Phil 210 Exam 3 Study Guide

Categorical Proposition

A proposition that relates two classes (or categories); letter names of; standard form of.

Relate subject terms and predicate terms

Either all or part of the class denoted by the subject

Subject Term

In a standard-form categorical proposition, the term that comes immediately after the quantifier.

Predicate Term

In a standard-form categorical proposition, the term that comes immediately after the copula.

Four Types of Categorical Propositions

All S are P.

No S are P.

Some S are P.

Some S are not P.

Quantifier

In standard-form categorical propositions, the worlds “all,” “no,” and “some”.

Called this because they specify how much of the subject class is included in or excluded from the predicate class.

Copula

In standard-form categorical propositions, the words “are,” and “are not.”

Called this because they link (or “couple”) the subject term with the predicate term.

Standard Form

A proposition that has one of the following forms: “All S are P, “ “No S are P, “ “Some S are P, “ and “Some S are not P. ”

A standard-form categorical proposition is a statement having a quantifier and also subject and predicate terms that are linked by a copula.

Always has a subject terms that is a noun or noun phrase, always has a predicate terms that is a noun or noun phrase, always has a quantifier: “all,” “no,” or “some,”, always has a copula linking the subject and predicate terms: “are” or “are not.”

Quality

The attribute of a categorical proposition is either affirmative or negative depending on whether it affirms or denies class membership.

Quantity

The attribute of a categorical proposition is either universal or particular, depending on whether the statement makes a claim about every member or just some member of the class denoted by the subject term.

Distribution

An attribute possessed by a term in a categorical proposition if and only if the proposition makes a claim about all the members of the class denoted by the term.

A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term.

Letter Names of the Four Types of Categorical Propositions

A - All S are P.

E - No S are P.

I - Some S are P.

O - Some S are not P.

Distribution for A

A distributes subject (S).

Distribution for E

E distributes both S and P.

Distribution for I

I distributes neither.

Distribution for O

O distributes predicate (P).

Existential Import

An attribute for a categorical proposition by which it implies that one or more things denoted by the subject term actually exist.

The Aristotelean (Traditional) Persepctive: Universal propositions about existing things have existential Import, though those that are about non-existing things do not.

The Boolean (Modern) Persepctive: No universal propositions have existential Import.

Both standpoints recognize that particular (I and O) propositions make a positive assertion about existence.

Example:

All pheasants are birds. Implies the existence of pheasants from the Aristotelian standpoint.

All satyrs are vile creatures. Does not imply the existence of satyrs from the Aristotelian standpoint.

All trucks are vehicles. Does not imply the existence of trucks from the Boolean standpoint.

All werewolves are monsters. Does not imply the existence of werewolves from the Boolean standpoint.

Both standpoints would say that the statement “Some cats are animals” asserts that at least one cat exists that is an animal and that the statement “Some fish are not mammals” asserts that at least one fish exists that is not a mammal.

Aristotelian Standpoint

See Class Notes.

Boolean Standpoint

See Class Notes

Venn Diagram

See Class Notes

Vacuously True

A and E. These propositions are said to be this because their truth value results solely from the fact that the subject class is empty, or void of members.

(Universal propositions (A&E) that have as their subject a class of objects that don’t exist are true simply in virtue of the fact that they speak of an empty class.)

Vacuously False

I and O. Falsity that results merely from the fact that the subject class is empty.

(Particular propositions (I&O) that have as their subject a class of objects that don’t exist are false simply in virtue of the fact that they are the contrary of a true universal proposition as its subject a class of objects that is vacuous.)

Immediate Inference

An argument having a single premise, implication.

Existential Fallacy

A formal fallacy that occurs whenever an argument is invalid merely because the premise lacks existential Import. Such arguments always have a universal premise and a particular conclusion. The fallacy consists in attempting to derive a conclusion having existential import from a premise that lacks it.

Conversion

An operation that consists in switching the subject and predicate terms in a standard-form categorical proposition; to reduce number of terms in a syllogism.

Examples

Obversion

An operation that requires changing the quality (without changing the quantity) and replacing the predicate with its term complement (indicates everything outside of the term, the complement of the term “dog” is “non-dog).

Examples

Contraposition

An operation that requires switching the subject and predicate terms and replacing the subject and predicate terms with their term complements.

Examples

Converse Statement

For example, if the statement “No foxes are hedgehogs” is converted, the resulting statement is “No hedgehogs are foxes.” This statement is called the converse of the given statement.

Fallacy of Illicit Conversion

A formal fallacy that occurs when the conclusion of an argument depends on the conversion of an A or O statement.

Examples

All A are B.

Therefore, all B are A.

Some A are not B.

Therefore, some B are not A.

