Video Transcript: Eight Valid Forms of Inference

Hi, I'm David Feddes. And this logic talk is about eight valid forms of inference. And as we get  into the various valid forms of inference, I have Glad tidings we are about to escape truth  tables. Now truth tables have value truth tables are our only formal method of deciding  whether an argument is valid or invalid. proofs which we're about to get into, can prove that  an argument is valid. But they can't prove the argument to be invalid. So truth tables can do  some things that proofs can't, proofs can only prove that it's valid, but they can't prove that  it's invalid. But proofs are much much more efficient than truth tables are. Take this argument for instance, R or S entails T entails K second line, not K. Third premise R or S and then  conclusion is not T. Now if you want to figure out whether that's a valid argument and make a  truth table, you would need 16 rows because there are four different variables. And remember to to the nth power is how many rows you're going to need. So two to the fourth power would  be 16 rows. If you had an argument with five variables, you need 32 rows. If you had an argue with six variables, you need 64 rows to work through whether the argument was valid. Take  this argument that we see here with four variables, you'd have to have 16 rows and a bunch  of columns in your truth table and spend some time working it out. This argument can be  proved valid with only two additional lines, if you know how to do a proof. So a proof can be  much much more efficient than a truth table, especially when a lot of variables are involved.  We'll see a little bit later how you can prove this in just a couple of lines. But first, let's get  into our eight valid inferences. We're going to see that they have some kind of funny names.  In some cases, the first is modus ponens, then modus tollens, then hypothetical syllogism,  then simplification, conjunction, disjunctive syllogism, addition and constructive dilemma.  Don't those logic people have a way with words? Don't you just love the labels they give to  things? Well, sorry about that. But that's it we're stuck with this is what logic people have  labeled these eight forms of inferences. So I pretty much have to use what it's always called,  by the logic, folks. Anyway, these are the eight things that we need to know. And you need to  memorize, not just these long winded names for them, but even more importantly, what they  mean and kind of get them in your mind so that you can understand the really basic ways of  making a valid inference in logic. The first form is modus ponens, which means the way that is affirmative in Latin mode, that's where this comes from modus ponens. Anyway, modus  ponens takes the form that in the first premise, if P then Q, and your second premise is P. And  then your conclusion is therefore Q. Or to put it into a case, in English words, If Joey is a  poodle, then Joey is a dog. That's your first premise. Joey is a poodle. That's your second  premise. Therefore, Joey is a dog. Not terribly complicated. But you see the form that it takes  P entails Q, P, therefore Q. And that's true. Even if you have more complex propositions, if  your premises A and B, and then you have that entails C, and your next premises A and B,  then you can reach the conclusion. See, your first premise is a conditional. And no matter how long that antecedent is, it could be A and B, it could be a whole bunch of different things. But  as long as the antecedent is there in the first premise, and then the second premise just  states the antecedent, then you know that the consequence is true, the conclusion is valid. So that's modus ponens. Modus tollens, is also taken from a Latin phrase, and that Latin phrase  is the way that denies it uses the negation. So again, you have a premise to begin with. P  entails Q, or if P then Q. Your second premise is not Q. If Q isn't true, then you can conclude  that P isn't true either, because P entails Q. So if P is true, Q has got to be true. Q is not true.  That means P isn't true. If Joey is a poodle, then Joey has a dog Joey is not a dog. Therefore  Joey is not a poodle, you can't be a poodle without being a dog. And that, again, is a true  form, even if you're using complex propositions. So if you say if C, then E or F, not E or F, well, therefore, not see your first premise is a conditional. And the second premise is a negated  consequent. And so if you have a negated consequence of your conditional, then you know  that the antecedent of the conditional is false. P entails Q, not Q, therefore not P. That's how  modus ponens works. Now let's talk about constructing a proof a proof is a series of  statements, starting with the premises, and ending with the conclusion, where each additional statement after the premises is derived from some previous line of the proof using one of the  eight valid forms of inference. So you start out with a number of premises and you're  supposed to prove a conclusion. And what you have to do is go step by step where every  state, every statement that you add, is derived from previous statements, and you got your 

