for disjunction and the symbol for that, symbolic form of that is what is this, we also

13:29 have if then, if ade is beautiful, then is AND, if ade is beautiful, then is what is

13:37 AND, you can see that, so if then is also a conductive word, then how can we do that,

13:45 is an implication statement, if ade is beautiful, it implies that is AND, so and how do you do

13:56 that, we use this word, we use this to connect the symbolic form of that is P implies Q, then

14:03 equivalency, if ade is, if ade is beautiful, if and only if is beautiful, then is what,

14:12 is AND, so it means those two statements are what, they are equivalents, if for example,

14:19 if A, if A equals B, and B equals C, the equivalence of this one is what, A equals C, that is

14:33 implication, that's what it means, that is what we, that is equivalency, and we use that,

14:38 we use this to what to denote it, negation given that P is a proposition, the negation

14:47 of P is denoted by, that's not P, is the proposition that when P is true, and when P is false, example

14:59 1, consider the proposition P, Abhujah is in Nigeria, the negation of P is the proposition

15:05 that, not P, it is for that Abhujah is in Nigeria, or Abhujah is not in Nigeria, it is, it is,

15:17 it is false that, it is false, that Abhujah is in Nigeria, all you say, Abhujah is not in Nigeria.

15:26 Conduction, if P and Q are two given propositions, the conjunction is the compound proposition, denoted

15:35 by P and Q, which is read as what P and Q, the compound position is what, is P and Q is true,

15:45 both P and Q are true, so it is only, when you want to, when you have two statements, for example,

15:53 and those two statements that you want to say, both are true, for example, a day is unsome,

16:00 a day is beautiful, for us to have both to the world, so be true, and the false statements must

16:10 also be true, the seventh statement must also be true, one cannot be true, another is false,

16:16 and both by time you join them, both will be true, no, if both want to be true, then the false and

16:23 the second statement must be true, if otherwise is false. Example two, form the conjunction of P

16:35 and Q for each of the following, P, Q is intelligent, Q is ad walking, P, another example, the weather is

16:46 odds, Q is, rail is falling, now the solution of this is, P, if you want to, if you want to form

16:57 a compound statement, you know what I'm going to say, Mules is intelligent and is ad walking,

17:03 Q is what is intelligent and is ad walking, and this statement can only be true, when the two statements,

17:12 when the statement, when the two statements are what are true, if otherwise is false,

17:18 take note of that, the second one is the weather is odds and is, and rail is falling,

17:24 for this statement to be true, for us to have this true statement, the two statements

17:30 must be what must be also true, if otherwise it's false like we said, the junction,

17:35 given to proposition P and Q, the disjunction of P and Q is compound, proposition P or Q,

17:44 and it reads that what is written as P or Q, the proposition P or Q is true, if at least one of P

17:56 or Q is true, so it's necessary, it's possible for us to have, for us to have a junction,

18:02 a compound position, usually a disjunction sign to form the compound position,

18:09 for us to have a compound position to be true, at least one of the two statements must be true,

18:15 so that is for us to have a true statement, is that that P is true or one is not, our Q is not true,

18:23 is that P is not true or P is true, so at least one of them must be what, our both is even true,

18:29 at least one must, most of our both, one must be true, for us to have a, for us to have,

18:35 a first is true, if at least P or Q is true, but if false, both P and Q are false,

18:47 so example of that is, consider the following proposition, Lagos is in Nigeria and Acra is in Ghana,

18:53 Q, Lagos in Nigeria and Acra is in Kenya, Lagos is in Egypt and Acra is in Ghana,

19:02 Lagos is in Egypt and Acra is in Kenya, let's look at this, the first statement, this false statement,

19:10 in the fourth position above, only A is true, whereas the rest are false, look at it, only A that is only

19:19 number one is false, it's true, the rest are false, Lagos is in Nigeria and Acra is in Nigeria,

19:25 if look at the rest, our words, when we are using A, our statement to, you know, the our statement,

19:33 I mean the our statement to denote that, Lagos in Nigeria, this is true, Acra is in Kenya,

19:40 it's not true, Lagos is in Egypt, it's not true, Acra is in Ghana, that is true, Lagos is in Egypt,

19:49 it's not true, Acra is in Kenya, it's not true, so you can see that in these four positions are both,

19:56 only Q, Lagos is true, whereas the rest are false, because A, Q, Lagos is both components of the

