Today’s Topics
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Introduction to Predicate Logic
Venn Diagrams
Categorical Syllogisms
Venn Diagram tests for validity
Rule tests for validity
Propositional logic is limited
• Some arguments that are clearly valid
cannot be shown valid in our system.
– “All fish have gills, all animals with gills have
hearts, so all fish have hearts” would be
symbolized F, G,  H, wich is non-valid.
• Propositional logic misses the internal
structure of sentences.
– ‘Al is taller than Bill’ implies that ‘Bill is not
taller than Al’ but propositional logic doesn’t
allow us to show this.
We need a new, more powerful,
tool: Predicate Logic.
• We divide predicate logic into two parts:
• Categorical (syllogistic) logic
– The logic of classes and terms
– Aristotelian logic
• Modern predicate logic
– The logic of properties and relations
Categorical (Syllogistic) Logic
(Chapters 5 & 6)
Propositional and full predicate logic are modern
inventions (post 1870)
Prior to the late 1800’s logic was a very narrow
discipline, concerned only with a special type of
sentence called a Categorical Proposition
Go to the Handouts link and download the handout
entitled Venn Study Guide. NOTE that you will
need to add some of your own diagrams.
A categorical proposition divides
the world into two classes (terms)
and then makes a claim about the
overlap in the membership of
those two classes.
Every categorical proposition has
four (4) parts:
A quantifier (all or some)
A subject class (subject term)
A copula (linking verb)
A predicate class
For any two terms, F and G, there
are four (4) possible categorical
propositions:
• Name
• A
• E
• I
• O
Quantifier Subject Copula Predicate
All
F
are
G
No
F
are
G
Some
F
are
G
Some
F
are not
G
Each categorical proposition has a
Quantity (universal or particular)
and a Quality (affirmative or
negative) and each term (subject
and predicate) is either distributed
or undistributed
Quantity and quality.
• Quantity is determined by the quantifier.
– If the quantifier is All the quantity is universal.
– If the quantifier is Some the quantity is particular.
• Quality is determined by whether the proposition
asserts or denies an overlap between the classes.
– If a proposition asserts an overlap named, the quality
of the proposition is affirmative.
– It a proposition denies an overlap, the quality is
negative.
Distribution of Terms
• Each term in a categorical proposition is
either distributed or undistributed.
• If the proposition refers to the entire class
named by a term, that term is distributed.
• If the proposition does not refer to the entire
class named by a term, that term is
undistributed.
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Name Quantity Quality Subject
A Universal Aff.
Dist
E Universal Neg
Dist.
I Particular Aff
Undist.
O Particular
Neg
Undist.
Predicate
Undist.
Dist.
Undist.
Dist.
The Square of Opposition
(Aristotle)
• Knowledge of the truth of one categorical
proposition allows us to make immediate
inferences about the truth of others.
• An A and an E proposition are contrary, at most
one can be true.
• I and O are sub-contrary, at most one can be false.
• A and O are contradictory, exactly one is true.
• E and I are contradictory.
Existential Import
• A and I and E and O are subalterns.
• Aristotle believed that a universal claim could be
true only if there were members of the subject
class (modern logicians do not accept this)
• SO, if an A is true, the subaltern I must be true.
Same for E and O.
• Similarly, if the particular is false, the universal
must be false as well.
Venn Diagrams for Categorical
Propositions
• John Venn discovered a very useful method
of diagramming the informational content
of categorical propositions, Venn diagrams.
• A Venn diagram for a categorical
proposition consists of 2 overlapping circles
with four (4) regions.
A Venn diagram for 2 classes, S
and P
Objects by region
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Region 1, things that are F and not G
Region 2, things that are both F and G
Region 3, things that are G but not F
Region 4, things that are neither F nor G.
Two simple rules govern Venn
diagrams:
Shade a region to show that it is empty.
Place an X in a region to show that it is
occupied.
A- All S are P
E-No S are P
I-Some S are P
O-Some S are not P
Try a few on your own
• Download the Handout entitled Venn
Worksheet and identify the form (A. E. I, or
O) of each proposition. Some of them are
tricky. Make sure that you know how to
diagram each type of proposition.
Venn Diagrams
• The logical implications which follow from
various propositions can be studied readily
according to the Square of Opposition (page 287)
• Another way to examine these immediate
inferences is through the use of Venn diagrams.
• Venn diagrams represent the logical relations that
obtain between classes in a categorical
proposition.
Categorical Syllogisms
A Categorical Syllogism is a special type of
argument.
A categorical syllogism consists of three
propositions, 2 premises and one
conclusion, each of which is a must be a
categorical proposition.
A categorical syllogism contains
exactly three (3) class terms:
• The major term is the predicate term of the
conclusion of the argument.
• The minor term is the subject term of the
conclusion of the argument.
• The middle term is the term that does not
occur in the conclusion of the argument.
In the following categorical
syllogism:
• All rotarians are patriots.
All patriots are Republicans.
So, all rotarians are Republicans.
the major term is 'Republicans', the minor
term 'rotarians', and the middle term
'patriots.'
The following rules apply to all
valid categorical syllogisms:
• RULE 1: The middle term must be distributed in
at least one premise.
• RULE 2: A term distributed in the conclusion
must be distributed in one of the premises.
• RULE 3: The number of negative premises must
be equal to the number of negative conclusions.
• RULE 4: A particular conclusion cannot be drawn
from two universal premises.
Another way to test for validity is
with a three (3) circle Venn diagram.
• A three circle diagram contains eight (8)
regions.
• The lower circle represents the MIDDLE
term.
• The upper left circle represents the MINOR
term (Subject of the conclusion).
• The upper right circle represents the
MAJOR term (predicate of the conclusion).
3 Circle Venn Diagram w/8
regions
2
1
5
3
6
4
7
8
Properties by Region (p 301)
• Region
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–
–
–
–
–
–
–
1
2
3
4
5
6
7
8
Major
Minor
Middle
no
yes
yes
yes
yes
no
no
no
yes
yes
no
no
yes
yes
no
no
no
no
no
yes
yes
yes
yes
no
Venn Diagram tests for validity:
• Diagram the first premise, paying attention only to
the circles that represent the terms in that premise.
• Next, diagram the second premise paying attention
only to the circles that represent the terms in that
premise.
• Now, examine the diagram and ask, “Does this
diagram represent the informational content of the
conclusion?”
• If YES, the argument is valid.
Consider the following argument:
• All fish have gills.
• All animals with gills have hearts.
• So all fish have hearts.
Diagram the first premise:
Diagram the second premise:
Ask: Does the diagram represent
the informational content of the
conclusion?
• Yes, because all the F’s in the universe are
in region 5 and everything in region 5 is an
F a G and an H, so all the F’s are H’s
• The argument is VALID
We are looking for an accurate
diagram of the conclusion of the
argument that follows from a
diagram of the premises.
• A categorical syllogism is valid if, but only
if, a diagram of its premises produces a
diagram that expresses the propositional or
informational content of its conclusion.
Try a few on your own
• Complete the Venn worksheet you
previously downloaded and test the
syllogisms at the bottom of the page for
validity using BOTH the rule and the Venn
diagram tests (make sure you understand
both methods)