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:


 * (P → Q) ↔ (~Q →  ~P)

The "→" is the symbol for material implication and the doubleheaded arrow "↔" indicates a biconditional relationship. The symbol "~" indicates negation. "P" and "Q" are components representing statements that form a truth functional compound proposition, where in a hypothetic proposition the first statement will be the antecedent and the last statement will be the consequent. The expression "truth function" has distinctive applications in philosophical logic and mathematical logic. This article concerns its philosophical application. (See also Transposition (mathematics))

Form of transposition
In the inferred proposition, the consequent is the contradictory of the antecedent in the original proposition, and the antecedent of the inferred proposition is the contradictory of the consequent of the original proposition. The symbol for material implication signifies the proposition as a hypothetical, or the "if-then" form, e.g. "if P then Q".

The biconditional statement of the rule of transposition (↔) refers to the relation between hypothetical (→) propositions, with each proposition including an antecent and consequential term. As a matter of logical inference, to transpose, or convert the terms of one proposition requires the conversion of the terms of the propositions on both sides of the biconditional relationship. Meaning, to transpose, or convert (P → Q) to (Q → P) requires that the other proposition, (~Q → ~P), be transposed or converted to (~P →  ~Q). Otherwise, to convert the terms of one proposition and not the other renders the rule invalid, violating the sufficient condition and necessary condition of the terms of the propositions, where the violation is that the changed proposition commits the fallacy of denying the antecedent or affirming the consequent by means of illicit conversion

The truth of the rule of transposition is dependent upon the relations of sufficient condition and necessary condition in logic.

Sufficient condition
In the proposition "If P then Q", the occurrence of 'P' is sufficient reason for the occurrence of 'Q'. 'P', as an individual or a class, materially implicates 'Q', but the relation of 'Q' to 'P', is such that the converse proposition "If Q then P" does not necessarily have sufficient condition. The rule of inference for sufficient condition is modus ponens, which is an argument for conditional implication:

Premise (1): If P, then Q

Premise (2): P

Conclusion: Therefore, Q

Necessary condition
Since the converse of premise (1) is not valid, all that can be stated of the relationship of 'P' and 'Q' is that in the absence of 'Q', 'P' does not occur, meaning that 'Q' is the necessary condition for 'P'. The rule of inference for necessary condition is modus tollens:

Premise (1): If P, then Q

Premise (2): not Q

Conclusion: Therefore, not P

Grammatically speaking
A grammatical example traditionally used by logicians contrasting sufficient and necessary conditions is the statement "If there is fire, then oxygen is present". An oxygenated environment is necessary for fire or combustion, but simply because there is an oxygenated environment does not necessarily mean that fire or combustion is occurring. While one can infer that fire stipulates the presence of oxygen, from the presence of oxygen the converse "If there is oxygen present, then fire is present" cannot be inferred. All that can be inferred from the original proposition is that "If oxygen is not present, then there cannot be fire".

Relationship of propositions
The symbol for the biconditional ("↔") signifies the relationship between the propositions is both necessary and sufficient, and is verbalized as "if and only if", or, according to the example "If P then Q 'if and only if' if not Q then not P".

Necessary and sufficient conditions can be explained by analogy in terms of the concepts and the rules of immediate inference of traditional logic. In the categorical proposition "All S is P", the subject term 'S' is said to be distributed, that is, all members of its class are exhausted in its expression. Conversely, the predicate term 'P' cannot be said to be distributed, or exhausted in its expression because it is indeterminate whether every instance of a member of 'P' as a class is also a member of 'S' as a class. All that can be validly inferred is that "Some P are S". Thus, the type 'A' proposition "All P is S" cannot be inferred by conversion from the original 'A' type proposition "All S is P". All that can be inferred is the type "A" proposition "All non-P is non-S" (Note that (P → Q) and (~Q → ~P) are both 'A' type propositions). Grammatically, one cannot infer "all mortals are men" from "All men are mortal". An 'A' type proposition can only be immediately inferred by conversion when both the subject and predicate are distributed, as in the inference "All bachelors are unmarried men" from "All unmarried men are bachelors".

Transposition and the method of contraposition
In traditional logic the reasoning process of transposition as a rule of inference is applied to categorical propositions through contraposition and obversion, a series of immediate inferences where the rule of obversion is first applied to the original categorical proposition "All S is P"; yielding the obverse "No S is non-P". In the obversion of the original proposition to an 'E' type proposition, both terms become distributed. The obverse is then converted, resulting in the contrapositive "No non-P is S", maintaining distribution of both terms. The contrapositive is again obverted, resulting in the obverted contrapositive "All non-P is non-S". Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory, and the predicate term of the resulting 'A' type proposition is again undistributed. This results in two contrapositives, one where the predicate term is distributed, and another where the predicate term is undistributed.

Differences between transposition and contraposition
Note that the method of transposition and contraposition should not be confused. Contraposition is a type of immediate inference in which from a given categorical proposition another categorical proposition is inferred which has as its subject the contradictory of the original predicate. Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory. This is in contradistinction to the form of the propositions of transposition, which may be material implication, or a hypothetical statement. The difference is that in its application to categorical propositions the results of contraposition has two contrapositives, each being the obvert of the other, i.e. "No non-P is S" and "All non-P is non-S". The distinction between the two contrapositives is absorbed and eliminated in the principle of transposition, which presupposes the "mediate inferences" of contraposition and is also referred to as the "law of contraposition".

Transposition in mathematical logic
See Transposition (mathematics), Set theory