Video Transcript: Material Equivalence

Hi, I'm David Feddes and this logic talk is about material equivalence as well as how we use  the word unless. So let's start with that word unless it's not always the easiest word to  translate or understand right away. We understand basically what it means, of course, but  how to put it into logical language, you will perish unless you repent. That could also be said,  unless you repent, you will perish that has two atomic propositions, you will perish will use the constant P and you repent is the constant R if you use those and basically translate unless as  if you don't, that's what we mean by unless, when it said you will perish unless you repent, we mean, you will perish if you don't repent. And the way to say that in symbolic language should be cute. And by that little phrase, If you don't, that's a negation. So the way to symbolize that then would be negation, R horseshoe P. If you don't repent, then you will perish. And that's  the way that you express unless, so the negation of our the negation of repenting is a  sufficient condition for perishing. Now, that's a mouthful of logical language to say that you  had better repent. And so we need to think about that word unless, and realize that it means  if you don't, unless also kind of means if it's not the case, just ways of negating. So if you  don't perish, you will repent till they are horse up. Now, you can also substitute, simply  substitute the word or for unless and you get the logical equivalent on last equals, or you  repent, or you perish, or as Jesus put it, repent or perish. So, R, or P is the same as not R  implies P, or to use the words of logic equivalent, not R implies P is equivalent to R or P. And  that can be shown to be materially equivalent propositions are materially equivalent if and  only if they have the same truth value for every assignment of truth values to the atomic  propositions. That is they have the same truth values on every row of a truth table. So, if you  have two different statements or propositions, and then you take all the possibilities for the  atomic propositions, then you'll find that those two statements always have the same truth  value. If they do, then they are called materially equivalent. I'll show you a little bit of what I  mean, you perish unless you repent. Or if you want to rephrase it, if you don't repent, you  perish. We translated that as tilde R horseshoe P, you repent, or you perish, R wedge P. So we  construct our truth table, the reference columns are P and R, we give all possible truth values, and combinations for P and R. And then we go to the column for not R entails P. First of all,  we've got to figure out what not R is. That's easy enough, just switch it from negate the value. So where R is true, not R is false, where R is false, not R is true, you go down the truth table  that way, and you have false true false true. Now the next step is to figure out what it means  once you add in the horseshoe P. And remember, when we're doing conditionals, that the only time a conditional is treated as false is when the antecedent is true, and the conclusion is  false. So, in the first one, not R is false. So we have to treat the conditional is true. In the  second one, we find that not R is true, but we find that P is also true and that means the  conditional is true. In the third one not R is false, but anytime not R is anytime the antecedent is false, it means the conditional is still treated as true. In the fourth row, you see that not R  was true, and that P is false. That means not are the antecedent is true in that case, and P the result the consequence is false. And whenever you have a true antecedent and a false  consequent T, that means that that the total conditional is treated as false. So the first three  rows are true the last condition, the last row, that conditional is false, then the R or P We've  done that a few times by now should be getting easier. If either one of the two is true, then R  or P is true. So in the first three rows of the truth table, you find that R or P is true, because in  the first row, they're both true in the second and third row. One of them's true. So that makes  the conditional true and only in the fourth row is the conditional false because both P and R  are false. Now, if you look and compare those two columns, the columns for not R horseshoe  P, and R or P, you find that they have identical truth values. Whenever P and R have a certain  truth value, then those two statements are exactly the same. They're false. When each other  is false. They're true when each other is true. And that's what it means to be materially  conditional. And that's why we can say on the basis of symbolic logic, if you don't repent, you  perish means the exact same thing as you repent or you perish. There's no other options. So  you know that if somebody doesn't repent, they will perish. unless Jesus was lying, or the laws of logic mean nothing. You even can get a truth table to show that those two statements are  equivalent. If you don't repent, you will perish, repent or perish. Well, neither nor is another  area where we can talk about material equivalence and expressing it in more than one way. P 

and Q are often used in logic. So I thought, well, I'll use a neither nor neither peace nor quiet  is here. So we'll use peace here and have constant P for that quiet is here, we'll have Q for  that. And you can translate that as a conjunction of two negations not P and not to not peace  and not quiet. You can also translate that as a negation of a disjunction, not peace or quiet.  not P orQq. So those two statements, not P and not Q are equivalent with not P or Q will work  out in a truth table. Again, will at peace here B P quiet as here B Q, make our reference  columns for P and Q, including all possible combinations of values, and then work our way  into the not P and not Q. Not P is simple enough, look at P and do the opposite. So where P is  true, not P is false, where P is false on the bottom two rows, then not P is true. Now we look  over at not Q and again, it's very simple where Q is true, not Q is false, where Q is false, not Q is true. And we have our column for not Q. Now with a conjunction, remember the rules for  conjunction. If either item is false, then the conjunction is false. So you look at the top row,  both items are false, so the conjunction is false second row, one item is false. So the  conjunction is false third row, one item is false, so the conjunction is false. And the fourth row, both items are true, not P is true, not Q is true, and therefore their conjunction is true. Now  we move to the next column, and we figure out what P or Q is, and if both P and Q are true,  therefore P or Q is true. Next, next two rows if one of them is true, then P or Q is true. And  only when both P and Q are false is P or Q false. So now we know what P or Q is, except now  we got to negate it. And that changes everything at what was true now becomes false. So not P and Q has these truth values false false, false true. And when you compare that with not P  and not Q you'll see that on the first row false and false second row false and false third row  false and false for throw true and true. For every possible combination of P and Q truth  values, you find that those two statements have the exact same truth value, and that means  they are materially equivalent. So it's materially equivalent to say, neither peace nor quiet is  here. You're saying peace is not here. Quiet is not here. Or you could say it. Not peace or quiet is here. So material equivalence can sometimes by people in logic be used as a shift truth  functional connected, we're not going to use it much in this course, we're not going to use this symbol, but we're just providing it so you know what it is? It's sometimes called the  biconditional and the biconditional or material equivalence is True when the atomic  propositions share the same truth value, so when P and Q always have the same truth value  in every circumstance, then P and Q are materially equivalent. And if the truth values ever  differ on P or Q, then they are not equivalent. So material equivalence is false when the truth  values are different, and the symbol for material equivalence is the tie bar. And the truth table for that tie bar which can also be expressed as if and only if, is such that the two values have  to match. So, where they're both true, then the tie bar is true. They're equivalent, where  they're different. They are not equivalent, so, it'd be false to say that they're equivalent. And  the same is true of the third row where P is false and Q is true. Again, obviously, those are not equivalent statements. Satan might want to tell you that false and true are equivalent. But  false and true aren't equivalent statements. And so in those middle two rows, you say that  they're both false. And then in the fourth, P and Q are both false. So again, they're equivalent. So when you have a truth table for that tribe, or or equivalent, then when the statements are  identical, it means that they're equivalent, but when they have different truth values, they are not equivalent statements.