Chapter 6: Categorical Propositions

Deductive Argument

An argument that claims to est. Its conclusion conclusively; one of the 2 classes of arguments

Valid Argument

A deductive argument in which, if all the premises are true, the conclusion must be true

Class (category)

Collection of all objects that have some specified characteristic in common

Categorical Proposition

Proposition used in deductive arguments that asserts a relationship between one category and some other category

Kinds of Categorical Propositions:

Universal Affirmative Proposition (A)

Propositions that assert that the whole of one class is included or contained in another class

All S is P

Universal Negative Proposition (E)

Proposition that assert that the whole of one class is excluded from the whole of another class

No S is P

Particular Affirmative Proposition (I)

Propositions that assert that 2 classes have some member or members in common.

Some S is P

Particular Negative Proposition (O)

Propositions that assert that at least one member of a class is excluded from the whole of another class

Some S is not P

Quality

An attribute of every categorical proposition determined by whether the proposition affirms or denies some form of class (+/-)

Quantity

An attribute of every categorical proposition, determined by whether the proposition refers to all members (universal) or only some members (particular) of the subject class

Distribution

A characterization of whether terms in a categorical proposition refer to all members of the class designated by that term

A proposition: subject term is distributed

E proposition: both subject and predicate are distributed

I proposition: both subject and predicate are not distributed

O proposition: predicate term is distributed

Universal propositions: subject term always distributed

Particular propositions: subject term always not distributed

Affirmative propositions: do not distribute predicate terms

Negative propositions: distribute predicate terms

The Square of Opposition:

A diagram showing the logical relationships among the four types of categorical propositions. Diff. from modern square of opposition

Opposition

Any logical relation among the kinds of categorical propositions exhibited on the Square of Opposition

Contradictories

2 propositions that cannot both be true and cannot both be false (A & O, E & I)

Contraries

2 propositions that cannot both be true but can both be false (A & E)

Subcontraries

2 propositions that cannot both be false but can both be true

Subalternation

Opposition between a universal proposition (superaltern) and its corresponding particular proposition (subaltern).

Classical logic: universal proposition implies the truth of its corresponding particular proposition

Modern logic: you cannot infer the truth of a particular proposition from its corresponding universal proposition but it may be done vice-versa

Immediate Inference

Inference drawn directly from only one premise

Mediate Inference

Inference drawn from more than 1 premise; conclusion is drawn from 1st premise through the mediation of the 2nd

Conversion

An inference formed by interchanging subject and predicate terms of a categorical proposition

S <-> P

Valid for E & I propositions

Complement of a Class

Collection of all things that do no belong to a class

e.g. voter and nonvoter

Obversion

An inference formed by changing the quality of a proposition and replacing the predicate term by its complement.

Valid for any standard-form categorical proposition

A: All S is P -> E: No S is non-P

E: No S is P -> A: All S is non-P

I: Some S is P -> O: Some S is not non-P

O: Some Sis not P -> I: Some S is non-P

Contraposition

Inference formed by replacing the subject term of a proposition with the complement of its predicate term and replacing the predicate term by the complement of its subject term

All S is P -> All non-P is non-S

Some S is not P -> Some non-P is not non-S

No S is P -> Some non-P is not non-S (contrapositive by limitation)

Not valid for I proposition

Venn Diagrams

A method of representing classes and categorical propositions using overlapping circles

A: Shade circle that refers to the subject (except middle part)

E: Shade middle part

I: Put x in middle part

O: Put x in circle that refers to the subject:

When making venn diagrams for syllogisms: if you cannot infer where to put the x in the circles, put it on the line in between them

Chapter 7: Categorical Syllogisms

Syllogism

Any deductive argument in which a conclusion is inferred form 2 premises

Has 3 propositions: 2 premises, 1 conclusion

Categorical Syllogism

A deductive argument consisting of 3 categorical propositions that together contain exactly 3 terms, each of which occurs in exactly 2 of the constituent propositions

Standard-form Categorical Syllogism

A categorical syllogism in which the premises and conclusions are all standard-form categorical propositions and are arranged with the major premise first, then the minor premise, and the conclusion

Major Term/premise

Term that occurs as the predicate of the conclusion in a standard-form syllogism

Major premise is the premise that contains the major term

Minor Term/premise

Term that occurs as the subject of the conclusion in a standard-form syllogism

Minor premise is the premise that contains the minor term

Middle Term

Term that occurs in both premises but not in the conclusion

Mood

One of the 64 3-letter characterizations of categorical syllogisms determined by the forms of the standard-form propositions it contains

e.g. AAA, EOE, etc.

