section_path: Related concepts and theories/Laws of thought
section_name: Laws of thought
paragraph_text: Laws of thought
  
  Traditionally, the term " laws of thought " refers to three fundamental laws of logic: the law of contradiction, the law of excluded middle, and the principle of identity. These laws by themselves are not sufficient as axioms of logic but they can be seen as important precursors to the modern axiomatization of logic. The law of contradiction states that for any proposition, it is impossible that both it and its negation are true: ¬ (p ∧ ¬ p) {\displaystyle \lnot (p\land \lnot p)}. According to the law of excluded middle, for any proposition, either it or its opposite is true: p ∨ ¬ p {\displaystyle p\lor \lnot p}. The principle of identity asserts that any object is identical to itself: ∀ x (x = x) {\displaystyle \forall x(x=x)}. There are different conceptions of how the laws of thought are to be understood. The interpretations most relevant to thinking are to understand them as prescriptive laws of how one should think or as formal laws of propositions that are true only because of their form and independent of their content or context. Metaphysical interpretations, on the other hand, see them as expressing the nature of "being as such".
  
  While there is a very wide acceptance of these three laws among logicians, they are not universally accepted. Aristotle, for example, held that there are some cases in which the law of excluded middle is false. This concerns primarily uncertain future events. On his view, it is currently "not... either true or false that there will be a naval battle tomorrow". Modern intuitionist logic also rejects the law of excluded middle. This rejection is based on the idea that mathematical truth depends on verification through a proof. The law fails for cases where no such proof is possible, which exist in every sufficiently strong formal system, according to Gödel's incompleteness theorems. Dialetheists, on the other hand, reject the law of contradiction by holding that some propositions are both true and false. One motivation of this position is to avoid certain paradoxes in classical logic and set theory, like the liar's paradox and Russell's paradox. One of its problems is to find a formulation that circumvents the principle of explosion, i.e. that anything follows from a contradiction.
  
  Some formulations of the laws of thought include a fourth law: the principle of sufficient reason. It states that everything has a sufficient reason, ground, or cause. It is closely connected to the idea that everything is intelligible or can be explained in reference to its sufficient reason. According to this idea, there should always be a full explanation, at least in principle, to questions like why the sky is blue or why World War II happened. One problem for including this principle among the laws of thought is that it is a metaphysical principle, unlike the other three laws, which pertain primarily to logic.