In traditional logic, a contradiction occurs when a statement conflicts with itself or with known facts. It is often used to identify dishonest ideas or unfair opinions. This shows a general pattern in how logic is applied. The law of noncontradiction explains that "a thing cannot be both true and false at the same time in the same way."

In modern formal logic and type theory, the term "contradiction" usually refers to a single statement that is always false. This is often shown using the symbol ⊥ (called "falsum"). A statement is a contradiction if it leads to a false conclusion based on the rules of logic. This idea can also apply to a group of statements, which is said to "contain" a contradiction if at least one statement in the group is false.

History

In Plato's Euthydemus dialogue, a paradox is used to show the importance of understanding opposite ideas. During the conversation, Dionysodorus claims that "there is no such thing as false opinion" and "no such thing as ignorance." He challenges Socrates to "refute me." Socrates replies with a question: "But how can I refute you, if, as you say, it is impossible to tell a falsehood?"

In formal logic

In classical logic, a statement is a contradiction if it leads to an impossible situation. This means that if a statement is false, then any other statement can be proven from it. This idea is called the "principle of explosion" or "ex falso quodlibet," which means "from falsity, anything follows." In a complete logic, a statement is contradictory if it cannot be true in any situation.

If a set of rules (called premises) is consistent and a statement is true under those rules, then combining the rules with the opposite of the statement leads to a contradiction. This method of proving a statement by showing that its opposite creates a contradiction is called "proof by contradiction." This technique is widely used in mathematics, especially when proving statements that are hard to verify directly, such as showing that the square root of 2 is not a fraction.

In minimal logic, which is similar to classical logic but avoids certain rules like proof by contradiction, different principles can be added to create new systems. For example:
1. Double-negation elimination allows removing double negatives (e.g., "not not A" becomes "A"). Adding this to minimal logic creates classical logic.
2. Ex falso quodlibet allows proving any statement from a contradiction. Adding this to minimal logic creates intuitionistic logic.
3. Peirce's rule is a way to prove statements without directly using contradictions. Adding it with other rules creates classical logic.
4. Gödel-Dummett axiom introduces a rule about ordering truth values. Adding this to minimal logic creates Gödel-Dummett logic.
5. Law of the excluded middle states that every statement is either true or false. Adding this to minimal logic with other rules creates classical logic.
6. Weak law of the excluded middle allows some classical-like reasoning but avoids certain contradictions.

In mathematics, symbols like ↯, ⊥, and others are used to show contradictions. These symbols can also represent "false," as in Boolean algebra. After a contradiction is shown, mathematicians often write "Q.E.D." to signal the end of a proof.

A consistency proof requires two things:
1. A set of basic rules (axioms).
2. A demonstration that both a statement and its opposite cannot be proven from those rules.

Consistency proofs depend on the idea of contradiction, which must be defined outside the system itself. Emil Post, in 1921, showed that proving the consistency of logic systems requires careful definitions of truth values. He used classes (K₁ and K₂) to organize formulas and prove that only statements that are always true (tautologies) can be derived from consistent axioms. A tautology is a statement that is always true, no matter how its parts are interpreted.

Philosophy

People who follow the theory of coherentism usually say that for a belief to be justified, it must be part of a system of beliefs that do not contradict each other. Some thinkers called dialetheists, such as Graham Priest, argue that coherence might not always require complete agreement between beliefs.

A pragmatic contradiction happens when the way an argument is presented directly opposes the point it is trying to make. In this case, the contradiction comes from how the argument is expressed, not from the actual ideas being discussed.

In dialectical materialism, which is based on ideas from Hegelianism, contradiction refers to a conflict that naturally exists within a single area, force, or object. Unlike metaphysical thinking, which sees contradictions as impossible, dialectical materialism views these opposing forces as real and existing in the world. These forces do not cancel each other out but instead help define each other's existence. According to Marxist theory, an example of this can be found in the tension between different aspects of society, such as:

Hegelian and Marxist theories suggest that the natural progression of history will eventually resolve contradictions through a process called sublation, or synthesis. Marx believed that this process would lead capitalism to change into a socialist society, where the tools and resources used to create goods and services would be shared equally among all workers. This would resolve the earlier conflict between two opposing ideas.

Outside formal logic

In everyday language, people may say that two actions or statements are opposite when they are based on assumptions that conflict logically. In mathematics, a method called proof by contradiction is used to create proofs by showing that assuming the opposite of what is being proven leads to a logical inconsistency.

China's contradictions

The word “矛盾” (contradiction) comes from a story in the “Nan I” chapter of Han Feizi. It describes a man from the State of Chu who sold a spear that could pierce any shield and a shield that could stop any spear. When a customer asked what would happen if the spear were used against the shield, the man could not answer. If the spear pierced the shield, the shield’s claim to be unbreakable would be wrong. If the spear could not pierce the shield, the spear’s claim to be unstoppable would be wrong. This situation shows that the seller’s statements cannot both be true at the same time.

Later, Han Fei used the term “矛盾” in a story to challenge Confucian ideas, especially the belief that rulers like Yao and Shun were perfect. Confucianism teaches that Yao, a wise ruler, gave his throne to Shun because Shun fixed problems and helped the people.

However, Han Fei argued that if Yao had been a great ruler, Shun would not have needed to improve things. If Shun’s changes were necessary, that would mean Yao’s rule was not perfect. This made the idea that both Yao and Shun were flawless and ruled perfectly impossible. By using the metaphor of “contradiction,” Han Fei criticized Confucian teachings and supported his own Legalist ideas, which focus on laws and rules rather than moral virtues.