Abstract
Logic is concerned with reasoning and with establishing the validity of arguments. It allows conclusions to be deduced from premises according to logical rules, and the logical argument establishes the truth of the conclusion provided that the premises are true. The origins of logic are with the Greeks who were interested in the nature of truth. Aristotle developed syllogistic logic, where a syllogism consists of two premises and a conclusion. The Stoics developed an early form of propositional logic, where the assertibles (propositions) have a truth-value such that at any time they are either true or false. Boole’s symbolic logic and its application to digital computing are discussed, and we consider Frege’s work on predicate logic.
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Notes
- 1.
The origin of the word Stoic is from the Stoa Poikile (Στοα Ποιλικη), which was a covered walkway in the Agora of Athens. Zeno taught his philosophy in a public space at this location, and his followers became known as Stoics.
- 2.
De Morgan was a nineteenth British mathematician based at University College London. De Morgan’s laws in Set Theory and Logic state that: (A ∪ B)c = Ac ∩ Bc and ¬ (A ∨ B) ≡ ¬A ∧ ¬ B.
- 3.
Finite differences are a numerical method used in solving differential equations.
- 4.
Victor Shestakov at Moscow State University also proposed a theory of electric switches based on Boolean algebra around the same time as Shannon. However, his results were published in Russian in 1941, whereas Shannon’s were published in 1937.
References
G. O’Regan, Guide to Discrete Mathematics. (Springer, 2016)
J.L. Ackrill, Aristotle the Philosopher. (Clarendon Press Oxford, 1994)
G. Boole, The calculus of logic. Cambridge and Dublin Math. J. III(1848), 183–198 (1848)
G. Boole, An Investigation into the Laws of Thought. Dover Publications. 1958.(First published in 1854)
D. McHale, Boole. (Cork University Press, 1985)
G. O’ Regan, Giants of Computing. (Springer, 2013)
C. Shannon, A Symbolic Analysis of Relay and Switching Circuits. Masters Thesis, Massachusetts Institute of Technology, (1937)
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O’Regan, G. (2017). A Short History of Logic. In: Concise Guide to Formal Methods. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-319-64021-1_5
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DOI: https://doi.org/10.1007/978-3-319-64021-1_5
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