The principle of contradiction or the principle of excluded contradiction says that two statements contradicting each other in the same respect cannot apply at the same time. In the course of the history of philosophy and science and from different theoretical points of view, the principle of contradiction has been related to different kinds of opposites and has been understood in different ways as an ontological , epistemological or logical principle.

The principle of contradiction must be distinguished from the principle of excluded third parties and the principle of bivalence .

## logic

In logic, the proposition of contradiction is often, in modern formal logic always, related to a statement and its proposition negation. Here the sentence says that a statement cannot apply simultaneously with its opposite (its sentence negation). So it is not possible, for example, that the earth is also a disk and that it is not the case that the earth is a disk.

In propositional logic , this sentence is represented by the formula

${\ displaystyle \ neg (A \ wedge \ neg A)}$
Literally: It is not the case ("¬") that the statement 'A' applies and ("∧") that the statement 'not ("¬") A' applies.

expressed.

The principle of contradiction is a basic principle of classical logic .

This sentence is also accepted in many non-classical logical systems and can be derived. However, there are also logical systems in which the principle of contradiction does not apply.

## philosophy

In philosophy , the principle of contradiction (also called the principle of contradiction or non-contradiction principle) is one of the most important statements of epistemology and traditional logic, where it is considered one of the laws of thought ; sometimes it is also viewed as an ontological principle. Aristotle formulates in his metaphysics :

“But the most certain principle of all is that in which a deception is impossible [...] But what that is, we want to state: Because it is impossible that the same thing should and should not happen to the same thing in the same relationship. [...] But we have just assumed that it is impossible for something to be and not to be at the same time. "

- Aristotle : Metaphysics 1005b

## theology

In the encyclical Fides et ratio by Pope John Paul II , the sentence of contradiction under the designation "Principle of non-contradiction" is included in the core of philosophical knowledge that is constantly present in the history of thought. This core represents something like a spiritual legacy of humanity. These core components of an "implicit philosophy" would - albeit possibly in an indistinct, unreflective form - shared by everyone and, in the opinion of the Pope, should represent a point of reference for the various philosophical schools.

## Discussion of the sentence

The principle of contradiction is counted - especially by the realists - among the evidences . These are the first truths or basic truths on which all other truths are built or which are implied by every other individual truth. The principle of contradiction shows the unprovability and irrefutability of the evidence particularly clearly. Any attempt to prove or disprove it would always presuppose it, because every statement or argument should convey itself and not its opposite.

The principle of contradiction also applies to Immanuel Kant's analytical judgments . The statement: “A body is extended” is “analytical” because the concept of the extended is already contained in the concept of the body. Analytical knowledge is a pure dissection of knowledge or concepts. In contrast, there are synthetic judgments, in which the content of a concept or knowledge is expanded, e.g. For example: “The body is red.” Here too, however, the principle of contradiction applies, because the body is not red and is not red at the same time and in the same relationship. The analytical truth is ultimately based on the principle of contradiction ( Ernst Tugendhat ).

In contrast to the principle of the excluded third party , the principle of contradiction also applies in intuitionist logics. In general, however, the sentence is not suitable for discriminating against paraconsistent logics, since these are often not completely agnostic towards contradictions.

Georg Wilhelm Friedrich Hegel tried to enforce the dialectic he advocated (in the sense of a doctrine of the opposites in things), to expand the logic in a way (so-called dialectical logic) that invalidates the principle of contradiction. In doing so, Hegel tried to circumvent Kant's refutation of so-called 'dogmatism in metaphysics'. The scientific theorist Karl Popper comments: “Hegel regards this refutation [Kant] as valid only for systems that are metaphysical in its narrower sense, but not for dialectical rationalism, which takes the development of reason into account and therefore does not need to fear contradictions. By avoiding Kant's criticism in this way, Hegel throws himself into an extremely dangerous adventure which must lead to catastrophe; for he argues something like this: 'Kant refuted rationalism by stating that it must lead to contradictions. I admit that. But it is clear that this argument derives its strength from the law of contradiction: it refutes only those systems that accept this law, that is, those that intend to be free of contradictions. The argument is not dangerous for a system like mine that is ready to accept contradictions - that is, for a dialectical system. ' There is no doubt that Hegel's argument establishes a dogmatism of an extremely dangerous kind - a dogmatism that need not fear any attack [see immunization strategy ]. Because every attack, every criticism of any theory must be based on the method of showing any contradictions, either in a theory itself or between a theory and some facts [...]. "