20:05 position are true, where others, provision at least, one component of each provision is false, a true

20:14 table of however we are talking about, we also have a true table for each of these, either negation,

20:21 either disjunction, conjunction, hand, so A, let's define, a true table is a table for evaluating the

20:30 true value of a true functional component sentence, on the basis of the true values of its parts,

20:38 a true table is used to define the true functional conductives, and also to test the validity,

20:45 meaning that a true table can also be used to, to test the validity of, of, of, of, of any statement,

20:56 so the following are the true table for negation, conjunction, and discussion, so we have if P is a

21:01 statement, and if P is true, the negation of that statement is going to be false, if P is a false

21:07 statement, and the negation of that statement is going to be true, for, for, for, for, for,

21:13 conjunction, if P, if we have the true statement P and Q, if the false statement P is true,

21:20 and the second statement is true, the second statement is true, then both statements are going to be true,

21:25 if P is false, the first statement is false, I mean is true, and the second statement is false,

21:32 the both statements are going to be false, are going to be false, just like we said,

21:38 Nigeria is in, now, is in, is in, are, are Nigeria is in Nigeria, and you also have acra is in Kenya,

21:45 Nigeria, are, are Nigeria is in Nigeria is the true statement, both acra is in Kenya is not the true

21:51 statement, so for us to have these two statements together, it must be false, because the false,

21:58 one or two is what is false, then if the P is false, true is, the second one is true, we have the

22:06 first statement, if both are false, we have the first statement, this junction as well, if T,

22:11 the first statement is true, and the second one is, is true, we have the true statement,

22:16 and if the first one is true, and the second one is false, we have the true statement,

22:20 it's only when both statements are false that we can have a false statement in this junction,

22:26 please take note of that,

22:29 example number four, we can construct, construct the true table for the compound position,

22:34 for this compound position, we have the P and Q, what we want to do is, we have the P,

22:41 or negation of P and Q, P and Q, negation of P and Q,

22:54 negation of, so we can negation of Q, so if true, if P is true and Q is true,

23:07 the negation of true, the negation of Q when it is true is false, and when you have Q

23:17 and the negation of Q, for the AND, it's going to be false, don't forget, don't forget that

23:26 for the AND, it's only false, it's going to be false, when it's going to be true, when both are true,

23:34 when one of them is true, it's going to be false or true, so true, true, true, false, we have the

23:40 false, first true, we have the words, we have what we call, we have the false, true false, we have

23:47 false, then false true, we have false, you can see, for this, then what we have P and this,

23:53 for OR, when we have true or false, we have the true, when we have true or false, we have

24:05 a true, when we have false or false, we have the false, when we have false and false, we have the true,

24:10 so all these construction of these attributes trying to tell us is that how we can easily identify

24:16 when you have AND, or that is when you have the conjunction and we have, and when you have a

24:22 disjunction, totology and contradiction, a compliant proposition that is always true for all

24:31 possible true values of its variables is called totologies, meaning that when you have accomplished

24:38 a compliant proposition and all of them are true, in all the ones that we have had earlier,

24:47 that is the conjunction, not all of them are true and not all of them are false, at least one is

24:52 what, one is false and one, probably maybe one is true as well, so but when you have all the statement

25:01 to be true or false, then you have, or let's say, you have a totology, when a compliant proposition

25:11 that is always false for all the possible true values of its variables is called contradiction,

25:17 so that is when all the, for example, if for example, is just an example, we have a true table of

25:22 this form, we're going to look at example anyway, and we have this is P, and we now see that

25:29 we, this is false true, this is false true, this is false true, now if you look at this now, these are

25:38 words, these are all true, so these statements are what we call totologies, you can see that this

25:47 one, all these are words are also false, we can see that this is what is also contradiction,

25:53 so that is what we are talking about, that is what we are saying, totology and contradiction,

25:58 its logic has it, as a totology has only T in the last column of its true table, in the last

26:04 column of its true table, similarly a contradiction has only false in the last column of its true table,

26:11 the last column of its true table, you will see now example number four, show the following

26:15 proposition is a totology of contradiction, we want to show this, we have P or not P, we have P

26:23 and not P, P we have the true, not P, we have the false true, P is true, not P is false, false

26:38 P is false, not P is tot is true, we have P is false and not P is going to be true,

26:49 now if you want to combine these two statements together now, using the all connective,

26:58 then we have the true, true false, true false, false true, false true, false true, false true,

