Chapter Three
3.5 Categorical Logic
All of the following statements describe a biconditional relation, for the case in which if you eat your vegetables you will get dessert, and if you don’t eat your vegetables, you won’t.
the relation between the two claims that allows us to infer that “Benny is blue”. There are four main types of categorical statements. We will use “S” to indicate the subject of the proposition, and “P” to indicate the predicate we are attributing to the subject.
• You can have dessert if and only if you eat your vegetables. • You can have dessert exactly if you eat your vegetables.
Universal Affirmative: All S are P.
• You can have dessert precisely if you eat your vegetables.
Example: “All cats are fuzzy.” (S: cats. P: fuzzy things.)
• Your eating your vegetables is a necessary and sufficient Universal Negative: No S are P
condition for your getting dessert. • You can have your dessert, if you eat your vegetables, but only if you eat them.
Example: “No dogs are ten feet tall.” (S: dogs. P: things that are ten feet tall.)
• You can have your dessert just in case you eat your Particular Affirmative: Some S are P
vegetables.
Example: “Some skyscrapers are beautiful.”
We can summarize the truth of a biconditional statement in a table:
(S: skyscrapers. P: beautiful things.) Particular Negative: Some S are not P
A
B
A↔B
Example: “Some books are not meant for children.”
T
T
T
(S: books. P: things meant for children.)
T
F
F
F
T
F
F
F
T
3.5 Categorical Logic Categorical logic is a type of deductive logic introduced by Aristotle in the 4th century BC, according to which we can infer true statements from other true statements that state that some or all things of a category belong to another category. For instance, the statement that “All cats are blue” tells us that there is a category of cats, and a category of blue things, and that everything that is a cat is also blue. In categorical logic, we can divide a statement into parts, each part describing a category. This is something we cannot do if we are only evaluating statements as a whole. For instance, if I claim that “All cats are blue” and that “Benny is a cat”, then the logical inference we can make is that “Benny is blue”. But if we’re looking at the propositions as a whole, then we can’t see the relation between the two statements. That is, if we symbolized “All cats are blue” as “A”, and “Benny is a cat” as “B”, then we have lost
Modern logic has a similar “predicate logic”, which extends beyond the realm of this text. In fact, attempts to symbolize Aristotle’s logic have resulted in horrible difficulties and frustrated logicians all over the world for millennia. One particular difference between Aristotle’s logic and modern predicate logic we should note is that while modern logic would symbolize Aristotle’s universal statements as conditional statements, Aristotle did not use conditionals in his logic, as he believed a conditional statement did not properly express the relation between the antecedent and the consequent. The proper relation is that of belonging to a category. This is why you might see “All S are P” reinterpreted by modern logicians as something like, “If X is an S, then X is a P” (where X is some random individual). Modern logic also assumes that when we make a statement about a particular thing, that particular thing exists, but when we make a universal statement, the subject of that statement doesn’t necessarily exist. Thus particular statements are said to have “existential import” that universal statements do not.
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