Kara Sears
The traditional laws of thought are the three fundamental laws of logic: the law of contradiction, the law of excluded middle (or third), and the principle of identity. The laws of thought have traditionally been conceived of as descriptive, prescriptive, or formal. The expression laws of thought gained prominence through its use by George Boole (1815-1864) in his 1854 book, 'An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'. Boole was a professor of mathematics at Queen's College, Cork, now University College Cork, in Ireland.
| Characteristics | Values |
|---|---|
| Name | George Boole |
| Birth and Death Year | 1815-1864 |
| Occupation | Professor of Mathematics at Queen's College, Cork (now University College Cork, Ireland) |
| Publications | An Investigation of the Laws of Thought: on Which are Founded the Mathematical Theories of Logic and Probabilities (1854) |
| Other Notables | Gained prominence for his usage of the term "laws of thought" in his work; Endorsed Aristotle's logic |
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What You'll Learn
The three laws of thought
- The Law of Contradiction, or Non-Contradiction: This law states that for all propositions p, it is impossible for both p and not p to be true simultaneously. Symbolically, this can be represented as ∼(p · ∼p), where ∼ means "not." This law is found in ancient Indian logic and was also discussed by Socrates in Plato's Socratic dialogues. John Locke characterized this principle as "It is impossible for the same thing to be and not to be."
- The Law of Excluded Middle, or Third: This law is related to the principle of contradiction and asserts that two contradictory terms are mutually exclusive. In other words, if a statement is true, then its negation must be false, and vice versa. This law is also attributed to Aristotle and was later rejected by the Dutch mathematician L.E.J. Brouwer, who did not accept its use in certain mathematical proofs.
- The Law of Identity: This law states that anything that has been determined to be true must be identical to itself and different from other things. Symbolically, this can be expressed as 'X is X'. An example of this law in practice is the understanding that a non-venomous snake is not poisonous.
These three laws form the foundation of logical reasoning and have been influential in various fields, including philosophy, mathematics, and psychology. While they have been questioned and refined over time, they continue to serve as a reference for understanding and analyzing thought processes and logical arguments.
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The laws' philosophical origins
The laws of thought refer to the three fundamental laws of logic: the law of contradiction, the law of excluded middle (or third), and the principle of identity. These laws have been the subject of philosophical debate since Plato and have been discussed by ancient Indian logicians and several Western philosophers, including Aristotle, Euclid, Gottfried Wilhelm Leibniz, and George Boole.
In one of Plato's Socratic dialogues, Socrates described three principles derived from introspection: firstly, that nothing can become greater or less, either in number or magnitude, while remaining equal to itself; secondly, that without addition or subtraction, there is no increase or diminution, only equality; and thirdly, that what was not before cannot be afterward without becoming and having become. The law of non-contradiction, a fundamental principle in logic, is also found in ancient Indian logic, appearing in the Shrauta Sutras, the grammar of Pāṇini, and the Brahma Sutras attributed to Vyasa.
Aristotle, a pioneer of Western logic, discussed the laws of thought in his logical and metaphysical works. He viewed the laws as primarily descriptive of "being as such" and only secondarily as standards of correct thinking. Aristotle produced seven "proofs" to demonstrate the indispensability of the law of contradiction. He also formulated four propositional forms of logic, which Boole later reduced to formulas in the form of equations.
George Boole, an Irish professor of mathematics, published "An Investigation of the Laws of Thought: On Which are Founded the Mathematical Theories of Logic and Probabilities" in 1854. In this work, Boole endorsed Aristotle's logic but sought to provide it with mathematical foundations and extend its range of applications. Boole's system could handle multi-term propositions and arguments, whereas Aristotle's system was limited to two-termed subject-predicate propositions. Boole's work gained prominence and contributed to the development of modern Boolean algebra.
The laws of thought have traditionally been conceived of as descriptive, prescriptive, or formal. As descriptive laws, they describe the nature of "being as such," the subject matter common to all sciences, or the activity of thinking or reasoning. As prescriptive laws, they express absolute or conventional standards of correct thinking or reasoning. As formal laws, they are propositions that are true independently of their content, true in all possible worlds, or true of any objects, regardless of their existence.
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Boole's 'An Investigation of the Laws of Thought'
The laws of thought are fundamental axiomatic rules upon which rational discourse is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic.
