Today’s Topics• 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 Quantifier Subject Copula Predicate

• A All F are G

• E No F are G

• I Some F are G

• O 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.

• Name Quantity Quality Subject Predicate

• A Universal Aff. Dist Undist.

• E Universal Neg Dist. Dist.

• I Particular Aff Undist. Undist.

• O Particular Neg 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

• 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

12

3

4

56

7 8

Properties by Region (p 301)

• Region Major Minor Middle– 1 no yes no– 2 yes yes no– 3 yes no no– 4 yes no yes– 5 yes yes yes– 6 no yes yes– 7 no no yes– 8 no no 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)