A Concise Introduction to Logic, Chapter 4 (Categorical Propositions)

What are the universal propositions?

"All S are P" and "No S are P" each assert something about every member of the S class and thus are universal propositions.

What are the particular propositions?

"Some S are P" and "Some S are not P" assert something about one or more members of the S class and hence are particular propositions.

What are quantifiers?

"all," "no," or "some" they specify how much of the subject class is included in or excluded from the predicate class

What is the copula?

"are" or "are not"

What is the distribution diagram of "Some S are not P"?

( " * " asterisk represented S outside the P circle) The particular negative (O) proposition asserts that at least one member of S is not a member of P. Since the other members of S may or may not be outside of P, it is clear that the statement "Some S are not P" does not make a claim about every member of S, so S is not distributed. But, as may be seen from the diagram, the statement does assert that every member of P is separate and distinct from this one member of S that is outside the P circle. Thus, in the particular negative (O) proposition, P is distributed and S is undistributed.

What is the distribution diagram of "Some S are P"?

( " * " asterisk represented S within the P circle) The particular affirmative (I) proposition states that at least one member of S is a member of P. Since the asterisk is inside the P class, it represents something that is simultaneously an S and a P; in other words, it represents a member of the S class that is also a member of the P class. Thus, the statement "Some S are P" makes a claim about one member (at least) of S and also one member (at least) of P, but not about all members of either class. Hence, by the definition of distribution, neither S nor P is distributed.

What is the distribution?

(1) 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; (2) a valid rule of inference that allows a conjunct/disjunct to be distributed through a disjunction/conjunction ________ A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term; otherwise, it is undistributed. Thus, if a statement asserts something about every member of the S class, then S is distributed; if it asserts something about every member of the P class, then P is distributed; otherwise S and P are undistributed

Four types of categorical propositions

(1) those that assert that the whole subject class is included in the predicate class, (2) those that assert that part of the subject class is included in the predicate class, (3) those that assert that the whole subject class is excluded from the predicate class, and (4) those that assert that part of the subject class is excluded from the predicate class.

What is an existential fallacy?

A fallacy that occurs whenever an argument is invalid merely because the premises lack 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.

Definition of a standard-form categorical proposition

A proposition that has one of the following forms: "All S are P," "No S are P," "Some S are P," "Some S are not P"

What is the definition of a categorical proposition?

A proposition that relates two classes, or categories The classes in question are denoted respectively by the subject term and the predicate term, and the proposition asserts that either all or part of the class denoted by the subject term is included in or excluded from the class denoted by the predicate term.

What is the conversion of the given statement form? All A are B

All A are B All B are A

What are some of the forms of existential fallacy?

All A are B. Therefore, some A are B. It is false that some A are not B. Therefore, it is false that no A are B. No A are B. Therefore, it is false that all A are B. It is false that some A are B. Therefore, some A are not B.

From the Boolean standpoint, the four kinds of categorical propositions have the following meaning

All S are P. = No members of S are outside P. No S are P. = No members of S are inside P. Some S are P. = At least one S exists that is a P. Some S are not P. = At least one S exists that is not a P.

Analyze the standard form categorical proposition

All members of the American Medical Association are people holding degrees from recognized academic institutions. quantifier: all subject term: members of the American Medical Association copula: are predicate term: people holding degrees from recognized academic institutions

What is the Aristotelian viewpoint of existential import?

All pheasants are birds. Implies the existence of pheasants. No pine trees are maples. Implies the existence of pine trees. All satyrs are vile creatures. Does not imply the existence of satyrs. The first two statements have existential import because their subject terms denote actually existing things. The third statement has no existential import, because satyrs do not exist.

What are immediate inferences?

An argument having a single premise Instead of reasoning from one premise to the next, and then to the conclusion, we proceed immediately from the single premise to the conclusion

What is existential import?

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

Definition of 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

What is the meaning of the categorical proposition in class notation? Some S are P

At least one member of the S class is a member of the P class.

What is the meaning of the categorical proposition in class notation? Some S are not P

At least one member of the S class is not a member of the P class.

What do the empty spaces in the Venn Diagrams represent?

Because there is no X in the diagrams that represent the universal propositions, these diagrams say nothing about existence. For example, the diagram for the A proposition merely asserts that nothing exists in the part of the S circle that lies outside the P circle. The area where the two circles overlap and the part of the P circle that lies outside the S circle contain no marks at all. This means that something might exist in these areas, or they might be completely empty. Similarly, in the diagram for the E proposition, no marks appear in the left-hand part of the S circle and the right-hand part of the P circle. This means that these two areas might contain something or, on the other hand, they might not.

