MLS 570Critical ThinkingReading Notes for Fogelin:

Categorical Syllogisms

We will go over diagrammingArguments in class.

Fall Term 2006North Central College

The difference … All squares are rectangles

All rectangles have parallel sides All squares have parallel sides

This argument cannot be written as p qq r . p r

This is because the premises in the argument are not compound, nor do they contain an “if … then” construction. [needed in order to use the conditional connective.]

Categorical Propositions All squares are rectangles

All rectangles have parallel sides All squares have parallel sides

Each premise asserts a relationship between the two terms. To understand this relationship we use a diagram of two overlapping circles.

This way of showing how Categorical Syllogisms work are called VennDiagrams

Diagramming propositions:All A are B All squares are rectangles – this says that there is

nothing that is a square that is not a rectangle. So we shade out the part of the diagram where nothing

exists. [the pink in this diagram]

Squares Rectangles

Diagramming propositions:No A are B Two groups or “classes” that have nothing in common

would be diagrammed like this. Again you shade in the area where there is nothing.

Triangles Squares

Diagramming propositions:Some A are B How do we handle “some”?

For example: Some aliens are spies.

We don’t want to shade in a whole area as that wouldmean “all”-- so we put an asterisk in the middle – thismeans that there is “at least one person who is an alienis also a spy”

aliens spies

Diagramming propositions:Some A are not B

Some aliens are not spies.

aliens spies

Diagramming the propositions:Some B are not A

Some spies are not aliens.

aliens spies

The 4 Basic Categorical Forms I

A: All S is P E: No S is PI: Some S is P O: Some S is not P

.

These are not propositions, but patterns for whole groups of propositions.

“Some spies are not aliens” is a substitution instance of the O propositional form.

The 4 Basic Categorical Forms IIOne more wrinkle ;)

A: Universal Affirmative E. Universal Negative

All S is P No S is P

I: Particular Affirmative O: Particular Negative

Some S is P Some S is not P

The 4 Basic Categorical Forms IIHow this looks in a table.

Affirmative Negative .Universal All S is P No S is P

Particular Some S is P Some S is not P

The four basic categorical forms All S is P [S=subject term, P=predicate term]

S P

The four basic categorical forms No S is P [S=subject term, P=predicate term]

S P

The four basic categorical forms Some S is P [S=subject term, P=predicate term]

S P

The four basic categorical forms Some S is not P [S=subject term, P=predicate

term]

S P

Exercise 1- #4: Indicate the information given in the diagram using the 4 basic propositions.

Some S is not PSome S is PSome P is not S[this is not one of S Pthe four forms, But is readable From the diagram]

Exercise 1- #8: Indicate the information given in the diagram using the 4 basic propositions.

Some S is PAll P is S[this is not one of the four forms, S Pbut is readable From the diagram]

“Contradictories”: E & I propositions These are pairs among the basic propositions that

can’t be true at the same time. Example: The E proposition says that there is

nothing that is both S & P, while the I proposition says that there is at least one thing that is both S & P

.

E: No S is P I: Some S is P

“Contradictories”: A & O propositions These are pairs among the basic propositions that

can’t be true at the same time. Example: The E proposition says that there is

nothing that is both S & P, while the I proposition says that there is at least one thing that is both S & P

.

A: All S is P O: Some S is not P

Validity for Arguments containing Categorical Propositions

An argument is valid if all the information contained in the diagram for the conclusion is included in the diagram for the premises.

[be sure to label the subject and predicate terms correctly.]

Some whales are mammals

Some mammals are whales

Validity for Arguments containing Categorical Propositions

You can [and should] generalize this to:

Some S is P

This argument is Some P is Svalid because thediagram for the conclusion is contained in the diagram for the premises.

Immediate Inferences These are arguments with a single premise constructed

from the A, E, I and O propositions. The simplest is conversion. I and E easily convert.

From an I proposition “Some S is P” you can infer its converse, which is “Some P is S”

From an E proposition “No S is P” you can infer its converse, which is “No S is P”

Neither of the O or A propositions can be automatically converted. “Some S is not P” does not infer “Some P is not S” “All S is P” does not infer “All P is S.”

The Theory of the Syllogism1. The argument has exactly two premises and one

conclusion.2. The argument contains only basic A, E, I, and O

propositions. 3. Exactly one premise contains the predicate term.4. Exactly one premise contains the subject term.5. Each premise contains the middle term.

The predicate term is the term in the predicate location in the conclusion.

The premise that contains the predicate term is called the major premise

The Theory of the Syllogism The predicate term is the term in the predicate

location in the conclusion. The premise that contains the predicate term is called

the major premise The subject term is the subject of the conclusion. The premises that contains the subject term is called

the minor premise.

All rectangles are things with 4 sides (Major premise)All squares are rectangles (Minor premise) All squares are things with 4 sides (Conclusion)

Subject term = “Squares”; Predicate term = “Things with 4 sides”Middle term = “Rectangles”

Venn Diagrams for determining the validity of a Categorical Syllogism

All rectangles have four sidesAll squares are rectangles All squares have four sides

Squares Things

having 4 sides

Notice that all the thingsthat are squares arecorralled into the region of all things that have 4 sides. This shows that this Rectanglessyllogism is valid

No ellipses have sidesAll circles are ellipses No circles have sides

Circles Sides

Conclusion Ellipses

You can see that the diagram for the conclusion is already present in the diagram for the premises.

Strategy: diagram a UNIVERSAL premise before a Particular one as it may tell you where the * should go. All squares have equal sidesSome squares are rectangles Some rectangles have equal sides.

The conclusion -- that there is something that is a Rectangle -- already appears in the diagram.

An Invalid argumentAll pediatricians are doctors

All pediatricians like children All doctors like children

Below: The diagram for theconclusion is not contained inthe diagram for the premises

Above: The diagram for the premises [ask: why ispart of the diagram darker?]

Diagramming “some”: when does the asterisk go on the line?Some doctors are golfersSome fathers are doctors Some fathers are golfers . The asterisk goes on the line when

you have no information about the relationship. For example in the above argument “Some

doctors are golfers” the premise says nothing about the relation of doctors to fathers. Thus the blue asterisk is on the line between D & F.

Likewise in the second premise nothing is said about golfers. So the red asterisk is on the line between F & G.

Diagramming “some”: Is the argument valid?Some doctors are golfersSome fathers are doctors Some fathers are golfers .

The argument is invalid becausethe diagram for the conclusion is not already contained in the diagram for in the premises.