Video Transcript: Conditionals

Hi, I'm David Feddes and this logic talk is about conditionals. conditionals are the fourth of the truth functional connectives. Thus far we've looked at conjunction, negation disjunction. And  now we get to the conditional, which is symbolized by the horseshoe and you often say P  entails Q or if P, then Q, we'll get into more detail here in a moment. The truth table for  conjunction going back through these various connectives looks like this, a conjunction is true only. If both of the conjuncts are true, otherwise, the conjunction is false. The truth table for  negation is very simple. If the proposition is true, then its negation is false. If it's false, then  the negation is true for a disjunction, the disjunction is true if either of the atomic propositions is false. So if they're both true, it's true. If either one is true, then the conjunct is true. And it's  false only if both are false. So we're looking at not just conjunction or negation, or disjunction  now, but at the conditional. And conditional is If then, for instance, here's a conditional  statement. If it is raining, then the ground is wet. You break that down into its atomic  propositions and you get it is raining R and the ground is wet. W you symbolize that by R  horseshoe, W Don't you love that horseshoe those people in logic, they're such exciting  creative people R horseshoe W is how you express the conditional if R then W. If it's raining,  that's the antecedent, then the ground is wet. That's the consequence. Sometimes people in  grammar will tell you these things as well in an English course, if it's raining, the antecedent  then the ground is wet. That's the consequence. And so in R horseshoe W, R is the  antecedent, W is the consequence means that whenever R is true, then W has to be true.  Whenever the antecedent is correct, then the W has to be the consequent has to be correct  as well. A conditional is a material conditional, which in the language of logic means that it's  true in every case, except when the antecedent is true, and the consequent is false. And it's  false. Therefore, when the antecedent is true, and the consequent is false, you get that, of  course, don't you? If you said, if something is true, but then it turned out to be false, then  you'd know that the one thing does not imply the other, you can't say that if one is true then  the other is true, if in fact, it turned out to be false. So whenever the consequent is false, but  the antecedent is true, then you know that you don't have the material conditional being true. Here's the truth table for a conditional. If R is true, if it's raining, and W is true, the ground is  wet, then it's true that if R then W if it's raining, then the ground is wet. If R is true, it's  raining, but it's false that the ground is wet. If R if it's raining and the ground is dry, then it is  false, that R horseshoe W, then it's false to say that if it's raining, the ground is wet, because  if you had it pouring rain, and the ground was wet, that statement would be false. What would happen if it wasn't raining? R is false. Would that mean that therefore the conditional is false.  You can have it not raining, and the ground being wet. And it still be true that R implies W or  that R entails W or if R then W if it's false, that it's raining, you still might have wet ground.  Maybe somebody had the sprinkler on. So a dry if it's not raining, and yet the ground is wet,  that doesn't say anything about the proposition whether R entails W and so you counted as  true. If R is false, if it's not raining, and the ground is dry, does that tell you anything about  whether if it's raining the ground as wet? No, if they're both false, it just might mean that it's  a sunny day, the ground is dry and in and the rain is not falling. So both of those things are  false. But it still might be true that if it's raining, the ground is wet. It just doesn't happen to  be raining. So you see what's going on the statement if it's raining The ground is wet is true. If it's raining and the ground is wet, it's false. If it's raining and the ground isn't wet, it's also  true if it's not raining period, because if it's not raining, then you haven't learned anything  about the proposition. If it's raining, then the ground is wet. So the conditional has to be true.  It's counted as true, except when you find the antecedent being true, and yet the conclusion  being false. Another example, if you pass all the exams, that's your antecedent, you will pass  the course consequence. That's nice to know. If you pass everything, then you ought to pass  the course. You symbolize that by E horseshoe C, where E is the antecedent and sees the  consequence? If E, then C. So here's the truth table. If E. If you pass all the exams, that's true, and you pass the course C is true, then E horseshoe C is true, if E then C. Now, yes, however,  you passed all your exams, and you flunked, you'd say what's wrong with that professor, but  you could also say, that statement, if you pass all the exams, you will pass the course is false, because I passed all the exams, and I still didn't pass the course. And my professor is not very fair, would be the other consequence of that. But that's not just logic. Anyway, moving on, if 