new statement, by using one of the eight valid forms of inference Thus far, we've only looked  at two valid forms of inference. But we can already do the proof of that example we saw  earlier. R or S entails that T and entails K, that's first premise, second premise, not K, third  premise, R or S. Conclusion, therefore, not T. And we'll just write the conclusion on the same  line as the last premise with the therefore sign. So that's what we want to prove. How do you  go about doing that? Well, the first thing you do is you notice that T entails K because we use  modus ponens. To get there. How do we do that? Again, our R or S is a premise in the third  line, right? But on the first line, we see that our R or S entails T or K. And so because our R or  S is true, and it entails T or K, therefore, T entails K is true, that's modus ponens, and you  write modus ponens. And then you write lines one, which was the conditional, and line three,  which was the antecedent, the antecedent means that T entails K. And so you can say that T  entails K. Our next step now that we know that T entails K, we look and we say, not T by  modus tollens. And we look at lines two and four. line two says not K. Line four says T entails  K, but remember, modus tollens means that if your consequence is negated, K is negated,  then T has to be negated. So if you have two premises, not K, and T entails K, then you know,  not T. And that's your conclusion. Because that's what you're supposed to prove. Therefore,  not T you've proven in just two lines, something that would have taken you 16 lines in a truth  table. Well, we've seen modus ponens and modus tollens. Let's move on to hypothetical  syllogism. hypothetical syllogism is also called the chain argument, and we'll see why in a  moment. Here's how it goes P entails Q, Q entails R. Therefore, P entails R or you could phrase it this way if P then Q, if Q then R. Therefore, if P then R. Or if you use an English example, If  Joey is a poodle, and Joey is a dog enjoys a dog enjoys an animal. Therefore, if Joey is a  poodle, then Joey is an animal. And this is true again no matter what form the antecedent or  the consequence takes it can be a complex proposition. So here's a hypothetical syllogism, A  or B entails not B, not D entails C and therefore, A or B entails C because the consequence of  the first premise is the antecedent of the second premise. And therefore, from that, you can  infer the consequence of the second premise P entails Q, Q entails R, therefore P entails R. It's also called the chain argument, because you can go on adding to it. Yes, P then Q. If Q then R, if R then S, if S then T. Therefore If P then T. See how that works? You're just going if then if  then if then if then well, then if the first item, the first antecedent is true, then the final  consequent is true. If you have that chain argument. simplification is another of the valid  forms of inference. And it is really simple, almost so simple and makes you say, duh. Why did  they even do that? Your first premise is P and Q. Therefore, P. You can also say, therefore Q,  basically, all you're saying is that, that if you have a conjunction of two propositions, then the  first proposition can be stated by itself to be true. And the second proposition Q can be stated by itself to be true. Joe is smart, and speedy. Therefore, Joe is smart. And you could also say,  therefore, Joe is speedy. And so it's one of those things where you say, duh, that is so obvious, why would anybody even say so. But that is really the whole point of logical argumentation is  to take something that's not very obvious, an argument that doesn't seem to be very obvious. And when you work it all out in small, obvious steps, you prove that it's true by a whole bunch of dust statements by things that take it by themselves or little tiny steps that absolutely  have to be true. And then you show that the whole big argument is true by taking it one little  step at a time. Now again, simplification, like all the others can have more complex  propositions, but it still holds true. Instead of just P and Q. You can have two complex  propositions that are conjunctions you can have A or B, and not quantity C and D. And then  you could say, well, therefore A or B, because it was one side of the conjunction, you could  also say therefore, not quantity, C and D, because that's the other side of the conjunction. So  if line one is a conjunction, then line two can infer either one of those two conjuncts. So that's  called simplification. We've seen modus ponens, modus tollens, hypothetical syllogism  simplification. Now let's move on to conjunction and don't confuse this with the logical  operator conjunction, although it does involve that. Here's how conjunction works on the first  line, P, the second premises Q, therefore P and Q. Another one of those does statements. The  first premise is P second premise is Q, and therefore you're entitled to infer the conclusion. P  and Q is a true statement. Joe is smart. That's your first premise. Joe is speedy. That's your  second premise. Therefore, Joe is smart and speedy, aren't you glad you took logic you are 

