 It's my first time at this park. I think I'm coming back. Okay, so we've dealt with Aristotle Square of opposition and you know you notice with those we just we had the subject of the predicate as four different kinds of categoricals and we looked at the truth relationships between them. All right, but it's always always always the same subject the same predicate. All right, the next three we're going to kind of mess things around a little bit. The conversion is first of all going to deal with conversion. Conversion of a categorical is when you leave the quantifier loan you leave the copular loan and you just simply take the subject and the predicate and switch places. So the conversion of all you know so we'll let s stand for the subject category and we'll let p stand for the predicate category. So the conversion of all SRP the universal affirmative the conversion of all SRP is all PRS. The conversion of the particular affirmative some SRP is some PRS. The conversion of the universal negative no SRP is no PRS and the conversion of some SRP some s are not p the particular negative is some p are not s. Okay it's time to introduce a new concept logical equivalence. Now earlier we talked about deductive inference as you know pretty much an inference or from premise to conclusion. So a valid deduct deductively valid inference is where the truth of the premises guarantees the truth of the conclusion and that's good. That's good as far as it goes. Logical equivalence is when that inference that valid inference runs from premise to conclusion and from conclusion to premise. The truth of the premise guarantees the truth of the conclusion and the truth of the conclusion guarantees the truth of the premise. Consequently these propositions have the exact same truth values. So we have deductive validity where the truth of the premises guarantees the truth of the conclusion. Logical equivalence is when that validity runs from the premise to the conclusion and from the conclusion to the premise. So looking at one of these conversions no SRP let's try an example no dogs are reptiles well that refers that no reptiles are dogs. So the the categorical claims that the subject predicate is not included excuse me the subject category is not included in the predicate category. They're apart right? Well if no dogs if dogs are not included in the predicate category of reptile well then the predicate category of reptile now the subject is not included in dogs. So that conversion we have a deductively valid inference from from the universal negative to its conversion. Similarly from the conversion we can also infer the first category we look at by the first universal negative that reptiles are not included in the predicate category of dog well that infers that dogs are not included in the predicate category of reptiles okay. So these two propositions invalidly infer each other they are logical equivalence. Look at the particular affirmative some SRP so some dog as a mammal will validly infers as some mammal as a dog I mean this seems kind of silly even saying it out loud but this is what's going on right that there's at least some members of dog included in the predicate category whether that infers at that at least some of the predicate category has at least one member that the dog right and that conversion also validly infers the first particular affirmative that we looked at. So these two propositions validly infer one another they are logically equivalent. Well let's look at the universal affirmative all dogs are mammals that is true that all dogs are mammals the conversion of the universal affirmative all mammals are dogs is false so the conversion of the universal affirmative is not deductively valid right we do not get to automatically infer the conversion of the universal affirmative the particular negative right is also not a valid the conversion of the particular negative is also not a valid conversion right so some mammals are not dogs that's true right some mammals are not dogs namely cats but the conversion that some dogs are not mammals that's false okay so that's conversion conversion is pretty simple you switch places with the subject in the predicate only the universal negative and only the particular affirmative are valid conversions right the other two are not valid and to show that they're not valid we provide a counter example and a counter example is using the same argument form where you have a true premise or premises and a false conclusion and we show that with the universal affirmative all dogs are mammals that's true to all mammals are dogs that's false the premise is true the conclusion is false that means it's not deductively valid the truth the premises does not guarantee the truth of the conclusion same thing with the particular negative some mammals are not dogs that is true the conversion some dogs are not mammals is false that is so that's not deductively valid all right so that's conversion let's look at some more our version is a little bit different from conversion and to explain our version i need to introduce a new concept the concept is the complementary category so the complementary category of a category is everything else besides what's in that category so the complementary category of dog is everything else besides a dog the complementary category of tree is everything else besides a tree and the way we usually express this in english anyway is to put you know one of the suffixes before that just kind of you know says not this so the complementary category of dogs is all non-dogs the complementary category of trees is all non-trees now with our version we're going to use this complementary category with our version you take the categorical you leave the quantifier alone you leave the subject alone leave the quantifier alone leave the subject alone you take the copula if the copula is an affirmative turn it into a negative if the copula is a negative turn it into an affirmative so if it's an is turn it into an is not if it's an is not turn it into it is if it's an r turn it into an are not if it's an are not turn it into an are more. Then you replace the predicate category with its complement category. So, practice with the universal affirmative. All dogs are mammals. The odd version of this is all dogs are not non-mammals. All dogs are not non-mammals. Take the particular affirmative. Some dogs are mammals. Its odd version is some dogs are not non-mammals. Take the universal negative. No dogs