George Boole's 1854 book, 'An Investigation of the Laws of Thought: on Which are Founded the Mathematical Theories of Logic and Probabilities', is the second of his two monographs on algebraic logic. Boole was a professor of mathematics at Queen's College, Cork, now University College Cork, in Ireland.
Boole's work fully accepted and endorsed Aristotle's logic, but with some differences. Firstly, Boole reduced the four propositional forms of Aristotle's logic to formulas in the form of equations. Secondly, Boole's addition of equation solving to logic involved the doctrine that Aristotle's rules of inference (the “perfect syllogisms”) must be supplemented by rules for equation solving. Thirdly, Boole's system could handle multi-term propositions and arguments, whereas Aristotle could only handle two-term subject-predicate propositions and arguments.
Boole's algebra differs from modern Boolean algebra. In Boole's algebra, A+B cannot be interpreted by set union, due to the permissibility of uninterpretable terms in his calculus. In Boole's account of his algebra, terms are reasoned about equationally, without a systematic interpretation being assigned to them.
Boole's work is considered to have outlined the laws that govern rational human intelligence in the brain. His eight propositions can all be expressed mathematically as either 'affirmative' or 'negative'.
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Logic and mathematics
The traditional "laws of thought" are often associated with the field of mathematics. George Boole, a professor of mathematics at Queen's College, Cork (now University College Cork, Ireland), is credited with popularising the expression "laws of thought". He used the term to refer to the theorems in his "algebra of logic", presented in his 1854 book, "An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities".
Boole's work laid the foundation for the discipline of algebraic logic and significantly influenced the development of modern Boolean algebra. However, it is important to note that Boole's algebra differs from modern Boolean algebra in certain interpretations and applications.
The historian of logic, John Corcoran, noted that Boole's work built upon and expanded Aristotle's logic. Aristotle, the famous Greek philosopher, is often regarded as the originator of the fundamental laws of logic, which include the principles of identity, non-contradiction, and excluded middle. These laws were considered necessary conditions for rational thinking and remained influential until the beginning of the 20th century.
In the early 20th century, Bertrand Russell and Alfred North Whitehead's seminal work, "Principia Mathematica", further explored the intersection of logic and mathematics. Russell defined Symbolic Logic or Formal Logic as "the study of the various general types of deduction". He also considered the inclusion of his principle of induction alongside the logical principles that encompass the "Laws of Thought".
Another notable contributor to the mathematical aspects of the "Laws of Thought" is Alfred Tarski, who, in his 1946 book, "Introduction to Logic and to the Methodology of the Deductive Sciences", presented a comprehensive list of universal laws and rules of sentential calculus, including three rules of inference and a fundamental law of identity.
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Aristotle's influence
The law of contradiction, also known as the principle of non-contradiction, states that it is impossible for something to be true and false at the same time. Aristotle provided seven "proofs" to demonstrate the indispensability of this law. The law of excluded middle, or the principle of excluded third, states that between two contradictories, one must be true, and the other must be false, excluding any third option. The principle of identity asserts that a thing is identical with itself, or in other words, a subject is equal to itself.
Aristotle's logic has been widely accepted and endorsed by later philosophers and mathematicians. George Boole, a 19th-century professor of mathematics, built upon Aristotle's logic by providing it with mathematical foundations and expanding its range of applications. Boole agreed with Aristotle's principles but also sought to extend and refine them. Boole reduced Aristotle's four propositional forms of logic to formulas, added equation solving to logic, and developed a system that could handle multi-term propositions and arguments, going beyond Aristotle's two-termed subject-predicate propositions.
Overall, Aristotle's influence on the laws of thought is significant. His formulation of the three fundamental laws of logic has shaped philosophical and mathematical thinking for centuries, and his ideas continue to be a source of inspiration and debate for modern scholars.
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Frequently asked questions
The laws of thought have been conceived of as descriptive, prescriptive, or formal, and are fundamental axiomatic rules upon which rational discourse is based.
The three laws of thought are the law of contradiction, the law of excluded middle (or third), and the principle of identity.
The laws of thought were frequently discussed from the time of the Greeks until the beginning of the 20th century. The term gained prominence through its use by George Boole (1815-1864) in his 1854 book, 'An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'.
The laws of thought are considered necessary conditions for thought and are still studied in philosophy and logic.
More recent developments in logic, such as intuitionistic logic, dialetheism, and fuzzy logic, have questioned or rejected the classical ideas of the laws of thought.
