Contraposition
Encyclopedia
In 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 (affirmation or negation). For its symbolic expression in modern logic see the rule of transposition
. Contraposition also has distinctive applications in its philosophical application distinct from the other traditional inference processes of conversion and obversion
where equivocation varies with different proposition types.
the process of contraposition is a schema composed of several steps of inference involving categorical proposition
s and classes
. A categorical proposition contains a subject
and predicate
where the existential impact of the copula implies the proposition as referring to a class with at least one member, in contrast to the conditional form of hypothetical
or materially implicative
propositions, which are compounds of other propositions, e.g. If P, then Q, where P and Q are both propositions, and their existential impact is dependent upon further propositions where in quantification existence is instantiated (exististential instantiation).
Conversion by contraposition is the simultaneous interchange and negation
of the subject and predicate, and is valid only for the type "A" and type "O" propositions of Aristotelian logic
, with considerations for the validity an "E" type proposition with limitations and changes in quantity. This is considered full contraposition. Since in the process of contraposition the obverse
can be obtained in all four types of traditional propositions, yielding propositions with the contradictory of the original predicate, contraposition is first obtained by converting the obvert of the original proposition. Thus, partial contraposition can be obtained conditionally in an "E" type proposition with a change in quantity. Because nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it can be either the original subject, or its contradictory, resulting in two contrapositives which are the obverts of one another in the "A", "O", and "E" type propositions.
By example: from an original, 'A' type categorical proposition,
which presupposes that all classes have members and the existential import presumed in the form of categorical propositions, one can derive first by obversion
the 'E' type proposition,
The contrapositive of the original proposition is then derived by conversion to another 'E' type proposition,
The process is completed by further obversion resulting in the 'A' type proposition that is the obverted contrapositive of the original proposition,
The schema of contraposition:
Notice that contraposition is a valid form of immediate inference only when applied to "A" and "O" propositions. It is not valid for "I" propositions, where the obverse is an "O" proposition which has no converse. The contraposition of the "E" proposition is valid only with limitations (per accidens). This is because the obverse of the "E" proposition is an "A" proposition which cannot be validly converted except by limitation, that is, contraposition plus a change in the quantity of the proposition from universal
to particular
.
Also, notice that contraposition is a method of inference which may require the use of other rules of inference. The contrapositive is the product of the method of contraposition, with different outcomes depending upon whether the contraposition is full, or partial. The successive applications of conversion and obversion within the process of contraposition may be given by a variety of names.
The process of the logical equivalence
of a statement and its contrapositive as defined in traditional class logic is not one of the axioms of propositional logic. In traditional logic there is more than one contrapositive inferred from each original statement. In regard to the "A" proposition this is circumvented in the symbolism of modern logic by the rule of transposition
, or the law of contraposition. In its technical usage within the field of philosophic logic, the term "contraposition" may be limited by logicians (e.g. Irving Copi
, Susan Stebbing
) to traditional logic and categorical propositions. In this sense the use the term "contraposition" is usually referred to by "transposition" when applied to hypothetical propositions or material implications.
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 is inferred having for its subject
Subject (philosophy)
In philosophy, a subject is a being that has subjective experiences, subjective consciousness or a relationship with another entity . A subject is an observer and an object is a thing observed...
the contradictory of the original 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...
, and in some cases involving a change of quality (affirmation or negation). For its symbolic expression in modern logic see the rule of transposition
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:...
. Contraposition also has distinctive applications in its philosophical application distinct from the other traditional inference processes of conversion and obversion
Obversion
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...
where equivocation varies with different proposition types.
Traditional logic
In traditional logicTerm 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...
the process of contraposition is a schema composed of several steps of inference involving 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....
s and classes
Class (philosophy)
Philosophers sometimes distinguish classes from types and kinds. We can talk about the class of human beings, just as we can talk about the type , human being, or humanity...
. A categorical proposition contains a 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...
and 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...
where the existential impact of the copula implies the proposition as referring to a class with at least one member, in contrast to the conditional form of hypothetical
Hypothesis
A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...
or materially implicative
Material conditional
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...
propositions, which are compounds of other propositions, e.g. If P, then Q, where P and Q are both propositions, and their existential impact is dependent upon further propositions where in quantification existence is instantiated (exististential instantiation).
Conversion by contraposition is the simultaneous interchange and negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
of the subject and predicate, and is valid only for the type "A" and type "O" propositions of Aristotelian logic
Organon
The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
, with considerations for the validity an "E" type proposition with limitations and changes in quantity. This is considered full contraposition. Since in the process of contraposition the obverse
Obversion
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...
can be obtained in all four types of traditional propositions, yielding propositions with the contradictory of the original predicate, contraposition is first obtained by converting the obvert of the original proposition. Thus, partial contraposition can be obtained conditionally in an "E" type proposition with a change in quantity. Because nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it can be either the original subject, or its contradictory, resulting in two contrapositives which are the obverts of one another in the "A", "O", and "E" type propositions.
By example: from an original, 'A' type categorical proposition,
- All residents are voters,
which presupposes that all classes have members and the existential import presumed in the form of categorical propositions, one can derive first by obversion
Obversion
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...
the 'E' type proposition,
- No residents are non-voters.
The contrapositive of the original proposition is then derived by conversion to another 'E' type proposition,
- No non-voters are residents.
The process is completed by further obversion resulting in the 'A' type proposition that is the obverted contrapositive of the original proposition,
- All non-voters are non-residents.
The schema of contraposition:
Notice that contraposition is a valid form of immediate inference only when applied to "A" and "O" propositions. It is not valid for "I" propositions, where the obverse is an "O" proposition which has no converse. The contraposition of the "E" proposition is valid only with limitations (per accidens). This is because the obverse of the "E" proposition is an "A" proposition which cannot be validly converted except by limitation, that is, contraposition plus a change in the quantity of the proposition from universal
Universal (metaphysics)
In metaphysics, a universal is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For example, suppose there are two chairs in a room, each of...
to particular
Particular
In philosophy, particulars are concrete entities existing in space and time as opposed to abstractions. There are, however, theories of abstract particulars or tropes. For example, Socrates is a particular...
.
Also, notice that contraposition is a method of inference which may require the use of other rules of inference. The contrapositive is the product of the method of contraposition, with different outcomes depending upon whether the contraposition is full, or partial. The successive applications of conversion and obversion within the process of contraposition may be given by a variety of names.
The process of the logical equivalence
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...
of a statement and its contrapositive as defined in traditional class logic is not one of the axioms of propositional logic. In traditional logic there is more than one contrapositive inferred from each original statement. In regard to the "A" proposition this is circumvented in the symbolism of modern logic by the rule of transposition
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:...
, or the law of contraposition. In its technical usage within the field of philosophic logic, the term "contraposition" may be limited by logicians (e.g. Irving Copi
Irving Copi
Irving Marmer Copi was an American philosopher, logician, and university textbook author....
, Susan Stebbing
Susan Stebbing
L. Susan Stebbing was a British philosopher. She belonged to the 1930s generation of analytic philosophy, and was a founder in 1933 of the journal Analysis.-Biography:...
) to traditional logic and categorical propositions. In this sense the use the term "contraposition" is usually referred to by "transposition" when applied to hypothetical propositions or material implications.
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...
- ObversionObversionIn 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...
- OrganonOrganonThe Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
- Propositional calculusPropositional calculusIn mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
- 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:...