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Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 5 Logical Inference
In the last class we saw about logical quantifier, universal quantifier and existential quantifier and how to use them we also saw the rules regarding them as to whether you can interchange the order in which they are occurring and so on. In any logical system you have some axioms and rules of inference, and making use of the rules of inference from the axiom you try to derive true statements about the system. And if you use the rules in a proper manner you arrive at a correct argument and if you use the rules in improper manner the argument will be invalid. So next we shall see how to find out whether an argument is correct or wrong starting from the axioms and rules of inference. Towards the end of the last lecture we saw some rules of inference we have also seen them as logical identities earlier so let us recall those rules.
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From P you can conclude P or Q this is called addition and from P and Q you can conclude P this is called simplification, from P implies Q AND P you can conclude Q this is called Modus ponens, from P implies Q AND NOT Q you can conclude NOT P that is called Modus tollens and from P OR Q and not P you can conclude Q this is called Disjunctive syllogism and from P implies Q AND Q implies R you can conclude P implies R this is called Hypothetical Syllogism.
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And from P AND Q you can conclude P AND Q this is called Conjunction. Then from P implies Q AND R implies S AND P OR R you can conclude Q OR S this is called Constructive Dilemma and from P implies Q AND R implies S AND NOT Q or NOT S you can conclude NOT P or NOT R this is called Destructive Dilemma.
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Mostly we will be using these two rules Modus ponens and Modus tollens in our arguments. Let us see how we can prove whether an argument is correct or whether an argument is wrong and so on.
So in the logical representation how do we represent these statements? If horses fly or cows eat grass then mosquito is the national bird. That will be denoted by if horses fly or cows eat grass then mosquito is the national bird. Then second statement is, if mosquito is the national bird then peanut butter tastes good on hot dogs and that is denoted by this. But peanut butter tastes terrible on hot dogs and that is denoted by NOT P. So the conclusion is therefore cows do not eat grass therefore NOT of G. This is the argument. The argument you can represent in logical notation in this form. There are three premises and this is the conclusion. Does the conclusion logically follow from the premises? Are we using the correct rules of inference and arriving at the conclusion? This is what we want to check. Let us see whether from these three premises we will be able to prove the conclusion.
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Now look at statement one and two and if you use from one and two if you use Hypothetical Syllogism what do you get? H OR G implies M and M implies P so from that you can conclude H OR G implies P. Let us call it as statement four. The fourth assertion we have arrived at. Now from 4 and 3 what is 3? So 3 is NOT P so from H OR G implies P and NOT P what can you conclude? By using Modus tollens you can conclude NOT of H OR G. Let us call it as statement 5 and from statement 5 use De Morgan’s laws, this you can expand as NOT H AND NOT G call it as statement 6. Then from 6 BY simplification you get NOT of G. So you are able to arrive at the conclusion from the premises by using proper rules of inference. So even though these sentences may not convey any meaning it may look a silly argument but still from the premises using the rules of inference you can arrive at the conclusion. So this argument is a valid argument.
Now let us see some more arguments. It is not the case that IBM or Xerox will take over the copier market. If RCA returns the computer market then IBM will take over the copier market hence RCA will not return to the computer market. Let us see whether this
argument is correct. It is correct but let us see how it is. So again IBM will take over copier market denote it by I.
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It is not the case as Xerox, Xerox will take over the copier market denote it by X. RCA will return to the computer market. Now how do you represent the statements in logical notation. It is not the case that IBM or Xerox will take over the copier market. NOT I OR X is the first statement.
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I have test in Economics by E. Economics professor is sick by S. Then how will you represent the statements in logical notation?
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The first statement is if today is Tuesday then I have a test in Computer Science or in Economics. And second statement is, if my Economics professor is sick then I will not have a test in Economics. And third statement says today is Tuesday and my economics professor is sick. So these three are the premises from which the conclusion is therefore I have a test in Computer Science.
