6.1: The Basics
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Categorical logic was devised by Aristotle (384-322 BCE) and developed throughout Western history all the way up until the 19^{th} Century. This was the dominant system really up until the early 20^{th} Century.
Categorical logic concerns the relations between Categorical Propositions, which are propositions that relate categories of things.
For example:
All cats have fur
Is a false categorical proposition, but is a categorical proposition nonetheless.
Some cats are things without fur
Is a true categorical proposition, since Sphynx Cats don’t have fur (or at least have no fur coat).
Each of these relates the category cats with some other category like things with fur or things without fur. Some of these relations obtain in the real world (are true or real relations) and some of them do not obtain (they are false or fake or illusory relations). Cats, it turns out, don’t all have fur and so the relation of “all ____ are ____” doesn’t hold between cats and things with fur.
Standard Form
In Categorical Logic, we’re trying to get propositions into Standard Form so we can analyze all categorical propositions using the same basic set of tools of analysis. Standard form is a tool we use. When I express propositions using standard form, I enable myself to understand that proposition and its relation to other propositions more clearly.
Standard form consists of four elements:
1) Quantifiers: “All,” “No,” or “Some”
2) Subject Term: Noun or Noun Phrase
3) Copula: “are” or “are not”
4) Predicate Term: Noun or Noun Phrase
So, for example, any given categorical proposition will look like one of the following:
A: All birds are things that have wings
E: No chickens are things that can swim
I: Some bracelets are very expensive things
O: Some people named ‘Andrew’ are not dogs
The Categorical Propositions relate classes in different ways:
A) Total inclusion: the subject category (the first) is totally included in the predicate category (the second). Meaning: all members of the subject category are also members of the predicate category.
E) Total exclusion: the subject category and the predicate category don’t share any members in common.
I) Partial inclusion: at least one member of the subject category is included in the predicate category.
O) Partial exclusion: at least one member of the subject category is not in the predicate category.
Euler Diagrams:
An Euler diagram is a diagram that uses bubbles to represent categories and their relations to one another and to individuals. There’s no standard “empty” diagram, so the way we draw the bubbles is how we represent these relations. In contrast, we’ll see that Venn diagrams have a blank diagram that is always the same and then we use shading and X’s to represent the relationships between the categories.
Quality and Quantity
Categorical Propositions have quality and quantity.
Quality: are they affirmative or negative? Do they posit inclusion or deny inclusion?
Quantity: are they about all things in the subject category? Are they “total” inclusion or exclusion relations? Or are they about one or more things in the subject category? Are they “partial” inclusion or exclusion.
By answering these two questions (what is the quantity? and what is the quality?), we find out what sort of proposition we’re talking about:
Table \(\PageIndex{1}\)
(Quantity) | Affirmative | Negative | |
---|---|---|---|
(Quality ) | |||
Universal | A: All Humans are Mortal | E: No Humans are Mortal | |
Particular | I: Some Human is Mortal | O: Some Human is not Mortal |
This table looks just like what is called the Square of Opposition.
Square of Opposition
The Square of Opposition tells us about the different relations between propositions with the same subjects and predicates translated into different forms.
A and E, as we can see, are Contraries, meaning they can’t both be true.
FOR EXAMPLE, IT CAN’T BE THE CASE THAT ALL TEA CUPS ARE FRAGILE AND ALSO BE THE CASE THAT NO TEA CUPS ARE FRAGILE. IT COULD BE THAT SOME ARE FRAGILE AND SOME ARE NOT, SO THEY COULD BOTH BE FALSE. BUT THEY CAN’T BOTH BE TRUE.
A and O are Contradictories, meaning that they can neither be both false, nor can they both be true. One must be true and the other must be false. This is the purest opposition: they mean the exact opposite thing from one another.
E and I are also Contradictories.
FOR EXAMPLE, IT CAN’T BE THE CASE THAT ALL HUMANS ARE MORTAL, IF ONE HUMAN IS IMMORTAL (I.E. NOT MORTAL). SIMILARLY, IT CAN’T BE THE CASE THAT NO CHICKENS CAN SWIM, IF ONE CHICKEN CAN SWIM.
EITHER AT LEAST ONE HUMAN IS NOT MORTAL OR ALL HUMANS ARE MORTAL—THEY CANNOT BOTH BE TRUE AND THEY CANNOT BOTH BE FALSE.
EITHER AT LEAST ONE CHICKEN CAN SWIM OR NO CHICKENS CAN SWIM—THEY CANNOT BOTH BE TRUE AND THEY CANNOT BOTH BE FALSE.
WHY? WELL, WHAT IT MEANS FOR IT TO BE FALSE THAT ALL HUMANS ARE MORTAL IS SIMPLY FOR THERE TO BE SOME HUMAN THAT IS NOT MORTAL. SIMILARLY, WHAT IT MEANS FOR IT TO BE FALSE THAT NO CHICKENS CAN SWIM IS SIMPLY FOR THERE TO BE A CHICKEN THAT CAN SWIM.
I and O, as we can see, are Subcontraries, meaning they can’t both be false.
FOR EXAMPLE, IT CAN’T BE FALSE THAT SOME PUPPIES ARE RAMBUNCTIOUS AND ALSO FALSE SOME PUPPIES ARE NOT RAMBUNCTIOUS.
TO SEE THIS, THINK ABOUT THE CONTRADICTORIES OF EACH. IF IT’S FALSE THAT I AND IT’S FALSE THAT O, THEN THE CONTRADICTORIES MUST BE TRUE. IN THIS CASE, THAT MEANS THAT ALL PUPPIES ARE RAMBUNCTIOUS AND ALL PUPPIES ARE NOT RAMBUNCTIOUS. THAT IS CLEARLY NONSENSE!
Subalternation is strange, but we’ll go over it quite quickly. If the particular proposition is false, then the corresponding universal proposition is false. If the Universal proposition is true, then it follows that the corresponding particular proposition is true.
FOR EXAMPLE, IF ALL LEGO PIECES ARE RED, THEN IT FOLLOWS THAT THERE IS AT LEAST ONE RED LEGO PIECE (OBVIOUSLY, RIGHT?)
IF THERE ARE NO NEON GREEN LEGO PIECES, THEN IT FOLLOWS THAT AT LEAST ONE LEGO PIECE IS NOT NEON GREEN (LESS OBVIOUS, BUT STILL PRETTY CLEAR).
GOING THE OTHER WAY...
IF IT’S FALSE THAT SOME DAISY IS PURPLE, THEN IT MUST BE FALSE THAT ALL DAISIES ARE PURPLE.
IF IT’S FALSE THAT AT LEAST ONE ROBOT IS NOT A CONSCIOUS BEING, THEN IT MUST BE FALSE THAT NO ROBOTS ARE CONSCIOUS BEINGS (AFTER ALL, AT LEAST ONE ROBOT IS A CONSCIOUS BEING!)