In
traditional logic
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, th ...
, obversion is a "type of
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", the immediate inference can be made that "no toads are not green" or "no toads are non-green" ...
in which from a given
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
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
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
is changed but the
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false'').
Computing
In some progr ...
is the same 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
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
and a universal negative proposition the
subject term and the
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
term are both replaced by their
negated counterparts:
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
The cat (''Felis catus'') is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of ...
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
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
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 to proceed from "A" Propositions to "E" Propositions, as well as from "I" Propositions to "O" Propositions, and vice versa. However, although the resulting propositions from obversion are
logically equivalent
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
to the original statements in terms of truth-value, they are not semantically equivalent to their original statements in their standard form.
Proof that the truth-value of the original statement is preserved by an obversion operation [{{Cite web, url=https://learningpundits.com/module-view/68-syllogisms/1-logical-reasoning-tips---syllogisms/, title = Syllogism: Syllogism Meaning, Syllogism Questions, Tricks]
Consider all possible relationships between the Subject (S) and the Predicate (P) represented using sets:
Case 1: S = P (S and P perfectly overlap)
Case 2: S is a subset of P
Case 3: P is a subset of S
Case 4: S and P are two overlapping sets
Case 5: S and P are disjoint sets
Case 6: S is the universe with P being a subset of S
Case 7: P is the universe with S being a subset of P
Validity of statements after Obversion:
The obversion operation is performed by changing the quality of the statement and replacing the predicate with its complement.
1. Statement: All S are P (Applicable for Case 1, 2, and 7)
Obverse: No S are non-P
Validity: YES
2. Statement: No S are P (Applicable for Case 5)
Obverse: All S are non-P
Validity: YES
3. Statement: Some S are P (Applicable for Case 1, 2, 3, 4, 6 and 7)
Obverse: Some S are not non-P
Validity: YES
4. Statement: Some S are not P (Applicable for Case 3, 4, 5 and 6)
Obverse: Some S are non-P
Validity: YES
See also
*
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
*
Categorical proposition#Obversion
*
Contraposition
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The cont ...
*
Conversion (logic)
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
*
Inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
*
Syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
...
*
Term logic
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, th ...
*
Transposition (logic)
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the tru ...
Footnotes
Bibliography
*Brody, Bobuch A. "Glossary of Logical Terms". Encyclopedia of Philosophy. Vol. 5–6. Macmillan, 1973.
*Copi, Irving. ''Introduction to Logic''. MacMillan, 1953.
*Copi, Irving. ''Symbolic Logic''. MacMillan, 1979, fifth edition.
*
Stebbing, Susan. ''A Modern Introduction to Logic''. Cromwell Company, 1931.
Immediate inference