Law of thought

The laws of thought are fundamental logical rules, with a long tradition in the history of philosophy, which collectively prescribe how a rational mind must think. To break any of the laws of thought (for example, to contradict oneself) is to be irrational.

Plato
Socrates, in a Platonic dialogue, described three principles derived from introspection. He asserted that these three axioms contradict each other.

Aristotle
The three classic laws of thought are attributed to Aristotle and were foundational in scholastic logic. They are:


 * law of identity
 * law of noncontradiction
 * law of excluded middle

Locke
John Locke claimed that the principles of identity and contradiction were general ideas and only occurred to people after considerable abstract, philosophical thought. He characterized the principle of identity as "Whatsoever is, is." The principle of contradiction was stated 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
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 and 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 seventeenth, eighteenth, and (while subject to greater debate) nineteenth century. As turned out to be the case with another such (the so-called law of continuity), they involve matters which, in contemporary terms, are subject to much debate and analysis (respectively on determinism and extensionality). Leibniz's principles were particularly influential in German thought. In France the Port-Royal Logic was less swayed by them. Hegel quarrelled with the identity of indiscernibles in his Science of Logic (1812-1816).

Four Laws
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 of the Principle of Sufficient Reason, §33:

Also:
 * 1) A subject is equal to the sum of its predicates, or a = a.
 * 2) No predicate can be simultaneously attributed and denied to a subject, or a = &mdash; a = 0.
 * 3) Of every two contradictorily opposite predicates one must belong to every subject.
 * 4) Truth is the reference of a judgment to something outside it as its sufficient reason or ground.

To show that they are the foundation of reason, he gave the following explanation:

Schopenhauer's four laws can be schematically presented in the following manner:
 * 1) A is A.
 * 2) A is not not-A.
 * 3) A is either A or not-A.
 * 4) If A then B.

Two Laws
Later, in 1844, Schopenhauer claimed that the four laws of thought could be reduced to two. "It seems to me," he wrote in the second volume of The World as Will and Representation, Chapter 9, "that the doctrine of the laws of thought could be simplified by our setting up only two of them, namely, the law of the excluded middle, and that of sufficient reason or ground." Here is Law 1:

Law 2 is as follows:

He further asserted that "Insofar as a judgment satisfies the first law of thought, it is thinkable; insofar as it satisfies the second, it is true … ."

Boole
The title of George Boole's 1854 treatise on logic, An investigation on the Laws of Thought, indicates an alternate path. The laws are now incorporated into his boolean logic in which the classic Aristotelian laws come down to saying there are two and only two truth values. The Leibnizian principles are ignored, at the algebraic level, absent second-order logic.

Welton
In the 19th century the Aristotelian, and sometimes the Leibnizian, laws of thought were standard material in logic textbooks, and J. Welton described them in this way:

Russell
Bertrand Russell discussed only the three classic Aristotelian laws of thought in his 1912 book The Problems of Philosophy. At this point, in the early twentieth century, the laws of thought were sliding out of pedagogy in the field of logic, and the law of excluded middle was shortly to be questioned by intuitionistic logic.