Law of noncontradiction
In logic, the Law of noncontradiction (LNC; also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that for any given proposition, the proposition and its negation cannot both be simultaneously true, e.g., the proposition “the house is white” and its negation “the house is not white” are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one of two propositions like “the house is white” and “the house is not white” holds.
One reason to have this law is the principle of explosion, which states that anything follows from a contradiction. The law is employed in a reductio ad absurdum proof.
To express the fact that the law is tenseless and to avoid equivocation, sometimes the law is amended to say “contradictory propositions cannot both be true ‘at the same time and in the same sense’”.
It is one of the so called three laws of thought, along with its complement, the law of excluded middle, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or De Morgan’s laws.
The law of non-contradiction and the law of excluded middle create a dichotomy in a so-called logical space, the points in which are all the consistent combinations of propositions. Each combination would contain exactly one member of each pair of contradictory propositions, so the space would have two parts which are mutually exclusive and jointly exhaustive. The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle is an expression of its jointly exhaustive aspect.
The Principle of Non-Contradiction (PNC) is a fundamental principle in Aristotelian logic and philosophy, stating that contradictory propositions cannot both be true in the same sense at the same time. In simpler terms, it means that something cannot both be and not be at the same time and in the same respect. This principle is considered a foundational law of thought and a cornerstone of scientific inquiry, reasoning, and communication.
Here’s a more detailed explanation:
- Ontological PNC: This aspect of the principle deals with the nature of being and reality. It asserts that a thing cannot possess a characteristic and lack that same characteristic simultaneously. For example, a glass of water cannot be both hot and cold at the same time and in the same way.
- Logical PNC: In logic, the PNC states that a proposition and its negation cannot both be true at the same time. For instance, the statements “The sky is blue” and “The sky is not blue” cannot both be true simultaneously.
- Psychological PNC: This aspect relates to the realm of belief and cognition. It suggests that one cannot hold contradictory beliefs at the same time.
- Significance: The PNC is not just a logical rule; it’s considered by Aristotle to be a fundamental principle that underpins our ability to understand and interact with the world. Without it, Aristotle argued, meaningful discourse, reasoning, and even knowledge would be impossible.
- Debate and Challenges: While the PNC is widely accepted, it has faced challenges and been the subject of philosophical debate. Some have questioned its application in certain contexts, particularly in areas like quantum mechanics or paraconsistent logic, which allow for contradictions. However, such challenges often involve reinterpreting or reformulating the principle rather than outright rejecting it, according to the Stanford Encyclopedia of Philosophy.