language and those principles that must inhere naturally within them: The law of non-contradiction The law of non-contradiction (alternately the 'law of contradiction'): 'Nothing can both be and not be.' In other words: "two or more contradictory statements cannot both be true in the same sense at the same time": ¬(A∧¬A). In the words of Aristotle, that "one cannot say of something that it is and that it is not in the same respect and at the same time". As an illustration of this law, he wrote: The law of excluded middle The law of excluded middle: 'Everything must either be or not be.' In accordance with the law of excluded middle or excluded third, for every proposition, either its positive or negative form is true: A∨¬A. Regarding the law of excluded middle, Aristotle wrote: Rationale As the quotations from Hamilton above indicate, in particular the "law of identity" entry, the rationale for and expression of the "laws of thought" have been fertile ground for philosophic debate since Plato. Today the debate—about how we "come to know" the world of things and our thoughts—continues; for examples of rationales see the entries, below. Plato In one of Plato's Socratic dialogues, Socrates described three principles derived from introspection: Indian logic The law of non-contradiction is found in ancient Indian logic as a meta-rule in the Shrauta Sutras, the grammar of Pāṇini, and the Brahma Sutras attributed to Vyasa. It was later elaborated on by medieval commentators such as Madhvacharya. Locke John Locke claimed that the principles of identity and contradiction (i.e. the law of identity and the law of non-contradiction) were general ideas and only occurred to people after considerable abstract, philosophical thought. He characterized the principle of identity as "Whatsoever is, is." He stated the principle of contradiction as "It is impossible for the same thing to be and not to be." To Locke, these were not innate or a priori principles. Leibniz Gottfried Leibniz formulated two additional principles, either or both of which may sometimes be counted as a law of thought: principle of sufficient reason identity of indiscernibles In Leibniz's thought, as well as generally in the approach of rationalism, the latter two principles are regarded as clear and incontestable axioms. They were widely recognized in European thought of the 17th, 18th, and 19th centuries, although they were subject to greater debate in the 19th century. As turned out