Video Transcript: Constructing Proofs

Hi, I'm David Feddes and this logic talk is about constructing proofs again based on the work  of Dr. Matthew Van Cleave. We've seen that there are eight valid forms of inference when  constructing proofs and it's very important to memorize or internalize these modus ponens  modus tollens, hypothetical syllogism, simplification conjunction, disjunctive syllogism,  addition, constructive dilemma. If you don't know what these are, if you haven't already  memorized them and worked on them, some you might as well just hit the stop on this video  right now, and go back and study them some more, we're going to use these eight valid forms of inference to construct some proofs. Proofs derive the conclusion from the given premises.  And they do that using only the eight valid forms of inference. Now you have eight valid forms available to you. But not every proof has to use every one of those eight rules, you can use  any of the eight rules as you move along. And you can use a strategy where you're kind of  working your way backwards or working your way forwards. In order to get a proof. We'll see a little bit more of how that works. First of all, let's look at working backward. What rule can we  use to derive the sentence that we need to derive? So working backward means you look at  the conclusion, and then you try to kind of think backward? How would I get to that  conclusion? Here's an example. Premise number one, R and S, premise number two, T is true,  therefore T or L and R and S. Now how do we work on that? Well, the conclusion of that proof  is or it's not a proof yet, but the conclusion of that argument is a conjunction. And before we  prove it, we just want to realize it is a conjunction. And what rule leads to a conjunction or  yields a conjunction. In this case, it's the rule that we call conjunction. If P is true, in one  premise, and Q is true as another premise, then we can say P and Q is true. So using that, we  say the rule of conjunction combines two conjuncts on separate lines. And so R and S is on a  line as premise one. And we'll need to get T or L on its own line. So when you look at the  conclusion you're trying to get R and S is already there. And it's a conjunct with T or L. So  we've got I or S we, R and S, we need to get T and L somehow on its own line in order to  reach a conclusion with that conjunct of the two statements. What rule yields a disjunction  because T or L is a disjunction. How do we get a disjunction? Well, you can do it using  constructive dilemma. Or you can do do it using addition. And in this case, it can't be a  constructive dilemma. If there's no conditional, and there's no conditional in any of those. So  we've got to use addition, in order to get there. And addition remember is if you have a  statement P and it's true, then P and any P or any Q will be true because any disjunction is  true, as long as one member of it's true. So we use disjunction in this case. Addition can  disjoin any statement to an existing statement. So we've got premise two, it's T. So we disjoin  L two it T or L is true, since T is true. If T is true, then T or L has got to be true. We're using  addition to line two and we get T or L. And we now have each conjunct of the conclusion, see  what the conclusion was supposed to be T or L and R and S we have each conjunct to that  conclusion on separate lines, line one has the R and S part. Line three has the T or L part and  all we got to do is conjoined them using conjunction. So you write down T or L and R and S  and D right conjunction of one comma three those two lines were conjoined to give you a line  four and what is line four line four is the conclusion you are supposed to prove. So you have  now a completed proof. Now, working forward is a little different strategy with working  backward if the conclusion to be proved is a complex proposition. If it contains truth functional connectives, then working backward is a good strategy. However, if the conclusion to be  proved as an atomic statement, then working forward is often a better strategy and working  forward. You just kind of fishing around a little bit. What rules can we apply to the existing  premises to at least come up with Have something to derive something and maybe the  something that we derive will turn out to be something that gets us toward that atomic  statement that we're trying to prove. So here's an example, A and B, B implies C, therefore C.  Now we look at that at those two premises. And we see that the conclusion C is atomic. So we can't work backward, you work backward, when you have a complex proposition you work  forward, if you have an atomic proposition as your conclusion. So we look at we see there's a  C, we're going to have to work forward. Now in working forward, the easiest thing in all of  logic is to look at a conjunction and break it apart, you've got A and B, you can break that  apart using simplification. Remember the rule of simplification. It's if you have P and Q as a  premise, therefore P. And also, therefore Q, you can just simplify and say that one is true on 