you failed some exams, so you didn't pass all the exams, E is false. Does that mean therefore  that and yet it turned out that E passed the course does that mean that if you pass all the  exams, you'll pass the course is false? No, even if you don't pass all the exams, you might still pass the course. So if E is false, it doesn't really say anything about whether E implies C is  true. So you counted as true. Once again, if both are false, if you didn't pass all the exams,  and you flunked the course, well, then E implies C or E horseshoe C, if you pass all the exams, you'll pass the course, would be true. So again, you see that that statement is false. If you  pass all the exams, you will pass the course that's false only if you did pass all the exams and  still didn't pass the course E is true, C is false, then that would mean that E horseshoe C is  false. That's how the truth table for a conditional works. If you still don't have it, sunk in, read  the article, watch the video again, you'll you'll get a sense of why that's the case. Now how do you express conditionals we've seen if then, there's a lot of different ways to say it. It is  raining only if the ground is wet. The ground is wet, if it's raining, only if the ground is wet, is  it raining, that it is raining implies that the ground is wet, that it is raining entails that the  ground is wet. Those are all different ways of saying if it rains, the ground is wet or R  horseshoe W. There's more ways. As long as it is raining, the ground will be wet. So long as it  is raining, the ground will be wet. Whenever it is raining, the ground will be wet, the ground is  wet, provided that it is raining, and our good old fashioned, if it is raining, the ground is wet.  All of those are different ways of saying the same thing that rain entails wet ground, R  horseshoe W. Now, we're talking about a sufficient condition that if something is true than its  consequent has to be true a condition that suffices for something else to obtain or to be the  case is a sufficient condition. You don't need to know anything else. If something is a  sufficient condition. If you know that that condition is true, then you know that its consequent  has to be true. If the antecedent was a sufficient condition. X is a sufficient condition for Y  means that anytime X is true, Y will be true. That's what we mean when we say X horseshoe  Y. beheading is sufficient condition for death. If you're beheaded, you're dead. You don't have  to be shot. You don't have to have any other bad things happen to you. If you're beheaded,  you're dead. It's a sufficient condition. B horseshoe D. birth in the US is sufficient condition for being a US citizen. Doesn't matter who your parents were doesn't matter if they were in the  country illegally or if they weren't citizens. If you are born on US soil under the US  Constitution, you are a citizen of the United States and that means B horseshoe W. It's a  sufficient condition whatever else might be true of you. If you emerged from your mother's  womb on US soil you You have fulfilled the sufficient condition to be a US citizen. Anytime the  antecedent is true, the conclusion is true. That's what it means to have a sufficient condition.  Now a necessary condition is somewhat different. It's a condition that must be present in  order for something else to be the case. So it's necessary for it to be the case. But even if it  were there, it would not prove that the other thing follows X is a necessary condition for Y  means that if X were not present, Y would not be present either. So instead of say X  horseshoe Y, which is what happens when you have a sufficient condition, you could actually  express that as Y horseshoe X because you know that X is a necessary condition. So if you're  told that the consequent Y is true, and it was a necessary condition to have X, well, then if  you're told that Y is true, you know X is true. For instance, having a sibling is a necessary  condition for being a brother. Okay. So being a brother implies or entails that you have a  sibling, if you are a brother, you have a sibling. So but if you are a sibling doesn't necessarily  tie to the other brother. So you see, it's a necessary condition. Being a sibling is a necessary  condition for being a brother. So anytime somebody is known to be a brother, you know they  have a sibling. But if you know they have a sibling, that doesn't mean you know they have a  brother because they might have a sister. So necessary conditions are not the same as  sufficient conditions. Having a sibling is a necessary but not a sufficient condition. For having  a brother, you can't be a brother without having a sibling. But you can be a sibling without  having a brother, being a US citizen, is a necessary condition, but not a sufficient condition to  become president. So if somebody becomes President, you know that they're a US citizen. It's  a necessary condition. So if you heard President Obama is President of the United States, or  President Trump, or President Reagan, or whatever other president you want to talk about,  you know, that they're a US citizen. But if you say somebody, you know, Philip is a US citizen, 

that does not mean he is president of the United States. He has one of the necessary  conditions to be president, but that doesn't make him the president. birth in the US is a  sufficient condition, but not a necessary condition for being a US citizen. So you know, if  somebody is born in the US, they are a US citizen. But on the other hand, if you know  someone is a US citizen, you don't know they were born in the US, because there are people  who emigrate and become naturalized citizens, there's different paths to citizenship. So birth  in the US is sufficient, but not a necessary condition to be a US citizen. Beheading is a  sufficient condition for dying. If you're beheaded, you'll be dead. But it's not a necessary  condition for dying because most people die without being beheaded. So again, P horseshoe  Q is not the same thing. It doesn't have the same meaning as Q horseshoe P. P horseshoe Q,  means that P is a sufficient condition for Q, Q horseshoe P means that P is a necessary  condition for Q. So again, the truth table for conditional is that it's false only if the antecedent  is true, but the consequent is false. And that's it for conditionals. study it carefully. I hope  you're able to really let that sink in because arguing if then statements is one of the most  important aspects of logic