Video Transcript: Tautologies, contradictions and contingent statements

I'm David Feddes. And this logic talk is about tautologies. contradictions, and contingent  statements. And again, I'm using material from Dr. Matthew Van Cleave a tautology is a  statement. That's true in virtue of its farm. It's always true, it can't be false. It's truth doesn't  depend on what the facts actually are. Here's an example. Matt is either 40 years old or not  40 years old. He's one or the other. And that poll statement cannot be false because if he's 40 years old, it's true. If he's not 40 year old, 40 years old, it's true. So the a statement of the  form P OR NOT P is a tautology, it is always true because if one side of the disjunction is false, the other is always true. So the total disjunction is always true. A tautology is simply a  statement that cannot be false. Here are some examples God exists or God does not exist.  That statement Taken as a whole has to be true, no matter what the facts are. Jesus is God or  Jesus is not God, Jesus sinned, or Jesus did not sin, unrepentant sinners go to hell, or they do  not go to hell. Those are tautologies, where it must be true, taken as a whole, because it's  giving you a disjunction where one side is true, and the other has to be false. So tautologies  are statements that are always true just based on their form, regardless of what the facts are. A contradiction is a statement that's false. In virtue of its form, it's always false. It cannot  possibly be true. It's truth doesn't depend on what the facts actually are. So for instance, you  could say Matt is both 40 years old, and not 40 years old. Well, he can't be both, it's logically  impossible. A conjunction of two opposites is a contradiction. P and not P is always false. It is  absolutely impossible for P and not P, to be a true statement. You cannot assert a  contradiction, and be telling the truth. God exists, and God does not exist. It's impossible for  that statement to be true. Jesus is God and Jesus is not God. Jesus sinned, and Jesus did not  sin. unrepentant sinners go to hell, and they do not go to hell. Those statements are all  contradictions, saying two opposite things in the same statement. And there is no possible  world in which any of those statements could be true. A contingent statement is a statement  whose truth depends on the way the world actually is. So it could be either true or false. It just depends on what the facts actually are. Take the statement, Matt is either 39 years old, or 40  years old, if he's 39. That statements true. If he's 40. That statements true. If he's 32, that  statement is false. If he's 23, that statement is false. So it's a contingent statement. And it  depends on what the facts actually are not on just the form of the statement. Here are some  examples of contingent statements. People evolved from monkeys or God created people.  Well, that statement depends on what the facts actually are. You might say, well, the  statement Taken as a whole has got to be true, because either people evolved from monkeys  or God created people. But that's not really so there are other possibilities, at least logically to consider. Maybe people evolved from kittens. Maybe God created people through evolving  them from monkeys. So there's our there are contingent statements that depend on the way  the world actually is. And they aren't inherently a contradiction, or a topology. The flood  covered the world, or there was no great flood. Well, maybe the flood covered part of the  world a big part of the world. I'm not arguing for that. Now, to make a case regarding the  biblical flood. I'm just saying that as a logical statement, it is contingent, to say the flood  covered the world or there was no great flood, there may be other possibilities, such as a  really big flood that didn't cover the whole world. Jesus disciples stole his body, or he rose  from the dead. Well, maybe his body was disposed of in some other way that people lost track of. Maybe there's other possibilities that we haven't thought of. But that's a contingent  statement. That depends on the way things really are. It's not inherently false based on its  form. It's not a contradiction. It's not a tautology. It's a statement of fact, and it depends on  what the facts Fact of the matter are in this case, of course, the fact is that he did rise from  the dead. You might say Paul, or Barnabas wrote Hebrews, but that's a contingent statement.  It would be true if Paul wrote it, it would be true if Barnabas wrote it. That statement wouldn't  be true if apologists wrote Hebrews, or as Peter wrote Hebrews, or if somebody else wrote  Hebrews, you see, it's not a contradiction. And it's not a tautology, it depends on who actually did write Hebrews. So contingent statements depend on the facts, not on the form, they're  not a contradiction. They're not a tautology. They're saying something that will be true or  false based on the facts in the world. So we have tautologies that are always true, just due to  their form. There's a contradiction, which is false. Due to its form, it's trying to say two things  at the same time, that are the opposite of each other, and they can't both be true at the same