Now let us see whether this is a correct argument and how do we arrive at the conclusion from the premises. Now from 3 by simplification you can get T and you can get S. And from 1 and 4 T implies CS OR E AND T by using Modus ponens you will get CS OR E. Similarly, from 2 and 5 S implies NOT E and S by again using Modus ponens you will get NOT E. This is again by the application of Modus ponens to 2 and 5. Now you have CS OR E and NOT E so from 6 and 7 use Disjunctive syllogism that is from CS OR E and NOT E you can conclude CS which is the conclusion.
So starting from the premises by the application of proper rules of inference you are able to get the conclusion and the argument is valid. So this is the sort of a proof you give saying that this argument is correct or not. Now if some argument is not correct you must give a counter example. What you mean by that. There will be some premises and conclusion and if you want to show that the argument is not correct you must give the truth values to the statements involved to the assertions involved such that the premises become true and the conclusion is false.
Usually people make two types of fallacies frequently and let us see what they are. One is called the fallacy of affirming the consequence.
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So, from P implies Q and Q you cannot conclude anything. For Modus ponens from P implies Q and P you can conclude Q. For Modus tollens from P implies Q AND NOT Q you can conclude NOT P. But from P implies Q and Q you cannot conclude anything.
Suppose if you make an argument like this; if the butler is nervous then he committed the murder. Or I will put it this way, if the butler committed the murder he will be very nervous. The butler is nervous so he committed the murder, is this argument correct? This is not a correct argument because here the statement is, if the butler committed the murder he will be nervous and because he is nervous he committed the murder that is a
And from for all of x P(x) you can conclude P(c) where C is an arbitrary element of the universe. This sort of a root is called Universal instantiation.
Now if you have P(c) for some C belonging to the universe it is not arbitrary but for some C belonging to the universe if you have P(c) then from that you can conclude there exist x P of x. This rule is called existential generalization. And from there exist x P(x) you can conclude therefore P(c) where C is some element not arbitrary belonging to the universe. This rule is called existential instantiation.
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So for using quantifiers we have four types of rules; Universal generalization where if you have P(c) for some arbitrary element you can conclude for all of x P(x). Then you have Universal instantiation where from for all of x P(x) you can conclude P(c) where C is an arbitrary element of the universe. If P(c) is true for some C belonging to the universe then you can conclude therefore there exist P(x) and this rule is called existential generalization. And from there exist x P(x) you can conclude P(c) where C is some element this rule is called existential instantiation.
Now let us see how we can give some arguments using quantifiers. Now let us see some arguments involving quantifiers. The first example which is very common stated in all logic books is like this. All men are mortal. Socrates is a man therefore Socrates is mortal.
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Obviously it is a correct argument and how do we prove this? Now man of x denotes X is a man X is a man. Mortal of x denotes X is mortal. So the argument is like this, if you transfer to logical notation it will be like this for all of x man of x implies mortal of x. Then Socrates is a man, man (Socrates) the conclusion is therefore Socrates is mortal i. e. mortal (Socrates). This is the argument let us see how you prove it.
There are two premises and the conclusion. So from first using Universal instantiation from using one and from using Universal instantiation you get man (Socrates) implies mortal (Socrates). That is x is taking the value of Socrates and the rule applied is Universal instantiation. And call this as statement 3 then from 3 and 2 by using Modus ponens that is p implies q and p you can conclude q so from this you can conclude mortal (Socrates). So this is the way you have to give the argument.
rules of inference that is what we want to check. Actually if you look at this in this case all the three statements are really true but whether the conclusion which is some trigonometric functions are periodic will follow from the premises. The premises here are some trigonometric functions are continuous and some continuous functions are periodic.
So if you denote by trigonometric function x is a trigonometric function T(x), x is trigonometric function and C(x) continuous function x is a continuous function P(x) x is a periodic function. The statements will be; there exist x T(x) AND C(x) this is the premise. And the second statement is there exist x C(x) AND P(x). The conclusion will be; there exist x T(x) AND P(x) this is the conclusion.
But you can see that this will not follow from this by the rules of inference in this case you can give a counter example like this; some trigonometric functions are continuous using the Venn diagram this may represent something like this, some trigonometric functions are continuous the intersection is nonempty, the intersection of this is nonempty.
