Contradiction

Dialectic (Ancient Greek: διαλεκτική, romanized: dialektikḗ; German: Dialektik), also known as the dialectical method, refers originally to dialogue between people holding different points of view about a subject but wishing to arrive at the truth through reasoned argument. Dialectic resembles debate, but the concept excludes subjective elements such as emotional appeal and rhetoric; the object is more an eventual and commonly-held truth than the 'winning' of an (often binary) competition. It has its origins in ancient philosophy and continued to be developed in the Middle Ages.

Hegelianism refigured "dialectic" to no longer refer to a literal dialogue. Instead, the term takes on the specialized meaning of development by way of overcoming internal contradictions. Dialectical materialism, a theory advanced by Karl Marx and Friedrich Engels, adapted the Hegelian dialectic into a materialist theory of history. The legacy of Hegelian and Marxian dialectics has been criticized by philosophers, such as Karl Popper and Mario Bunge, who considered it unscientific.

Dialectical materialism is a materialist theory based upon the writings of Karl Marx and Friedrich Engels that has found widespread applications in a variety of philosophical disciplines ranging from philosophy of history to philosophy of science. As a materialist philosophy, Marxist dialectics emphasizes the importance of real-world conditions and the presence of contradictions within and among social relations, such as social class, labour economics, and socioeconomic interactions. Within Marxism, a contradiction is a relationship in which two forces oppose each other, leading to mutual development.

The first law of dialectics is about “the unity and conflict of opposites”. It explains that all things are made up of opposing forces, not purely "good" nor purely "bad", but that everything contains internal contradictions at varying levels of aspects we might call "good" or "bad", depending on the conditions and perspective. An example of this unity and conflict is the negative and positive particles that make up atoms.

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory ${\displaystyle T}$ is consistent if there is no formula ${\displaystyle \varphi }$ such that both ${\displaystyle \varphi }$ and its negation ${\displaystyle \lnot \varphi }$ are elements of the set of consequences of ${\displaystyle T}$. Let ${\displaystyle A}$ be a set of closed sentences (informally "axioms") and ${\displaystyle \langle A\rangle }$ the set of closed sentences provable from ${\displaystyle A}$ under some (specified, possibly implicitly) formal deductive system. The set of axioms ${\displaystyle A}$ is consistent when there is no formula ${\displaystyle \varphi }$ such that ${\displaystyle \varphi \in \langle A\rangle }$ and ${\displaystyle \lnot \varphi \in \langle A\rangle }$. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

In a sound formal system, every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete. The completeness of the propositional calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.

A self-refuting idea or self-defeating idea is an idea or statement whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are therefore almost always controversial, with defenders stating that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below are unambiguously or incontrovertibly self-refuting. These ideas are often used as axioms, which are definitions taken to be true (tautological assumptions), and cannot be used to test themselves, for doing so would lead to only two consequences: consistency (circular reasoning) or exception (self-contradiction).

Infinite regress is a philosophical concept to describe a series of entities. Each entity in the series depends on its predecessor, following a recursive principle. For example, the epistemic regress is a series of beliefs in which the justification of each belief depends on the justification of the belief that comes before it.

An infinite regress argument is an argument against a theory based on the fact that this theory leads to an infinite regress. For such an argument to be successful, it must demonstrate not just that the theory in question entails an infinite regress but also that this regress is vicious. There are different ways in which a regress can be vicious. The most serious form of viciousness involves a contradiction in the form of metaphysical impossibility. Other forms occur when the infinite regress is responsible for the theory in question being implausible or for its failure to solve the problem it was formulated to solve.

This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.