Categorical propositions and classes Basic connectives and truth

Categorical propositions and classes

Basic connectives and truth • Assertions, called statements or propositions, are declarative sentences that are either true or false • New statements can be obtained from existing ones in two ways. – Transform a given statement p into the statement p – Combine two or more statements into a compound statement

LIKE • “If I weigh more than 120 pounds, then I shall enroll in an exercise class” • p: I weigh more than 120 pounds • q: I shall enroll in an exercise class • the four cases of p q

A word of caution • In our everyday language, we often find situations where an implications is used when the intention actually calls for a biconditional. • If you do your homework, then you will get to watch the PK movie.

SIMPLE SENTENCES • noun-verb-direct object • structure with a quantifying prefix.

A categorical proposition and A Class • Any proposition that can be interpreted as asserting a relation of inclusion or exclusion, complete or partial, between two classes. • A class is defined as a collection of all objects which have some specified characteristic in common. This is no more complicated than observing that the class of "lightbulbs" all have the common characteristic of "being a lightbulb. "

• Categorical propositions are simple sentences composed in a noun-verb-direct object structure with a quantifying prefix. The first noun is the subject (S), the direct object is the predicate (P). The verb is called the copula and is almost always a conjugation of the verb "to be. " In most cases sentences using other verbs can be rewritten using the verb "to be. " The predicate (also known as a direct object) is

Categorical propositions • The building blocks of categorical logic, which goes back to Aristotle’s fundamental work in the 4 th century BC. Aristotle developed his logic as a foundation for science. His system of logic was based on classification: what is the relationship between one class of objects and another?

• A proposition that relates two classes, or categories, is called a categorical proposition. The two classes in question are denoted by the subject term and the predicate term. Categorical propositions assert 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.

Types of categorical propositions Four types 1. All stand-up 2. No stand-up 3. Some stand-up 4. Some stand-up

we can have four class relations in the various kinds of categorical propositions: • Utilizing the classes, "people" and "good beings": • a. complete inclusion>>>"All people are good beings. “ • b. complete exclusion>>>"No people are good beings. “ • c. partial inclusion>>>"Some people are good beings. “ • d. partial exclusion>>>"Some people are not good beings. "

Standard-form categorical propositions • • universal affirmative A universal negative E particular affirmative I particular negative O

Standard forms of categorical propositions • • 1. All S are P (A proposition) 2. No S are P (E proposition) 3. Some S are P (I proposition) 4. Some S are not P (O proposition)

• 'S' represent the subject and 'P' • predicate terms. Note that this is different from the grammatical subject and predicate. The grammatical subject contains the quantifier (All), while the subject term does not. 'No', 'All', and 'Some' are quantifiers. 'Are' and 'are not'

The quantifier • Indicators of "how much" are called quantity indicators (quantifiers)and three quantifiers are used in categorical propositions: 1. All 2. none and 3. some

QUALIFIERS • Indicators of affirmative and negative are quality indicators (qualifiers) and specifically are "are, " "are not, " "is not, " and "no, “ • Note that "no" is both a quantifier and a qualifier.

QUALITY • The quality of a categorical proposition is determined by whether the asserted class relation is one of exclusion or inclusion (i. e. , affirmative or negative).

QUALITY • Categorical propositions can have one of two qualities. The proposition is either affirmative or negative. Affirmative propositions are represented by the letters A and I from the Latin. Aff. Irmo. Negative propositions are represented by the letters E and O from the Latin n. Eg. O.

QUANTITY • The quality of a categorical proposition is determined by whether or not it refers to all members of its subject class (i. e. , universal or particular). • The question "How many? " is asking for quantity.

Quantity • Categorical propositions may refer to all, some, or no members of a category. Propositions which refer to all or none are termed universal propositions. Propositions which refer to some are termed particular propositions.

All The diagram for "all" is an open, unfilled circle. ALL

None • The diagram for "none" is an open, filled, darkened, or cross-hatched circle. That "all" is an empty circle and "none" is a filled in circle can be confusing later on in the course. The filled-in circle is equivalent to an "erased" circle. That category no longer exists. In these documents the circle is a darkened circle, on a white board or sheet of paper the category is usually "cross-hatched" with diagonal lines.

• NONE

Some The diagram for "some" is an open, unfilled, circle with a single "X" in the circle.

Some not The diagram for "some (are) not" a member is an open, unfilled, circle with a single "X" outside the circle