Obversion
Encyclopedia
In traditional logic, obversion is a "type of immediate inference
in which from a given proposition
another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, and whose quality is affirmative if the original proposition's quality was negative and vice versa". The quality of the inferred categorical proposition
is changed but the truth value is equivalent to the original proposition. The immediately inferred proposition is termed the "obverse" of the original proposition, and is a valid form of inference for all types (A, E, I, O) of categorical propositions.
In a universal affirmative
and a universal negative proposition the subject
term and the predicate
term are both replaced by their complements:
The universal affirmative ("A" proposition) is obverted to a universal negative ("E" proposition).
"All S are P" and "No S are non-P"
"All cat
s are animals" and "No cats are non-animals"
The universal negative ("E" proposition) is obverted to a universal affirmative ("A" proposition).
"No S are P" and "All S are non-P"
"No cats are friendly" and "All cats are non-friendly"
In the particular affirmative the quantity of the subject term remains unchanged, but the predicate term of the inferred proposition negates the complement of the predicate term of the original proposition. The particular affirmative ("I" proposition) is obverted to a particular negative ("O" proposition).
"Some S are P" and "Some S are not non-P"
"Some animals are friendly creatures" and "Some animals are not unfriendly creatures."
In the obversion of a particular negative to a particular affirmative the quantity of the subject also remains unchanged, and the predicate term is changed from simple negation to a term of the complementary class. The particular negative ("O") proposition is obverted to a particular affirmative ("I" proposition).
"Some S are not P" and "Some S are non-P"
"Some animals are not friendly creatures" and "Some animals are unfriendly creatures."
Note that the truth-value of an original statement is preserved in its resulting obverse form. Because of this, obversion can be used to determine the immediate inferences of all categorical propositions, regardless of quality or quantity.
In addition, obversion allows us to navigate through the traditional square of logical opposition
by providing a means for us to proceed from "A" Propositions to "E" Propositions, as well as from "I" Propositions to "O" Propositions, and vice versa. However, it must be noted that although the resulting propositions from obversion are logically equivalent
to the original statements in terms of truth-value, they are not semantically equivalent to their original statements in their standard form.
Immediate inference
An immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green." we can make the immediate inference that "No toads are not green." This new statement is known as the contrapositive of the original statement...
in which from a given proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, and whose quality is affirmative if the original proposition's quality was negative and vice versa". The quality of the inferred categorical proposition
Categorical proposition
A categorical proposition contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics....
is changed but the truth value is equivalent to the original proposition. The immediately inferred proposition is termed the "obverse" of the original proposition, and is a valid form of inference for all types (A, E, I, O) of categorical propositions.
In a universal affirmative
Term logic
In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
and a universal negative proposition the subject
Subject (grammar)
The subject is one of the two main constituents of a clause, according to a tradition that can be tracked back to Aristotle and that is associated with phrase structure grammars; the other constituent is the predicate. According to another tradition, i.e...
term and the predicate
Predicate (grammar)
There are two competing notions of the predicate in theories of grammar. Traditional grammar tends to view a predicate as one of two main parts of a sentence, the other being the subject, which the predicate modifies. The other understanding of predicates is inspired from work in predicate calculus...
term are both replaced by their complements:
The universal affirmative ("A" proposition) is obverted to a universal negative ("E" proposition).
"All S are P" and "No S are non-P"
"All cat
Cat
The cat , also known as the domestic cat or housecat to distinguish it from other felids and felines, is a small, usually furry, domesticated, carnivorous mammal that is valued by humans for its companionship and for its ability to hunt vermin and household pests...
s are animals" and "No cats are non-animals"
The universal negative ("E" proposition) is obverted to a universal affirmative ("A" proposition).
"No S are P" and "All S are non-P"
"No cats are friendly" and "All cats are non-friendly"
In the particular affirmative the quantity of the subject term remains unchanged, but the predicate term of the inferred proposition negates the complement of the predicate term of the original proposition. The particular affirmative ("I" proposition) is obverted to a particular negative ("O" proposition).
"Some S are P" and "Some S are not non-P"
"Some animals are friendly creatures" and "Some animals are not unfriendly creatures."
In the obversion of a particular negative to a particular affirmative the quantity of the subject also remains unchanged, and the predicate term is changed from simple negation to a term of the complementary class. The particular negative ("O") proposition is obverted to a particular affirmative ("I" proposition).
"Some S are not P" and "Some S are non-P"
"Some animals are not friendly creatures" and "Some animals are unfriendly creatures."
Note that the truth-value of an original statement is preserved in its resulting obverse form. Because of this, obversion can be used to determine the immediate inferences of all categorical propositions, regardless of quality or quantity.
In addition, obversion allows us to navigate through the traditional square of logical opposition
Square of opposition
In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
by providing a means for us to proceed from "A" Propositions to "E" Propositions, as well as from "I" Propositions to "O" Propositions, and vice versa. However, it must be noted that although the resulting propositions from obversion are logically equivalent
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
to the original statements in terms of truth-value, they are not semantically equivalent to their original statements in their standard form.
See also
- AristotleAristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
- ContrapositionContrapositionIn traditional logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality . For its symbolic expression in modern logic see the rule...
- Conversion (logic)
- InferenceInferenceInference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
- SyllogismSyllogismA syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
- Term logicTerm logicIn philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
- Transposition (logic)Transposition (logic)In the methods of deductive reasoning in classical logic, transposition is the rule of inference that permits one to infer from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. Its symbolic expression is:...