time. And a contingent statement depends on the way the world actually is. Now, with truth  tables, we can determine whether something's a tautology, or contradiction in a tautology,  every row under the main operator of a statement will be true in its truth table. For a  contradiction, every row under the main operator of the statement will be false. And for a  contingent statement, there will be a mixture of true and false under the main operator of the statement. So let's look at the truth table for a tautology. And then we'll look at the truth  table. For a contradiction, we'll get an example of each A entails B, or A, that's what you see  in the far right column. You could also phrase if A then B, or A, so you have a conditional. And  then taking that conditional in parentheses, you have a disjunction with a so let's first get the  truth values for the conditional. You remember that with a conditional, the only time it's false  is if the antecedent is true, but the consequent is false. So only if A is true, and B is false, is  the conditional false. So we see in the second row, that the conditional is false, and on the  other three rows, it's true. Now let's work out the truth values for the whole statement. The  value under the main operator, which is the disjunction with a disjunction. Remember, if  either side of the disjunction is true, then the disjunction as a whole is considered to be true.  So in the first row, A is true, and the conditional is true. So the disjunction Taken as a whole is  true. In the second row A is true, and the disjunction and the conditional is false. But  remember with a disjunction is if one is true, and the other is false, then the disjunction as a  whole is still true. In the third row. The conditional is true, but A is false. But it doesn't matter  because we're the conditional if either part is true, then the entire disjunction is true. So in  this case, A entails B or A is still true. And in the fourth row, you see that A is false. But you  see that the conditional is true, and therefore the disjunction Taken as a whole is true. You  notice what you've got in that whole column then is true, true, true, true. And that's the sign  of a tautology. If, under the main operator, the truth value of the whole statement is true,  then you have a tautology, because for every value of A and B No matter what value of A or B or combination of them you pick. It always comes out true. That's the sign of a tautology. This  is the truth table for a tautology. Now if you were to consider a contradiction, you're going to  wind up with the main operator having false all the way down and will work out if you just  glanced at this. You wouldn't necessarily think oh, that's obviously a contradiction. If you  thought about it. You might realize it's a contradiction, but we'll work it out in terms of a truth  table. Start with A or B. Remember again with a disjunction if either member of the  disjunction is true, then the disjunction as a whole is true. So in the first row both are true  therefore disjunction is true second row, one is true, the other false so the disjunction Taken  as a whole is still true. Third, row A is false, but B is true. So the disjunction is still true. And  only in that fourth row where both A and B are false is the disjunction Taken as a whole false.  Now we'll move over, we'll look at not a which is very easy to figure out look at the A column  it's true. So not A is false. Second row A is true, not A is therefore a false third row is false. So  not A as true fourth row is false. So not A is true so much for that column. Let's look at not B  Well in the first column B is true, so not B's false, second, Column B is false or second row B is false, and so not B is true, third, row B is true, so not B is false. And then again on that fourth  row, very simple, B is false. So not B is true. Now let's get the conjunction of not A and not B.  And you remember with the conjunction, both members have to be true in order for the  conjunction to be true. So in that first row, both of those things are false, therefore, the  conjunction is false. In the second row, one is false and the other true, so that means the  conjunction Taken as a whole is false. In the third row, one member is true, the other member  is false. Therefore, the conjunction as a whole is false. And in the fourth row, both members of the conjunction are true. Therefore, the conjunction as a whole is true. And now we're not  quite done. Now we have to get the main operator of the whole thing, the main operator that  connects A or B with not A and not B. And when we do that, then we look again at how a  conjunction works. When there's a conjunction, both members have to be true in order for the conjunction to be true. In the first row, one member is true, but the other member is false,  and therefore the conjunction is false. The second row, A or B is true, but not A and not B is  false, and therefore the conjunction of the two statements is false. Third row A or B is true,  but again, you've got a false statement for not A and not B, therefore, the total conjunction is  false. And in the fourth row, you have A or B was false. Not A and not B was true, but that 

doesn't do any good. To make it true because with a conjunction, they both have to be true,  and therefore the conjunction is false. So in that truth table under the main operator, it's false false, false false, and that is the sure sign of a contradiction. No matter what value has for A  and B, the statement Taken as a whole is false. So you know that it is a contradiction.