another line, and Q is true on the other line. So in we look at it, we break apart that  conjunction A and B by using simplification. And so that gives us line three a simplification of  line one, four is B simplification of line one, you just took the two parts of that conjunct and  put them on separate lines as being true. Now, what lines can we apply to derive something  or other? That's the question we asked, we look at what we have so far, A and B, B entails C.  And we also have the line A is true, and B is true. But now, when you see that B is true, and  you also see that B entails C, you've got something you can work with line two is a  conditional. And line four, B is the antecedent of that conditional a number two. And so when  you have a conditional and its antecedent, what do you do you think modus ponens P entails  Q, and P is true. So therefore, Q is true. And you apply that in this case, line four is the  antecedent of line to B as true, and B implies C or entail C. So therefore, C is true by modus  ponens, using lines two, and four. But now you're done. Because you just proved C.  Remember, therefore C was what we were trying to prove. And so you have a completed  proof, starting with those two premises, and now going through a couple of extra steps,  yielding that C is proved to be true. Well, let's try a longer proof. And as I get into the longer  proof, just a word of sympathy and warning, some people find that this is not their thing. They don't have a great knack for abstract ideas or abstract methods of logic. Others find it really  something that they're very good at, when I was studying philosophy, and getting into these  kinds of things. Symbolic Logic was actually the course I did better in than any other course  ever took. I did well in courses, but I was literally perfect in Symbolic Logic, because when you get this stuff, right, you know, it's right, you proved it. And so you just you get it, all right. And for some of you, you might get it all right, or you might get almost nothing, right? Because  this method of arguing is very hard for you. When I first studied symbolic logic, I was also in  the process of having studied some advanced mathematics and doing some computer  science and high level programming courses. And so if you're, if you're somebody who's really wired for math, and for programming, then this kind of logic would come pretty easily to you  have now I'm out of that kind of stuff I've gotten into ministry, and preaching and some of you may be more into ministry, preaching other kinds of things that didn't have a lot to do with  math and computer programming. And so this kind of logic may be a challenge. Just to word  to you. Don't give up. Don't throw up your hands. You've come this far in the course, you don't want to quit now. We'll make the quizzes something that are probably passable. Even if you  struggle with this stuff. Just try to remember it in the big picture. There's a lot of different  ideas of logical out different kinds of fallacies, things of inductive logic now with proofs. In the  total big picture of studying this course, you're going to learn some things that have helped  you to be a better and clearer thinker. So if it's not your thing, don't get discouraged. If it is  your thing. Well enjoy it and take it step by step. Let's look at this longer proof now, with five  premises. first premise not A or B, entails l second premise not B. Third premise A entails B  fourth premise This L entails not R or D, and the fifth premise, not D, and R or F. And the thing to be proved is therefore L or G and not R. So let's just dive into that. And we'll think about  how we can get it done. You've got your five premises, and we're going to start with the  easiest thing you can do in logic. If you have a conjunction, just break it down using  simplification. Premise five is not the end R or F. So let's just say not D, simplification of five,  and R or F. Again, a simplification of line five. Now what do we have? Well, lines two and six  are negated atomic propositions, not B, and not B. And when you see a negated atomic  proposition, then you start thinking, well, we see those sometimes when you're using the rule  of modus tollens, or disjunctive syllogism, that often will give you some results, if you have  the negation of a proposition. So what can we derive? We look at modus tollens P entails Q,  and if you have not Q, therefore, you know, not P. So let's see where that gets us. In this  particular attempt at a proof, we see line two is not B. And we see line three is A entails B. So  by modus tollens, you know, not A, because the consequence of line three is B, but premise  two is not B and therefore not A by modus tollens. Using lines two and three. Well, now we've  got our next step, is there anywhere that not A can be used to help because again, that's A  negated? That's A negated atomic proposition. Is it any use anywhere? Well, we, we noticed  that in line one, not A is in a disjunction with B. And you can always once you know something is true, as we know from line eight that not A is true. Now we can say that not A or B is true 

