Video Transcript: Categorical Logic

Hi, I'm David Feddes and this talk is about categorical logic. categorical logic involves  categorical statements and a categorical statement is a statement about a category or a type of thing. And categorical logic is the logic that deals with the logical relationships between  categorical statements. So you have statements made about various categories or kinds of  things. And categorical logic describes the relationship between those categories. And  categorical logic doesn't exactly fit with propositional logic, the kind of logic that we've been  looking at in some previous talks of propositional logic isn't adequate to express an argument  like this. All humans are mortal. All mortal things die. Therefore all humans die. If you use just  propositional logic and its methods for that, it can't show that. But if you look at it, you just  intuitively know by informal tests of validity that that's a good argument. All humans are  mortal. All mortal things die, therefore all humans die. So when we think about categorical  logic, we are dealing with something that's a little different than what some of the  propositional logic had been, like. categorical logic uses as its two main logical terms, all and  some. And when you're dealing with categorical logic, you use capital letters to stand for  categories of things, not for atomic propositions, as you do in propositional logic. So if you say all humans are mortal, the way of expressing that in categorical logic is all H or M, where H is  things that are human, and M is things that are mortal. So all humans are mortal as expressed as all H or M. Now, when we use categorical logic, a good tool for helping us to understand it  is a Venn diagram. And a Venn diagram uses circles, and uses overlapping circles to show the  relationship between different categories. In the case of the statement that all humans are  mortal, you have two categories, the category humans and the category things that are  mortal. And then you look at the intersection of those two categories. Now, in this Venn  diagram, you'll notice that the category humans is entirely shaded or blacked out, except for  the portion that overlaps with a category, things that are mortal that's meant to show that  anything in the human category that's not mortal doesn't exist, the entire portion of the non  mortal area of humans is blacked out because all humans are mortal. And so the only humans that are exist, will overlap with the category things that are mortal. Now, you'll also notice  that the circle things that are mortal has areas that don't include things that are human. And  of course, there are cats, dogs, bears, lots of kinds of animals that are mortal, that aren't  human. So that part of the category things that are mortal, doesn't have to include humans in order for it to have other things. But the blacked out portion of the category humans means  that there are no humans that don't overlap with the category. Things that are mortal, all  humans are mortal would have this Venn diagram. There are four basic categorical forms,  where and those are expressed using s&p as your constants. When you're talking about the  general rules. S is the subject term for the subject category. P is the predicate term for the  predicate category. And so you got four of these all S are P. That's a universal Affirmative, no S are P. That's a universal negative. Some S are P is a particular Affirmative, and some S or not  P is a particular negative. Here are examples of each of those. All humans are mortal all S are  P that's the that's the form. No, reptiles give live birth. No reptiles are among the things that  give live birth. Some birds are taller than six feet. So that takes the form some S are P where  the category P is things that are taller than six feet. And then finally, some birds don't fly or to take the more classic form some S are Not in the category birds that don't fly, some S are not  P. So you have those four categorical forms. And when you're dealing with the English  language, you need to translate the categorical and when you do that, you need to translate  so that each term is a category and that each term is a noun in particular. So if you take the  statement, all humans are mortal. Mortal is actually an adjective. So you have to translate  that to be a noun. All humans are things that are mortal. That's a category that's a noun  category, things that are mortal. So you translate No, reptiles give live birth as No, reptiles  are things that give live birth. Because again, getting live birth is not a noun. So you use the  word things and you're using it to describe a category again, a proper name would not be a  category. So you're using a noun and it's a cat, a noun that describes a category of things.  That's what you're doing when you're translating into categorical logic. Now, let's look again  at the Venn diagrams. The Venn diagram for all S are P, a universal affirmative is something  that we've already looked at all humans are mortal. So if you have them of all S are P, then  you black out the part of us that's not included in P. And you have the entire circle of P is 