Figure (\||/ or W)

The logical shape of a syllogism determined by te position of the middle term in its premises

1. Middle term is the subject term of the major premise and the predicate term of the minor premise

2. Middle term is the predicate of both premises3. Middle term is the subject term of both premises4. Middle term is the predicate of the major premise and subject of the minor premise

Venn Diagram for Categorical Syllogisms

Used to test validity

1. Draw 3 circles (Minor term, major term, middle term).2. Label the circles.3. Mark circles according to the propositions (2 propositions at a time). Don’t diagram

conclusion. When using this to diagram one universal and one particular premse, diagram the universal premise first.

4. Argument is valid if the diagram of the premises implies or entails the truth of the conclusion

Syllogistic Rules and Fallacies

1. Fallacy of Four Terms

Avoid 4 terms

2. Fallacy of the Undistributed Middle

Distribute the middle term in atleast one premise

3. Fallacy of an Illicit Major

Any term distributed in the conclusion must be distributed in the premises

4. Fallacy of an Illicit Minor

Any term distributed in the conclusion must be distributed in the premises

5. Fallacy of Exclusive Premises

Avoid 2 negative premises

6. Fallacy of Drawing an Affirmative Conclusion from a Negative Premise

If either premise is negative, the conclusion must be negative

7. Existential Fallacy

From 2 universal premises, no particular conclusion may be drawn

15 Valid Forms of Standard-Form Categorical Syllogism

First Figure

AAA-1 Barbara

EAE-1 Celarent

AII-1 Darii

EIO-1 Ferio

Second Figure

AEE-2 Camestres

EAE-2 Cesare

AOO-2 Baroko

EIO-2 Festino

Third Figure

AII-3 Datisi

IAI-3 Dusaus

EIO-3 Ferison

OAO-3 Bokardo

Fourth Figure

AEE-4 Camenes

IAI-4 Dimaris

EIO-4 Fresison

Chapter 8: Syllogisms in Ordinary Language

Syllogistic Argument

Argument that is a standard-form categorical syllogism, or can be formulated as one without any change in meaning

Reduction to Standard Form

Reformulation of a syllogistic argument into standard form

Singular Proposition

Proposition that asserts that a specific individual belongs or does not belong to a particular class

Unit Class

A class with only one member

Exclusive Proposition

A proposition asserting that the predicate applies only to the subject named

Exceptive Proposition

A proposition making 2 assertions that all members of some class – except for members of one of its subclasses – are embers of some other class

Parameter

An auxiliary symbol that aids in reformulating an assertion into standard form

Uniform Translation

Reducing propositions into a standard-form syllogistic argument by using paramets or other techniques

Enthymeme

An argument containing an unstated proposition

First-order

An incompletely stated argument in which the proposition taken for granted is the major premise

Second order

An incompletely stated argument in which the proposition taken for granted is the minor premise

Third-order

An incompletely stated argument in which the proposition that is left unstated is the conclusion

Sorites

An argument in which a conclusion is inferred from any number of premises through a chain of syllogistic inferences

Disjunctive Syllogism

A form of argument in which 1 premise is a disjunction and the conclusion claims to the truth of one of the disjuncts. Not all are valid.

e.g. I’m trying to cover up her illegal peccadillo or stonewall her way out of it, she was driven either by stupidity or arrogance. She’s obviously not stupid. Her plight, then, must result from her arrogance.

Hypothetical Syllogism

A form of argument containing at least 1 conditional proposition as a premise.

Can be pure (where all premises are conditional) or mixed (where one premise is conditional and the other is not)

Modus Ponens

A valid hypothetical syllogism in which the categorical premise affirms the antecedent of the conditional premise and the conclusion affirms its consequent

If Bacon wrote Hamlet, then Bacon was a great writer. Bacon was a great writer. Therefore Bacon wrote Hamlet.

Fallacy of Affirming the Consequent

A formal fallacy in a hypothetical syllogism in which the categorical premise affirms the consequent rather than the antecedent of the conditional premise

If the one-eyed prisoner saw 2 red hats, then he could tell the color of the hat on his own head. The one-eyed prisoner could not tell the color of the hat on his own head. Therefore, the one-eyed prisoner did not see 2 red hats.

Modus Tollens

A valid hypothetical syllogism in which the categorical premise denies the consequent of the conditional premise and the conclusion denies its antecedent.

If Carl embezzled the college funds, then Carl is guilty of a felony. Carl did not embezzle. Therefore, Carl is not guilty of a felony.

Fallacy of Denying the Antecedent

A formal fallacy in a hypothetical syllogism in which the categorical premise denies the antecedent rather than the consequent of the conditional premise

Dilemma

A common form of argument in ordinary discourse in which it is claimed that a choice must be made between 2 (usually bad) alternatives

If the blessed in heaven have no desires, they will be perfectly content. So they will be also if their desires are fully gratified. But either they will have no desires or have them fully gratified. Therefore, they will be perfectly content.

Simple/complex Dilemma

Simple: conclusion is a single categorical proposition

Complex: conclusion itself is a disjunction