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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.
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 negated counterparts:
An early occurrence of proof by contradiction can be found in Euclid's Elements, Book 1, Proposition 6: [7] If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. The proof proceeds by assuming that the opposite sides are not equal, and derives a contradiction.
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as the type of object that declarative sentences denote. For instance, the sentence "The sky is blue" denotes the proposition that the ...
The principle that for any proposition, either that proposition is true or its negation is true, with no middle ground. exclusion negation In three-valued logic, form of negation that strictly excludes the possibility of something being true, as opposed to constructive negation which asserts the truth of an opposite proposition. [125] [126]
In modern formal logic and type theory, the term is mainly used instead for a single proposition, often denoted by the falsum symbol ; a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition).
In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form , the inverse refers to the sentence .
The converse, which also appears in Euclid's Elements (Book I, Proposition 48), can be stated as: Given a triangle with sides of length , , and , if + =, then the angle opposite the side of length is a right angle.