discovering things you never would have thought of on your own. But that's how this proof  kind of logic works. It seeks to help us break things down into very small, simple obvious  steps. And then we can show that more complicated things are true as well. And again,  conjunction can involve complex propositions. If A then B is your first premise C or D is your  second premise. Therefore, if A or B and C or D. So you have the conjunction of A entails B  with C or D. Yes, you assert two propositions no matter how long winded or complex. Those  two propositions are. You can also assert the conjunction of those two propositions. disjunctive syllogism begins with a disjunction P or Q. The second premise is not P. And then the third is  there for Q, because in the first line, you say P or Q is true. And if one side of that isn't true,  you know, the other side has to be true. In English, Joe or Jane drives, Joe does not drive.  Therefore Jane drives. And again, using more complex propositions you could have not A or B  and C. Second premises, not not a now that's bad grammar to have double negatives, but bad grammar doesn't mean bad logic. Sometimes people in logic use double negatives not not A  so anyway take my word for it. Not not A is okay to say in logic. So anyway, if your first line is  a disjunction of not A with B, and C, and your second line is not not A then you can say  therefore B and C is true. The second part of the disjunct is true because the first part of the  disjunct is false. The first premise is a disjunction. The second premise is the negation of the  left disjunct and premise one. So if one side of the disjunct is false, the other side has to be  true. If the disjunct as a whole is true, that's what a disjunctive syllogism is P or Q, not P  therefore, Q or you could also phrase it P or Q, not Q. Therefore, P. It doesn't matter what  order those are in if you have a disjunction, and one member of the disjunction is false, the  other member of it has to be true. Modus ponens, modus tollens, hypothetical syllogism,  simplification, conjunction, disjunctive syllogism. And now just two more and remember,  these are to be memorized, I don't expect you to get them all from just watching this video  and have them instantly imprinted on your mind study Dr. Van Cleves article about it. Watch  this video a time or two more that get into your mind what these eight valid forms of  inference are addition, your first premise is P. And then your conclusion is simply P or Q. In  English Jane drives, therefore Joe or Jane drives, you can add anything once you know that  something is true, you can add the word OR. And even if whatever you add is false. The  disjunction is going to be true, because you know that the first part is true. So anytime you  know that P is true, then you can say P or anything else. And the statement P or Q is going to  be true simply because you know that P is and that's true of any complex proposition as well  as atomic proposition. So if you say A or B is true, then you can say A or B or not C or D. It has the same form. You see line two is a disjunction of line one and an additional statement. And  so if line one is true, then line two must be true. Even if the second disjunct is false. It doesn't  matter how false it is, if the first disjunct is true, and that's why addition works. You say why  would anybody even want to do that? If you have P why would you want to say P or Q no  matter what Q might happen to be? Well, the answer for now is it helps you do proofs. And  sometimes it helps you to be able to do this method of addition to prove something. So  anytime you have a statement that is true, you can just add the wedge and add any other  proposition you want. And the disjunct is always going to be true. That's the method of  addition, when you're doing logical proofs. And the final one is constructive dilemma, which is  a little more complicated than the ones we've been looking at. But it's not terribly  complicated. We'll start with an English example. The killer is either in the attic or the  basement. If he's in the attic, he is above me. If he's in the basement, he's below me.  Therefore the killer is either blank, or blank. How do you finish that? He's either in the attic or  the basement, if he's in the attic, he's above me. If he's in the basement, he's below me.  Therefore, he's either above me or below me. Wow, that is tremendous logic. He's either  above me or below me. But that's the way a constructive dilemma works, you start with a  disjunction P or Q, then you have two conditionals P entails R, Q entails S and then your  conclusion is R or S must be true. Take it in English Joe or Jane will come if Joe comes, He will  bring burgers, if Jane comes, she will bring salad. Therefore Joe will bring burgers or Jane will  bring salad, you know, based on constructive dilemma that if you have a disjunction and then  to consequence with the antecedent of those two concepts of those two conditionals if you  have the antecedent of two conditionals. And you know that one of those conditionals is true 

por que then you know that one or the other of the consequence is going to be true. So, you  have these eight valid forms of inference modus ponens, modus tollens, hypothetical  syllogism, simplification, conjunction, disjunctive syllogism, addition and constructive  dilemma. You will need to know them to get anywhere on proofs to be able to look at some  premises into our try to work out what needs to be done. So I encourage you just watch the  video another time. Read the article a couple more times. If you learn better by reading, then  by watching and listening and get these eight methods of drawing valid inferences into your  mind. They will serve you well as you move into your more logical future.