by addition. Because if not A is true than not A or anything else is true. So therefore we can  say not A or B is true. And we want to say not or not A or B is true, because we see that it's  appearing in line one as a disjunct that would lead to a different conclusion. And so it's  something we can use to come up with other ideas and other conclusions. So we've got nine,  and then we move on to 10. We know that L is true, because in line nine, we said not A or B is true. And since line one says that not A or B entails L. If not A or B, then L, we can say that L  is true by modus ponens. Using lines nine, and one. Well, we're looking at our conclusion for a moment. And our conclusion is a conjunction of L or G. And not R, L or G is a conjunct that we  need in our conclusion. And we know that L is true from line 10. So now by addition, because  L is true, L or anything else is true. So we can say that L or G is true by simple addition using  line 10. So now we have one side of the conjunct that we need for the conclusion L or G has  been proven. Now we need to come up with not R not R is the other conjunct that we need for the conclusion that we're trying to prove. And let's look around for A not R somewhere line  four has A not R in it. And line four has that as part of its consequence, not R or D. So what  can we do with that fact? Well, not R or D is something that we can get at by modus ponens.  Because remember, if we look at line four, it says L entails not R or D, the line 10 says L. So if  L is true, then not R or D has to be true because line four says that l entails not R or D. So now we know by modus ponens from lines four and 10 that not R and D is true. Next step. How do  we show that not R is true? Because remember, in the conclusion, not R is the second part of  the conjuncture that we're trying to prove. How do we show that not R is true? Well let's do it  by disjunctive syllogism, because you notice in line 12 We have not R or D. disjunctive  syllogism says that if you have P or Q, and you know that P is true, if you know not P,  therefore Q is got to be true. So if you have a disjunction, and you know that one side of the  disjunction is a negation, then you know that the other side is true. And when we apply that,  we see that not R or D is true on line 12. But we see on line six, that not D is true, you see  that, so, you have the disjunct not R or D. But if D isn't true, because we have a negation of D being true, then we know that not R is true by disjunctive syllogism using lines six and 12.  Now, what do we have? Well, in line 13, we have not R, which is the second half of the  disjunct we were trying to prove in line 11. We have L or G, which is the first part of the of the  conjunction we were trying to prove. And so, in order to get that conjunction, we just take line 11 L or G. And line 13, not , we can join them using the raised dot. And we have L or G and  not R using the conjunction of lines 11 and 13. And you're done. The lines here that are shown in yellow are the lines that were added using the various valid forms of inference based on  the original premises that we got. And you'll notice that line 14 L or G, and not R is exactly the same as the line that we were originally trying to prove using those five premises that were  given to us. Let me again, say that if you learn this, and if you were to move into a course in  Symbolic Logic, then you would have to develop this much more as you use these eight forms of inference. And as you do a lot of proofs in this course, you're only going to be doing a few  proofs, you're going to be getting the hang of it, I hope a little bit, but we're not concentrating on it hugely. We're trying to learn a lot of different elements of logic in this course. So do your  best in this week of the eight valid forms of inference and constructing proofs. If you survive  great. If this week is a struggle for you. There's other weeks where your quiz scores will help  your final grade even out pretty well. So you're going to have a number of proofs to work on  as exercises. And you'll be given those proofs in advance you won't be under the clock ticking as you have the quiz. There'll be a separate file that's presented with the proofs that you're  supposed to do. You can work on those for a while. And then you'll take the quiz, which will  ask some questions about the proofs that you were working on. So again, this is the thing I  hate to say it because some of you will be very mad at me. This is the thing that I'm better at  than almost anything else in all of academia. And it is the hardest, hardest hardest thing to  teach. Because it's something that a lot of people just struggle with and don't have a knack  for and others just get it. It's okay. If you don't get it and try your best. Keep learning. And  remember what we started this course with one of the talks was, think humbler. If you try to  do proofs, and it's not working, you may be tempted to say, That's so stupid. And the people  who came up with this are so stupid. Well, maybe so but it might be just a little hint that there are some forms of thinking of logic of reasoning that I'm not so good at, or at least so well 

trained in and if nothing else, if I don't get better at logic, just get a little better at humility.  Well, God bless you as you try to learn these things and and God bless you as you seek to  think clearer. Think harder as you become more and more a person who understands and  communicates God's truth.