PHL 3000 Lecture 8: Module 8: The Elements of Categorical Logic
Module 8
Lecture 8.1: The Elements of Categorical Logic
The Need for Another System
• Consider the following argument:
o All humans are mortal
o Bob is human
o Therefore, Bob is mortal
• Clearly valid; but according to (PL), thought its
o P
o Q
o R
▪ Invalid according to PL
• Another set of purely logical words: All, No, Some
o All humans are mortal
o Some rectangles are squares
o No weekdays are fun
Categorical Logic
• Originally developed by Aristotle, and the only system of logic until very recently
o PL was developed in the early part of the 1900s
• Called categorical logic because instead of examining the logical relationships that exist
between statements, we will be examining the logical relationships that exist between
categories of things, such as the category of all dogs, or all children, or all tables, etc.
• Categorical statement: any statement that relates two classes of things, or categories, to
each other
• Terms are words that refer to the two classes
• There are two terms in a categorical statement: subject and predicate term
• Categorical statements assert that either all of some of the class denoted by the subject
term are either included o\or excluded from the class denoted by the predicate term
• Examples of categorical statements:
o Students want to get good jobs
o Stories about reality TV should not be on the news
o Some of these apples are rotten
o Socrates was a philosopher
Standard-Form Categorical Statements
• Four standard-form categorical propositions:
o All S are P
o No S are P
o Some S are P
o Some S are not P
o In order to be legitimate they have to have all no or some
o Only way to conjoin these two categories are are or are not
o Hard to make all English sentences fit into one of these four
▪ Then it will likely work better with PL
o All S are not P is not standard form, because it is ambiguous between No S are P
and Some S are not P.
• The elements of a standard-form categorical statements
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o Quantifier: all, no, some
o Subject term: must be a noun phrase
o Copula: are, are not
o Predicate term: must be a noun phrase
Quantity and Quality
• Quantity and Quality are properties of categorical statements
o Quantity: either universal or particular, depending on whether the statement is
about all of the members of S, or only about some of the members of S
o Quality: either positive or negative, depending on whether the statement affirms
or denies that S is a part of P
Distribution
• Distribution is a property of the two terms in categorical statements
o A term is distributed in a categorical statement if the statement asserts something
about every member of the term
o A term is undistributed in a categorical statement if the statement does not assert
something about every member of that term.
Naming the Four Statements
• Since there are only four possible standard form categorical statements, logicians have
given them names (single letters derived from Latin words long ago)
o A: All S are P
o E: No S are P
o I: Some S are P
o O: Some S are not P
A Statements
• A Statement: All S are P
o Quality: Affirmative (asserts that S are within P)
o Quantity: Universal (is about every member of S)
o Distribution: S term is distributed; P term is not
▪ Every member of S is within P
▪ Do not know about P
E Statements
• E statement: No S are P
o Quality: Negative (denies that S are with P)
o Quantity: Universal (is about every member of S)
o Distribution: S term is distributed; P term is distributed
▪ Know something about every member of S and every member of P
I Statements
• I statement: Some S are P
o Quality: Affirmative (asserts membership of S within P)
o Quantity: Particular (is about only some members of S)
o Distribution: S term is undistributed; P term is undistributed
▪ Both S and P terms don’t know about all things.
O Statements
• O statement: Some S are not P
o Quality: Negative (denies membership of S within P)
o Quantity: Particular (is about only some members of S)
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o Distribution: S term is undistributed; P term is distributed
▪ Don’t know something about every S
▪ Somewhat confusing
Existential Import
• Consider the following statement made by a professor:
o Everyone who cheated on the last test will be expelled from school!
• Does the truth of this statement imply that anyone cheated?
o We assume no existential import for the universal statements. “All S are P” and
“No S are P” might be true, even if there is nothing in S.
o All round squares are impossible objects
Lecture 8.2: Venn Diagrams and Three Operations
Venn Diagrams for Standard Form Categorical Statements
• A venn diagram is a way of representing the information contained in a standard form
categorical statement. It is a picture of the information: represents it spatially, not just
symbolically. We begin by representing classes with circles:
• The four standard form categorical statements we will study all assert some kind of
relation to exist between two classes of things. We can create a diagram with two
overlapping circles to represent all of the possible logical relations that exist in the
following way:
•
• Markings:
o Shading = empty
o Star = something goes in the circle
o Blank = I know nothing
• A Statements:
o
• E Statements:
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Document Summary
The need for another system: consider the following argument, all humans are mortal, bob is human, therefore, bob is mortal, clearly valid; but according to (pl), thought its, p, q, r. Invalid according to pl: another set of purely logical words: all, no, some, all humans are mortal, some rectangles are squares, no weekdays are fun. No s are p might be true, even if there is nothing in s: all round squares are impossible objects. Venn diagrams for standard form categorical statements: a venn diagram is a way of representing the information contained in a standard form categorical statement. It is a picture of the information: represents it spatially, not just symbolically. We begin by representing classes with circles: the four standard form categorical statements we will study all assert some kind of relation to exist between two classes of things. If the two pictures are the same, it is valid.