All cats are animals. (True)

Therefore, all animals are cats. (False)

Some animals are not dogs. (True)

Therefore, some dogs are not animals. (False)

Class Complement

The complement of a class is the group consisting of everything outside the class.
For examle, the complement of the class of dogs is the group that includes everything that is not a dog (cats, fish, trees, and so on).

Term Complement

The word or group of words that denotes the class complement.

For terms consisting of a single word, the term complement is usually formed by simply attaching the prefix “non” to the term.

The Contrapositive of a Given Statement

For example, if the statement “All goats are animals” is contraposed, the resulting statement is “All non-animals are non-goats.” This new statement is called this.

Fallacy of Illicit Contraposition

A formal fallacy that occurs when the conclusion of an argument depends on the contraposition of an E or I statement.

Examples

Some A are B.

Therefore, some non-B are non-A.

No A are B.

Therefore, no non-B are non-A.

No dogs are cats. (True)

No non-cats are non-dogs. (False)

Some animals are non-cats. (True)

Some cats are non-animals. (False)

Traditional Square of Opposition

An arrangement of lines that illustrates logically necessary relations among the four kinds of categorical propositions. Since the Aristotelian standpoint recognizes the additional factor of existential Import, the traditional square supports more inferences than does the modern square.

Contradictory Relation

Opposite truth value (A and O, E and I).

Subcontrary Relation

At least one is true (I-O).

Subalternation Relation

Truth “trickles down” from A to I and E to O (If A is true, then so is I. If E is true, then so is O).

Falsity “Floats Up” from I to A and O to E (If I is false, then so is A. If O is false, then so is E).

Illicit Contrary

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the contrary relation.

Examples

It is false that all A are B.

Therefore, no A are B.

It is false that no A are B.

Therefore, all A are B.

Illicit Subcontrary

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the subcontrary relation.

Examples

Some A are B.

Therefore, it is false that some A are not B.

Some A are not B.

Therefore, some A are B.

Illicit Subalternation

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the sub alternation relation.

Examples

Some A are not B.

Therefore, no A are B.

It is false that all A are B.

Therefore, it is false that some A are B.

Conditionally Valid

Valid from the Aristotelian standpoint on condition that the subject terms of the premise (or premises) denotes actually existing things.

Unconditionally Valid

Valid from the Boolean Standpoint; Valid regardless of whether the terms denote actually existing things.

Contrary Relation

At least one is false (A and E).

Nonstandard Verbs

Statements that incorporate forms of the verb “to be” other than “are” and “are not” or omit it entirely need to have the implicit “are” or “are not” made explicit.

Examples

Some dogs would rather bark than bite.

Some dogs are animals that would rather bark than bite.

All ducks swim.

All ducks are birds that swim.

Some college students will become educated.

Some college students are people who will become educated.

Some dogs would rather bark than bite.

Some dogs are animals that would rather bark than bite.

Some birds fly south during the winter.

Some birds are animals that fly south during the winter.

All ducks swim.

All ducks are swimmers

or

All ducks are animals that swim.

Singular Propositions

A proposition that makes an assertion about a specific person, place, or thing (or singular nouns) and can be transformed into universals by means of a “parameter.”

Parameter

A phrase that, when introduced into a statement affects the form but not the meaning.

Some parameters that may be used to translate singular propositions are these:

people identical to

places identical to

things identical to

cases identical to

times identical to

Example

Socrates is mortal.

All people identical to Socrates are people who are mortal.

Unexpressed Quantifiers

Implicit quantifiers muyst be made explicit. To determine the quantity you must be sensitive to the most probable meaning of the statement.

Examples

Emeralds are green gems.

All emeralds are green gems.

There are lions in the zoo.

Some lions are animals in the zoo.

Nonstandard Quantifiers

There are quantifiers other than “all,” “some,” or “no.” Examples include few, a few, every, etc.

Example

Nonstandard quantifier: A few soldiers are heroes.

Standard quantifier: Some soldiers are heroes.

Conditional Statements

The verbiage following “if” becomes the subject of a universal proposition. Such statements are always translated as universals. Language following the word “if” goes in the subject terms of the categorical proposition, and language following “only if” goes in the predicate term. The verbiage following “unless” becomes the negated subject of a universal proposition.

Examples

If it is a basset hound, then it is adorable.

All basset hounds are dogs that are adorable.

It’s a Harley only if it leaks oil.

All Harleys are motorcycles that leak oil.

Unless you feed a cat, she’ll eat the houseplants.

All cats that are not fed are cats who will eat the houseplants.

Exclusive Propositions

“Only,” “None but,” “None except,” etc. The verbiage following an exclusive term goes in the predicate of the categorical proposition.

Example

None but the elect will be saved.

All people who will be saved are people who are elect.

Exceptive Propositions

Take the form “All except S are P” or “All but S are P.” These must be translated as two conjoined categorical propositions.

Example

All except students are invited.

No students are invited people, and all nonstudents are invited people.