are reptiles. No dogs are reptiles. So, we're leaving the quantifier alone. We're changing the copula. Now, don't get confused here. We're changing it from the universal negative to the universal affirmative. So, no dogs are reptiles is now all dogs are non-reptiles. All dogs are non-reptiles. Particular negative. Some dogs are not reptiles. Turns into some dogs are non-reptiles. Some dogs are non-reptiles. Okay. Every aversion is deductively valid. Every aversion is deductively valid. Now, if you'd like to try to just prove that, you're welcome to give it a shot. If you can come up with a counter example for an aversion, I'll be impressed. So, an aversion. Leave the quantifier alone. Leave the subject alone. Reverse, right? Turn the copula into its opposite. If it's an affirmative, turn it to a negative. It's a negative turned to affirmative. And then use the complement category for the predicate. Okay. One left. Okay. Last one. Contra positive. For a contra positive, it sounds a little odd. For a contra positive, you leave the quantifier alone. You leave the copula alone. You replace the subject category with a complementary predicate category. You replace the predicate category with the complement subject category. So, all dogs are mammals. All dogs are mammals. Okay. That contra positive is all non-mammals are non-dogs. All non-mammals are non-dogs. So, let's see which of these categoricals are equivalent to their contra positive. Let's start with the universal affirmative, all SRP. So, for example, let's try all dogs are mammals. What does that, does that validly infer all non-mammals are non-dogs? Well, all dogs are, what all, what all dogs are mammals are saying is everything inside the subject category of dog is also included in the subject category of mammal. Well, then what that, that does mean that everything outside the subject category of, of mammal. Okay. So, we have the, excuse me, the complement category. Everything outside the complement category, everything in the complement category of, of mammal is also in the complement category of dog. So, everything besides a mammal is also outside of dog. Okay. So, that, that valid, that reference does work. Well, then let's try it the other way. So, everything outside of mammal, other than mammal is also outside of dog. Well, then that does mean, right, that everything that, that is a dog is also inside of mammal. So, by saying everything that's not a mammal is also not a dog, that implies that everything that is a dog is a mammal. So, these two propositions do validly infer one another. The universal affirmative is logically equivalent to its contrapositive. Okay. Well, let's try the particular negative. So, some dog is not a reptile. Well, that infers that there's something that's in the complement category of reptile, but that is not included in the complement category of dog. So, it's something other than a reptile that is excluded from the complement of dog. So, yes, some dog is not a reptile that doesn't fur that some non-reptile is not a non-dog. Okay. Well, then let's try it the other way. Some non-reptile, so there's something, you know, other than a reptile is not a non-dog. So, it's excluded from the complement category dog, or in other words, that's in the complement category, is that in the category of dog. To say that something is excluded from the complement category means that it's included in that category. Well, then that means that some dog is not a reptile. So, for the particular negative, it is logically equivalent to its contrapositive. Okay. The two propositions validly infer one another. Well, that's two categoricals. Let's look at the other two. Let's take a look at no dogs or reptiles. Well, the contrapositive for no dogs or reptiles is no non-reptiles or non-dogs. So, in other words, there's nothing out, you know, there's nothing in the complement category of reptiles, right, that's also in the complement category of dogs. But, you know, there's lots of things, right, cats. For instance, cats are both non-reptiles and non-dogs. So, already, right away, the contrapositive is not equivalent to the universal negative, or I should say the universal negative is not equivalent to its contrapositive since the proposition doesn't infer from the universal negative to the contrapositive. Let's look at the universal, excuse me, particular affirmative. Now, so we have, let's try this one. Some mammal is a dog, does not infer that some dog is a non-mammal. Well, now, that, you know, kind of looks like it might. But, you know, you might just have to take my word on this because what this is saying, right, is that, you know, since there's something that is a dog and a mammal, there's something that is not, that is neither a dog nor a mammal, and that, that kind of makes sense. But, if you get your categories broad enough, right, you're going to start inventing whole new categories of things. If you're talking about something that is in the two broadest categories, then you might have a third object that's just completely outside both. And, you know, regardless, you're just simply saying, well, there's something that is, you know, a dog and a mammal, therefore, there's something that's neither, I'm not going to tell you what it is, it's just out there. Okay, that's not real helpful. So, you know, for at least for now, just take my word for it that some dog is a mammal, does not infer that some non-dog is a non-mammal. What about the other way, some non-dog is a non-mammal, does not infer that some dog is a mammal? Well, now that looks like it really does, because it's obviously some dog is a mammal. But, remember, what we're looking for is a count, for example, we're trying to figure out whether this form of inference is invalid. And this form of inference here is going from a contra-positive of a particular affirmative to the particular affirmative. Well, here's a contra-positive. Some non-truck is a non-shark. That's true. There's things out there that are neither trucks nor sharks, but that does not mean we get to infer that some shark is a truck. So, at the very least, at the very least, the contra-positive of a particular affirmative does not infer the particular affirmative, so these two propositions do not validly infer one another, so they are not equivalent. Okay, there we go. For the homework, you'll be expected to not only identify which inferences we make here, but you'll be required to identify valid inferences versus invalid.