What are conversion, obversion, and contraposition?

Conversion, obversion, and contraposition are operations that can be performed on a categorical proposition, resulting in a new statement that may or may not have the same meaning and truth value as the original statement. Venn diagrams are used to determine how the two statements relate to each other.

Identify the letter name, quantity, and quality. Then state whether the subject and predicate terms are distributed or undistributed. No vampire movies are films without blood.

E proposition, universal, negative, subject and predicate terms are distributed.

What is the meaning of the categorical proposition in class notation? All S are P

Every member of the S class is a member of the P class; that is, the S class is included in the P class.

Identify the letter name, quantity, and quality. Then state whether the subject and predicate terms are distributed or undistributed. Some hospitals are organizations that overcharge the Medicare program.

I proposition, particular, affirmative, subject and predicate terms undistributed.

What is the definition of a predicate term?

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

What is the definition of a subject term?

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

How do we spot an existential fallacy?

Look for a pair of diagrams in which the premise diagram contains shading and the conclusion diagram contains an X. If the X in the conclusion diagram is in the same part of the left-hand circle that is unshaded in the premise diagram, then the inference commits the existential fallacy.

What is the conversion of the given statement form? No A are B

No A are B No B are A

What is the meaning of the categorical proposition in class notation? No S are P

No member of the S class is a member of the P class; that is, the S class is excluded from the P class.

What is the Boolean viewpoint of existential import?

No universal propositions have existential import. Such statements never imply the existence of the things talked about: All trucks are vehicles. Does not imply the existence of trucks. No roses are daisies. Does not imply the existence of roses. All werewolves are monsters. Does not imply the existence of werewolves.

Identify the letter name, quantity, and quality. Then state whether the subject and predicate terms are distributed or undistributed. Some Chinese leaders are not thoroughgoing opponents of capitalist economics.

O proposition, particular, negative, subject term undistributed, predicate term distributed.

Identify the quantifier, subject term, copula, and predicate term: All animal-rights activists are individuals motivated by empathy.

Quantifier: all subject term: animal-rights activists copula: are predicate term: individuals motivated by empathy.

Identify the quantifier, subject term, copula, and predicate term: No fast-food restaurants are establishments that promote public health.

Quantifier: no subject term: fast-food restaurants copula: are predicate term: establishments that promote public health.

Identify the quantifier, subject term, copula, and predicate term: No sex-education courses that are taught competently are programs that are currently eroding public morals.

Quantifier: no subject term: sex education courses that are taught competently copula: are predicate term: programs that are currently eroding public morals.

Identify the quantifier, subject term, copula, and predicate term: Some executive-pay packages are insults to ordinary workers.

Quantifier: some subject term: executive pay packages copula: are predicate term: insults to ordinary workers.

Identify the quantifier, subject term, copula, and predicate term: Some preachers who are intolerant of others' beliefs are not television evangelists.

Quantifier: some subject term: preachers who are intolerant of others' beliefs copula: are not predicate term: television evangelists.

Identify the quantifier, subject term, copula, and predicate term: Some presidential candidates are not threats to the establishment.

Quantifier: some subject term: presidential candidates copula: are not predicate term: threats to the establishment.

Examples of categorical propositions

Reality TV stars hope for recognition. Junk foods do not belong in school cafeterias. Many of today's unemployed have given up on finding work. Not all romances have a happy ending. Oprah Winfrey publishes magazines.

Why do we shade/mark the Venn Diagrams?

Recall that the A proposition asserts that no members of S are outside P. This is represented by shading the part of the S circle that lies outside the P circle. The E proposition asserts that no members of S are inside P. This is represented by shading the part of the S circle that lies inside the P circle. The I proposition asserts that at least one S exists and that S is also a P. This is represented by placing an X in the area where the S and P circles overlap. This X represents an existing thing that is both an S and a P. Finally, the O proposition asserts that at least one S exists, and that S is not a P. This is represented by placing an X in the part of the S circle that lies outside the P circle. This X represents an existing thing that is an S but not a P.

What is the conversion of the given statement form? Some A are B

Some A are B Some B are A

What is the conversion of the given statement form? Some A are not B

Some A are not B Some B are not A

Where do the Aristotelian and Boolean viewpoints diverge in regards to A and E statements?

