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- The Three Traditional Laws
- Plato
- Indian Logic
- Locke
- Leibniz
- Schopenhauer
- Welton
- Russell
- Ladd-Franklin (1914): "Principle of Exclusion" and The "Principle of Exhaustion"
- Post (1921): The Propositional Calculus Is Consistent and Complete
History
Hamilton offers a history of the three traditional laws that begins with Plato, proceeds through Aristotle, and ends with the schoolmen of the Middle Ages; in addition he offers a fourth law (see entry below, under Hamilton): 1. "The principles of Contradiction and Excluded Middle can be traced back to Plato: The principles of Contradiction and of Excluded Middle can both be traced back to Plato, by whom they were enounced and frequently applied; though it was not till long after, that either...
Three traditional laws: identity, non-contradiction, excluded middle
The following states the three traditional "laws" in the words of Bertrand Russell (1912):
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.
In one of Plato's Socratic dialogues, Socrates described three principles derived from introspection:
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.
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 impo...
Gottfried Leibnizformulated two additional principles, either or both of which may sometimes be counted as a law of thought: 1. 1.1. principle of sufficient reason 1.2. 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 ...
Four laws
"The primary laws of thought, or the conditions of the thinkable, are four: – 1. The law of identity [A is A]. 2. The law of contradiction. 3. The law of exclusion; or excluded middle. 4. The law of sufficient reason." (Thomas Hughes, The Ideal Theory of Berkeley and the Real World, Part II, Section XV, Footnote, p. 38) Arthur Schopenhauer discussed the laws of thought and tried to demonstrate that they are the basis of reason. He listed them in the following way in his On the Fourfold Root o...
Two laws
Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. In the ninth chapter of the second volume of The World as Will and Representation, he wrote:
In the 19th century, the Aristotelian laws of thoughts, as well as sometimes the Leibnizian laws of thought, were standard material in logic textbooks, and J. Welton described them in this way:
The sequel to Bertrand Russell's 1903 "The Principles of Mathematics" became the three-volume work named Principia Mathematica (hereafter PM), written jointly with Alfred North Whitehead. Immediately after he and Whitehead published PMhe wrote his 1912 "The Problems of Philosophy". His "Problems" reflects "the central ideas of Russell's logic".
At about the same time (1912) that Russell and Whitehead were finishing the last volume of their Principia Mathematica, and the publishing of Russell's "The Problems of Philosophy" at least two logicians (Louis Couturat, Christine Ladd-Franklin) were asserting that two "laws" (principles) of contradiction" and "excluded middle" are necessary to spe...
As part of his PhD thesis "Introduction to a general theory of elementary propositions" Emil Post proved "the system of elementary propositions of Principia [PM]" i.e. its "propositional calculus" described by PM's first 8 "primitive propositions" to be consistent. The definition of "consistent" is this: that by means of the deductive "system" at h...
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