In mathematical and philosophical reasoning, symbolic logic
entails the use of symbols to express concepts, relations, and positions.
Symbolic logic varies from (Aristotelian) syllogistic logic
in that it employs ideographs or a particular notation to "symbolize
exactly the item discussed" (Newman 1956, 1852), and it may be modified
according to precise rules.
Traditional logic investigated the truth and falsehood of
assertions, as well as their relationships, using terminology derived from
natural language.
Unlike nouns and verbs, symbols do not need interpretation.
Because symbol operations are mechanical, they may be
delegated to computers.
Symbolic logic eliminates any ambiguity in logical analysis
by codifying it entirely inside a defined notational framework.
Gottfried Wilhelm Leibniz (1646–1716) is widely regarded as
the founding father of symbolic logic.
Leibniz proposed the use of ideographic symbols instead of
natural language in the seventeenth century as part of his goal to
revolutionize scientific thinking.
Leibniz hoped that by combining such concise universal
symbols (characteristica universalis) with a set of scientific reasoning
rules, he could create an alphabet of human thought that would promote the
growth and dissemination of scientific knowledge, as well as a corpus
containing all human knowledge.
Boolean logic, the logical underpinnings of mathematics, and
decision issues are all topics of symbolic logic that may be broken down into
subcategories.
George Boole, Alfred North Whitehead, and Bertrand Russell,
as well as Kurt Gödel, wrote important contributions in each of these fields.
George Boole published The Mathematical Analysis of Logic
(1847) and An Investigation of the Laws of Thought in the mid-nineteenth
century (1854).
Boole zoomed down on a calculus of deductive reasoning,
which led him to three essential operations in a logical mathematical language
known as Boolean algebra: AND, OR, and NOT.
The use of symbols and operators greatly aided the creation
of logical formulations.
Claude Shannon (1916–2001) employed electromechanical relay
circuits and switches to reproduce Boolean algebra in the twentieth century,
laying crucial foundations in the development of electronic digital computing
and computer science in general.
Alfred North Whitehead and Bertrand Russell established
their seminal work in the subject of symbolic logic in the early twentieth
century.
Their Principia Mathematica (1910, 1912, 1913) demonstrated
how all of mathematics may be reduced to symbolic logic.
Whitehead and Russell developed a logical system from a
handful of logical concepts and a set of postulates derived from those ideas in
the first book of their work.
Whitehead and Russell established all mathematical concepts,
including number, zero, successor of, addition, and multiplication, using
fundamental logical terminology and operational principles like proposition,
negation, and either-or in the second book of the Principia.
The Principia showed how every mathematical postulate might
be inferred from previously explained symbolic logical facts.
Only a few decades later, Kurt Gödel's On Formally
Undecidable Propositions in the Principia Mathematica and Related Systems
(1931) critically analyzed the Principia's strong and deep claims,
demonstrating that Whitehead and Russell's axiomatic system could not be
consistent and complete at the same time.
Even so, it required another important book in symbolic
logic, Ernst Nagel and James Newman's Gödel's Proof (1958), to spread Gödel's
message to a larger audience, including some artificial intelligence
practitioners.
Each of these seminal works in symbolic logic had a
different influence on the development of computing and programming, as well as
our understanding of a computer's capabilities as a result.
Boolean logic has made its way into the design of logic
circuits.
The Logic Theorist program by Simon and Newell provided
logical arguments that matched those found in the Principia Mathematica, and
was therefore seen as evidence that a computer could be programmed to do
intelligent tasks via symbol manipulation.
Gödel's incompleteness theorem raises intriguing issues
regarding how programmed machine intelligence, particularly strong AI, will be
realized in the end.
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See also:
Symbol Manipulation.
References And Further Reading
Boole, George. 1854. Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities. London: Walton.
Lewis, Clarence Irving. 1932. Symbolic Logic. New York: The Century Co.
Nagel, Ernst, and James R. Newman. 1958. Gödel’s Proof. New York: New York University Press.
Newman, James R., ed. 1956. The World of Mathematics, vol. 3. New York: Simon and Schuster.
Whitehead, Alfred N., and Bertrand Russell. 1910–1913. Principia Mathematica. Cambridge, UK: Cambridge University Press.