27:07 false true, false true, false true, false true, false true, we have false, we have the true rather,

27:09 you can see that the last column of these true table is all false, I mean it's all true,

27:16 then we said this is what is totology, this is what is totology, then also

27:22 for AND, you can see that by the time you have this, for the AND construct, for the AND,

27:32 which is conjunction, all the last column here is also false, then this is what is contradiction,

27:42 since the last column of the true table contains only F, the provision of the law of the position,

27:47 P or not P is the contradiction, is contradiction, so that is when we have the contradiction or a

27:55 totology statement, arguments, arguments, and arguments is a sequence of proposition or statement,

28:05 or the position before the final one are called the premises, why the final proposition is called

28:13 the consequent, that is when you have a sequence of proposition, the first one, we call it premises,

28:20 the second one we call this consequent, or conclusion, and the symbols of that is therefore it

28:26 treats it, the therefore is what we use when we are having statements, now a valid argument is

28:36 one in which the conjunction of the premises implies the conclusion, that is when we have a

28:44 valid argument, if the argument is said to be valid, if the premises implies the conclusion,

28:51 the first statement implies the last statement, we may say that if an argument is a totology,

28:57 then it is called a valued argument, if all the last column in your compound position are

29:06 true, such statements are valid arguments, on the other hand, if an argument is not a totology,

29:16 we say it is what, it is invalid, a true table can be used to prove whether an argument is valid

29:23 or not, and that gives us an example number five, time whether or not the following argument is valid,

29:29 if it contests it will win, it has contested, therefore it will win, look at this statement,

29:37 there are three statements or four statements, if it contests it will win, it has contested,

29:44 therefore it will win, let p it contests and q denote it will win, the argument has the following

29:53 formats, it contests, if it contests it will win, you can see that the first the premises

29:59 give us the conclusion, which is the conclusion, because if it contests it will win,

30:08 then the argument has this, the form of this argument is what is that, e, p implies q,

30:15 and p all over q, that is, it has contested, therefore it will win, that is the last statement,

30:26 this is the very one, we must also symbolize the argument as this, p implies q, and p,

30:36 if it contests it will win, and it has contested, then what it will win, you can see that,

30:52 this one, the p, give us the conclusion of this, and between this,

30:59 therefore there is also give us q, so we can also represent it in this form,

31:06 and we can easily construct the true table for this, and the true table for this are shown,

31:11 yeah, we have a p q, p implies q, we have talked about this in ss1, p, when we have true true,

31:20 we have a true, when we have true false, when we have false true, we have a true,

31:24 when we have false true we have a true, then for this very one too, in between this,

31:29 the p and this, for the and, then for this very one as well we have this, so from the last column

31:38 of the true table, we see that the argument is a photology, if you look at the last statement here,

31:44 you see that the argument is what is photology, and hence the argument is valid,

31:49 and the abode argument as we have seen as the format of this, it is valid argument and it is called

31:54 modules ponens, or rule of detachment, the law of algebra, all these are at the law of algebra for

32:06 preposition that we have, we state all these rules, all these laws that prove, the law of algebra

32:13 for preposition, we have what we call an indipotence laws, p or p will give us equivalent to p, p and

32:21 p equivalent to p, compressive law p or q equivalent to q or p, likewise p and q equivalent to q

32:31 and p, then we also have an associative law, p or q or r, we have p or q or r, so we also have

32:42 distributive laws as well, then we have identity laws, just like we have law in our binary operation,

32:50 and I know that you must have been taught a binary operation, all these also, a binary operation

32:55 also goes for logic reasoning as well, we have infolutions law, then we have the Morgan's law,

33:05 so all these are the law, the algebra law of preposition that we have,

33:12 and that's because to conclude, the class of law and the evolution is what we have, the first one is,

33:20 was that the following preposition, obi is proud, obi is proud, wale is ombo, these are the two

33:27 statements p and q that we have, right each obi, each obi following compound preposition in symbolic

33:34 forms, obi is proud and wale is ombo, obi is ombo and wale is proud, obi is proud,

33:42 or wale is not ombo, we can also test the validity of the following preposition, if today is one day,

33:49 then tomorrow will be Tuesday, tomorrow is Tuesday, there for today is Monday,

33:54 so these are the logical way in which we can say that whether a statement is valid or is not valid,

34:02 whether the statement is totology or is contradiction, thank you for watching, my name is Aded O'Connor,

34:10 I feel once again, till next time.