Lecture Notes by Christopher Lay
Los Angeles Pierce College
Department of History, Philosophy, and Sociology
Mathew Van Cleave's 2016 Introduction to Logic and Critical Thinking
https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=457
Chapter 2 "Formal Methods of Evaluating Arguments," § 14 "Categorical Logic"
"In this section we [learn] what categorical statement are, how to translate categorical statements into one of the four categorical forms, and how to construct Venn diagrams for each of the four categorical forms."
Categorical Logic
"Categorical logic is the logic that deals with the logical relationship between categorical statements."
Categorical Statements
"A categorical statement is simply a statement about a category or type of thing."
Consider this argument:
1) All humans are mortal.
2) All mortal things die.
3) Therefore, all humans die.
"[T]he first premise of the above argument is a statement about the categories of humans and things that are mortal."
"The second premise is a statement about the categories of things that are mortal and things that die."
"Finally, the conclusion is a statement about humans and things that die."
Logical Terms in Categorical Logic
"In categorical logic, the logical terms ... are the terms 'all' and 'some.'
Placeholders / Symbols
"In ... categorical logic we will use capital letters to stand for categories of things in the world ... ."
"Thus, we can represent the statement:"
"All humans are mortal as"
"All H are M"
"where 'H' stands for the category of 'humans' and 'M' stands for the category, 'things that are mortal.'"
Nouns and Noun Phrases
"Notice that the categories are nouns or noun phrases."
"Thus, instead of saying that the category is 'mortal' I said the category is 'things that are mortal.'"
"It is important to recognize ... how the capital letters are being used in categorical logic ... ."
"In categorical logic, the capital letters stand for noun phrases that denote categories of things in the world—for example, 'cars' or 'things that are man-made' or 'mammals' or 'things that are red.'"
Venn Diagrams
"In categorical logic, we will use what are called Venn diagrams to represent the logical relationships between the different kinds of categorical statements."
"A Venn diagram is simply a way of graphically representing the logical relationship between two different categorical statements."
E.G. Venn Diagram
All Humans are things that are mortal.
Interpreting Venn Diagrams
"[T]he intersection of those two categories (i.e. the place where the two circles overlap) represents things that are both human and mortal."
"Any shaded portions of the Venn diagram (by 'shaded' I will mean 'blacked out') represent that there is nothing in that area of the category."
"So the above Venn says that there is nothing in the category 'humans' that is not also in the category 'things that are mortal.'"
"[T]he reason the category 'things that are mortal' is left unshaded is that in saying 'all humans are mortal' I leave open the possibility that there are things that are not human and yet mortal."
There could be (and indeed are), things that are mortal that are not human, like honey badgers.
The Four Categorical Forms' Placeholders / Symbols
"The way we will represent [the four, and only four,] categorical forms generally are with an 'S' (which stands for 'subject term') and a 'P' (which stands for 'predicate term')."
E.G. Categorical Form
"Thus, the categorical statement, 'all humans are mortal,' has the following categorical form: All S are P"
Universal Affirmative Statement Forms
The Universal Affirmative statement form is All S are P.
"This statement form is what we call a 'universal affirmative,' since it is a universal statement that does not contain a negation."
"The way we interpret statements of this form are as follows: everything in the category S is also in the category P."
The Four Categorical Statement Forms
"[A]ny categorical statement can be translated into one of these four forms."
All S are P (Universal Affirmative)
No S are P (Universal Negative)
Some S are P (Particular Affirmative)
Some S are Not P (Particular Negative)
Let's go over the last three in turn (as we've already gone over the first one).
Universal Negative Statement Forms
The Universal Negative statement form is No S are P.
"No reptiles give live birth."
"This categorical statement refers to two different categories: the category of 'reptiles' and the category of 'things that give live birth.'
(We've added "'things that...' to the predicate of the sentence ('give live birth') because 'give live birth' is not a description of a category. Rather, the way of describing the category is with the noun phrase, 'things that give live birth.'")
So, we get:
"No reptiles are things that give live birth."
"[T]here is nothing in the intersection of the two categories, 'reptiles' and 'things that give live birth.'"
"If you think about it, this is exactly what our original statement was saying: there isn’t anything that is both a reptile and gives live birth."
Particular Affirmative Statement Forms
The Particular Affirmative statement form is Some S are P.
Consider: "'[S]ome birds are taller than President Obama.'"
Translated, we get:
"Some birds are things that are taller than President Obama."
("By convention, an asterisk on the Venn diagram means that there is at least one thing in that category. By putting the asterisk in the intersection of the two categories, we are saying that there is at least one thing that is a bird and is taller than President Obama, which is exactly what our original sentence was saying.")
Particular Negative Statement Forms
The Particular Negative statement form is Some S are Not P.
Consider: "some birds don't fly."
"[W]e have to be ... careful with the 'P' term, since its predicate [seems to contain] a negation."
"We do not want any of our categories to contain a negation." (Do not forget this point.)
"[T]he negation is contained in the form." (Do not forget this point.)
Translated, we get:
"Some birds are not things that fly."
("By convention, an asterisk on the Venn diagram means that there is at least one thing in that category. By putting the asterisk inside the 'birds' category, but outside the 'things that fly' category, we are representing that at least one thing that is a bird isn’t a thing that flies. This is exactly what our original sentence was saying.")
On Translating into Categorical Statement Forms
"Translating categorical statements into their categorical form can be tricky."
"[I]t is probably one of the trickier things you’ll do in formal logic."
"There is no simple way of doing it other than asking yourself whether your translation accurately captures the meaning of the original English sentence."
E.G. "Tricky" Categorical Statement
"Nobody loves me but my mother."
"The first step is to ask what two categories are being referred to in this sentence."
We get:
"'things that love me'"
'"things that are my mother.'"
("Notice that the category couldn’t just be 'my mother' since that isn’t a category; it’s a particular thing.")
"The next question is: what is this sentence saying is the relationship between these two categories?"
"The sentence is saying that the only things that love me are things that are my mother."
"The categorical form of the statement is the “all S are P” form."
So, we get:
"All things that love me are things that are my mother."
Another E.G. "Tricky" Categorical Statement
"The baboon is a fearsome beast."
"Although the article 'the,' which often denotes particulars, may lead one to think that this is a particular affirmative form (some S are P), it is actually a universal affirmative form (all S are P)."
("This English sentence has the sense of 'baboons are fearsome beasts' rather than of 'that (particular) baboon is a fearsome beast.' English is strange, which is what makes translation one of the trickiest parts of logic.")
So, we get:
"All baboons are fearsome beasts."