The Aristotelian standpoint differs from the Boolean standpoint only with regard to universal (A and E) propositions. Taking the Aristotelian standpoint amounts to recognizing that universal statements about existing things convey evidence about existence. Conversely, for a statement to convey such evidence, the Aristotelian standpoint must be taken and the subject of the statement must denote actually existing things. Taking the Boolean standpoint, on the other hand, amounts to ignoring any evidence about existence that universal statements might convey.

What is the distribution diagram of "No S are P"?

The S and P circles are separate. This statement makes a claim about every member of S and every member of P. It asserts that every member of S is separate from every member of P, and also that every member of P is separate from every member of S. Accordingly, by our definition, both the subject and predicate terms of universal negative (E) propositions are distributed.

What is the distribution diagram of "All S are P"?

The S circle is contained in the P circle, which represents the fact that every member of S is a member of P. (Of course, should S and P represent terms denoting identical classes, the two circles would overlap exactly.) by the definition of "distributed term", S is distributed and P is not. In other words, for any universal affirmative (A) proposition, the subject term, whatever it may be, is distributed, and the predicate term is undistributed.

What is the quantity of a categorical proposition?

The attribute of a categorical proposition by which it is either universal or particular

How would we test this immediate inferences (one premise argument) with a Venn diagram? It is false that all A are B.

The first statement claims that "All A are B" is false. Thus, to diagram it, we do the exact opposite of what we would do to diagram "All A are B." To diagram "All A are B," we shade the left-hand part of the A circle. To diagram "All A are B," we shade the left-hand part of the A circle. To diagram "It is false that all A are B," we enter an X in the left-hand part of the A circle. Entering an X in an area is the opposite of shading an area. Any statement that is diagrammed by entering an X in an area is a particular proposition. Thus, as the diagram shows, "It is false that all A are B" is actually a particular proposition. By similar reasoning, "It is false that no A are B" is also a particular proposition.

Why the Aristotelian and Boolean viewpoints identical about I and O statements?

The two standpoints are identical with regard to particular (I and O) propositions. Both the Aristotelian and the Boolean standpoints recognize that particular propositions make a positive assertion about existence. For example, from both standpoints, the statement "Some cats are animals" asserts that at least one cat exists that is an animal. Also, from both standpoints, "Some fish are not mammals" asserts that at least one fish exists that is not a mammal. Thus, from both standpoints, the word "some" implies existence.

Why does the inference "All A are B; therefore, some A are not B" does not commit existential fallacy?

This inference is invalid because the conclusion contradicts the premise. Thus, to detect the existential fallacy, one must ensure that the invalidity results merely from the fact that the premise lacks existential import. This can easily be done by constructing a Venn diagram.

How would we test this immediate inferences (one premise argument) with a Venn diagram? It is false that some A are B.

To diagram "It is false that some A are B," we do the exact opposite of what we would do to diagram "Some A are B." For "Some A are B," we would enter an X in the overlap area. Thus, to diagram "It is false that some A are B," we shade the overlap area. Any statement that is diagrammed by shading an area is a universal proposition. Thus, "It is false that some A are B" is actually a universal proposition. By similar reasoning, "It is false that some A are not B" is also a universal proposition.

Mnemonic devices for distribution

Universals distribute Subjects. Negatives distribute Predicates. A distributes Subject. E distributes Both. I distributes Neither. O distributes Predicate.

How can we represent the four categorical propositions with Venn Diagrams?

We can now use Venn diagrams to represent the information expressed by the four kinds of categorical proposition. To do this we make a certain kind of mark in a diagram. Two kinds of marks are used: shading an area and placing an X in an area. Shading an area means that the shaded area is empty and placing an X in an area means that at least one thing exists in that area. The X may be thought of as representing that one thing. If no mark appears in an area, this means that nothing is known about that area; it may contain members or it may be empty. Shading is always used to represent the content of universal (A and E) propositions, and placing an X in an area is always used to represent the content of particular (I and O) propositions.

"I" proposition

particular affirmative (some S are P)

"O" proposition

particular negative (some S are not P)

What is the quality of a categorical proposition?

quality (affirmative propositions: All S are P; Some S are P) or (negative propositions: No S are P; Some S are not P) In universal propositions the quality is determined by the quantifier, and in particular propositions it is determined by the copula.

True or false: Particular propositions mean no more and no less than the meaning assigned to them in class notation. The statement "Some S are P" does not imply that some S are not P, and the statement "Some S are not P" does not imply that some S are P.

true

"A" proposition

universal affirmative (all S are P)

"E" proposition

universal negative (no S are P)