Venn Diagrams & the Modern Square of Opposition

logically undetermined truth value

A condition that exists when a certain statement is not necessarily either true or false, given the truth value of some related statement.

illicit conversion

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

logically unrelated

A statement converse

immediate inferences

Example: Some trade spies are not masters at bribery. Therefore, it is false that all trade spies are masters at bribery.

logically equivalent

I statement converse

Given Statement: All A are non-B. (F) Operation: Obversion

New statement after conversion: No A are B. Truth value: False

Convert the following statement and determine the truth value: Given Statement: No A are non-B (T) Operation: Conversion

New statement after conversion: No non-B are A Truth value: True

Boolean standpoint

No universal propositions have existential import 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. the Boolean standpoint is "closed" (unreceptive) to existence. When things exist, the Boolean standpoint does not recognize their existence, and universal statements about those things have no existential import. Conversely, when things do not exist, neither the Aristotelian nor the Boolean standpoint recognizes any existence

Contrapositive

The contrapositive of any categorical proposition is the new categorical proposition that results from putting the complement of the predicate term of the original proposition in the subject place of the new proposition and the complement of the subject term of the original in the predicate place of the new. Thus, for example, the contrapositive of "All crows are birds" is "All non-birds are non-crows," and the contrapositive of "Some carnivores are not mammals" is "Some non-mammals are not non-carnivores." -------------------------------------------------------- 1) Statement: All A are B. New statement: All non-B are non-A. 2) Statement: No A are B. New statement: No non-B are non-A. 3) Statement: Some A are B. New statement: Some non-B are non-A. 4) Statement: Some A are not B. New statement: Some non-B are not non-A. 5) Statement: All non-A are B New statement: All non-B are A. 6) Statement: No non-A are non-B. (F) New Statement: No B are A. Truth value: Undetermined

Conversion

The converse of any categorical proposition is the new categorical proposition that results from putting the predicate term of the original proposition in the subject place of the new proposition and the subject term of the original in the predicate place of the new. Thus, for example, the converse of "No dogs are felines" is "No felines are dogs," and the converse of "Some snakes are poisonous animals" is "Some poisonous animals are snakes." -------------------------------------------------------- 1) Statement: All A are non-B. (F) New Statement: All non-B are A. Truth Value: True 2) Statement: Some A are non-B. (F) New Statement: Some non-B are A Truth Value: False 3) Statement: All B are A. new statement: All A are B. 4) Statement: No B are A. new statement: No A are B. 5) Statement: some B are not A. new statement: some A are not B. 6) Statement: Some non-A are not B. new statement: Some B are not non-A.

Obversion

The obverse of a standard form categorical statement is the result of (i) changing its quality (from affirmative to negative or vice versa) and (ii) replacing the predicate term with its term- complement. -------------------------------------------------------- 1) Statement: No non-A are B. (F) New Statement: All non-A are non-B. 2) Statement: All A are B. New statement: No A are non-B. 3) Statement: No A are B. New statement: All A are non-B. 4) Statement: Some A are B. New statement: some A are non-B. 5) Statement: Some A are not B. new statement: Some A are non-B. 6) Statement: Some non-A are non-B. (T) new statement: some non-A are NOT B. truth value: True 7) Statement: Some A are not non-B. (T) New Statement: Some A are B. truth value: True

The Modern Square of Opposition

This relationship of mutually contradictory pairs of propositions is represented in a diagram called the modern square of opposition.

Aristotelian Standpoint

Universal propositions about existing things have existential import the Aristotelian standpoint is "open" (or receptive) to existence.Footnote When things exist, the Aristotelian standpoint recognizes their existence, and universal statements about those things have existential import.

parameter

When put in a statement, it affects the form but not the meaning

Existential Import

a proposition has existential import if it presupposes the existence of certain kinds of objects

term complement

the word or group of words that denotes the class complement

vacuously false

their truth value results solely from the fact that the subject class is empty, or void of members

vacuously true

their truth value results solely from the fact that the subject class is empty, or void of members.