open, because you're not saying anything that all P has to be S you're just saying all S are P  and therefore, the P has to the P That's not blacked out, or I mean, the S That's not blacked  out has to overlap with P That's how you use them for all SRP. The category of humans is  blacked out except for the portion of it that overlaps with the category things that are mortal.  How do we do a Venn diagrams for no S are P? Well, where the two circles of the categories  overlap? You're black that out? That means that there is no overlap between the two  categories? No S are P, no reptiles give live birth. So if you have the category, reptiles, and  the category things that give live birth, there is no overlap between them. And therefore, you  black out the overlapping part of the Venn diagram, anything that's in the category reptile  does not at all overlap with things that give live birth because reptiles lay eggs, they don't  give live birth. Some S are P. How do we represent that in a Venn diagram? How do we  represent a particular Affirmative? An example of such a statement would be some birds are  taller than President Obama? Well, to show that you don't have anything that's completely  blacked out, because you have the word some. And some means that no part of it is  completely shaded. All you're saying is that some things that are in the category S are also in  the category P, some birds, some things that are birds are also things that are taller than  President Obama, perhaps a large ostrich. And so in the area of the overlap of the two circles,  for some S are P, you put an asterisk and that little asterisk represents the fact that at least  one item, and maybe more than one, but at least one item is in both categories. The  categories overlap and have something in the overlapping area. Some s are not P is the  particular negative. And an example of that is some birds don't fly. So if you have two  categories, birds, and things that don't fly, and you draw the Venn diagram, you're not saying  anything about the overlap, it's possible that some birds do fly. It's possible there are things  that fly that aren't birds, such as bats. But what we are saying with this Venn diagram, is that  you know, the fact that there are some birds that don't fly, so there's at least one bird that  can't fly. And so you put an asterisk in the area of the category birds that doesn't overlap with things that fly. Now, if you wanted to say all birds fly, you'd have a different event. But if  you're saying some birds don't fly, some birds are not in the category of things that fly. Then  you put the asterisk in the category birds but not in the area that overlaps with things that fly. That's basically the four kinds of Venn diagrams that you use for these four categorical forms.  All S are P is a universal Affirmative, no S are P is a new versal negative, some S are P is a  particular Affirmative, some S are not P is a particular negative. And those are four major  categorical forms that we use in categorical logic. We'll get more in Venn diagrams and future  talks and future exercises. But an important part of categorical logic is simply translating in  the first place into nouns and into categories. Here's a statement. Nobody loves me but my  mother. Now, what are the two categories? In this statement, nobody loves me. But my  mother, well, things that loves me is one category. And things that are my mother is another  category, it sounds a little strange to say things that are my mother. But remember, we need  to translate into categorical logic and a noun, well, my mother's a noun, but not a category.  It's not an entire category. It's a specific thing or person. And so my mother isn't a category.  And in order to make mother, my mother a category, we call it things that are my mother. So  you've got two categories, things that love me, and things that are my mother. What's the  relationship between those two categories? The relationship is all S are P. All things that loves  me are things that are my mother. Now, that's a very strange way of saying it. But when  you're trying to state things in terms of categorical logic, and what fits in one category and  what doesn't in another, and you're using these statements, all S are P or no S are P or some  S are P or some S are not P, then you have to be able to translate into this kind of awkward  language. So you come up with a simple statement. Nobody loves me, but my mother being  turned into the categorical logic statement. All things that love me are things that are my  mother. Hey, that sounds almost better, doesn't it? No, but who loves me, but my mother  sounds so sad. Oh, things that love me are things that are my mother. sounds so much better. Well translate carefully. The baboon is a fearsome beast. Now, if you think of the baboon it  sounds like it's just a noun speaking of one baboon, but the English language is kind of a  funny thing. When you say the baboon what you really mean is, baboons in general, are  fearsome beasts in general. So what are the two categories, baboons and fearsome beasts? 

And what's the relationship between the two categories? If you say that baboon is a fearsome  beast, you're saying, All S are P. All baboons are fearsome beasts. So when you're doing  categorical logic, one of the first things is to translate an English sentence and make sure that the categories are indeed nouns that are categorical nouns where you're talking about a  category of things. And once you have done that, then you can get more into the various  operations. Some are some aren't all are all are not the Venn diagrams, all of that. So we're  going to have some practice translating as an exercise, and then we'll get more into Venn  diagrams and more of the operations of categorical logic.