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Then the second statement is, some continuous functions are periodic. So if you denote the periodic functions like this the intersection of C and P is nonempty. There are some functions which are continuous and periodic. So the first two statements may represent a situation like this; but you can see that this does not mean that T and P intersect. The intersection of T and P may be empty also by this argument. By this argument we cannot conclude that some trigonometric functions are periodic. So if you want to show that some argument is wrong then in that case you can draw a proper Venn diagram and show that it is correct. But for proving it is not enough if you just draw the Venn diagram you have to also use proper logical argument.
Let us see how you do this in this case. First you can check whether the argument is correct by drawing a Venn diagram but then you have to give the proper argument for this. Babies are illogical, so if you denote B of x to be x is a baby and Il of x x is illogical, D(x) x is despised and M(x) x can manage a crocodile.
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Using Venn diagram how will you represent this? Babies are illogical so babies will be inside illogical. And all illogical people are despised so that will be inside D and nobody you can manage a crocodile is despised that is the intersection of M and D is empty that means babies cannot manage crocodile that is B and M are disjoint, that is true from this diagram. So this argument is the correct argument. You can very it by drawing the Venn diagram but that is not the proof you have to prove step by step. But if you want to show that some argument is not correct it is enough if you draw the Venn diagram and show that there is a situation where the premises may be true but the conclusion is not true. Let me prove this now. Babies are illogical for all of x B(x) implies Il of x.
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So you can write the third state statement like this and you can also write it as for all of x D(x) implies NOT of M(x) because NOT P OR Q is equivalent to saying P implies Q. So the third one can be written in this form, I will write it as 4. So using Universal instantiation taking an arbitrary C you can write these statements 1, 2 and 4 in the following manner. B(c) implies Il of c Il of c implies D(c) and D(c) implies NOT M(c).
And using Hypothetical syllogism for these two using hypothetical syllogism you will get B(c) implies D(c) and from these two again using hypothetical syllogism you will get B of c implies NOT M(c). Using this and this and using hypothetical syllogism you get B(c) implies NOT M(c) because you have B(c) implies D(c) and D(c) implies NOT of M(c) you will get B(c) implies NOT of M(c). And here from this use Universal generalization you will get the conclusion for all of x B of x implies NOT(M(x)). Let us consider some more examples in logical inference.
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Take this problem, prove or disprove the validity of the following arguments: Take the first one a; Every living thing is a plant or animal. David’s dog is alive and it is not a plant. All animals have hearts. Hence David’s dog has a heart. Let us see whether this argument is correct by writing in logical notation. Let L(x) denote x is a living thing and P(x) denote x is a plant, A(x) denote x is an animal and H(x) denote x has a heart.
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And let David’s dog be denoted by d. Now what is the first sentence? The first sentence is every living thing is a plant or animal. You can write it in this way; for all of x L(x) implies P(x) OR A(x).
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Now the sixth statement is NOT Pd from NOT Pd and the eighth statement Pd OR Ad. From this, that is from 6 and 8 by Disjunctive syllogism what do you get? What is Disjunctive syllogism? That is if you have P OR Q and NOT P by Disjunctive syllogism you will get Q. So this is what we are going to use now. So 6 is NOT P of d and 8 is P of d or Ad so from 6 and 8 by using Disjunctive syllogism you will get Ad. And what is 7? So 7 is Ad implies Hd, 7th statement is Ad implies Hd. So from Ad and Ad implies Hd that is from 7 and 9 by Modus ponens you get Hd so this is the conclusion. So you are arriving at a conclusion from the premises so this argument is correct.
In a similar manner you can prove this part b also. All clear explanations are satisfactory some excuses are unsatisfactory. Hence some excuses are not clear explanations. This is a correct argument. Look at these two things; the following propositions involve predicates that define sets. Use the properties to conclude relationships between these sets. Use Venn diagram to check the validity of the arguments.
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So you can use the Venn diagram to show that some thing is not correct some argument is not correct. Consider the first one; some Scientists are not Engineers, some Astronauts are not Engineers hence some Scientists are not Astronauts. Let me draw a Venn diagram, if you draw a Venn diagram which will satisfy the premises but not the conclusion then the argument is not correct then the conclusion cannot be inferred from the premises.
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Now the first statement is some Scientists are not Engineers. You can have something like this:
