FINAL PROJECT
on
HISTORY OF LOGIC
submitted towards the partial fulfillment of
the requirement for the award of the degree of
Bachelor of Technology
In
Mathematics and Computation Engineering
(Semester-l)
Submitted by
Shobhit Saha-2K20/A8/04
Under the Supervision
Of
Prof. DEEPTI SINHA
FEC-32 Logical Reasoning ( Slot-4 )
Delhi Technological University
Bawana Road. Delhi –11004
HISTORY OF LOGIC
SHOBHIT SAHA
ACKNOWLEDGEME
NT
I would like to express my special thanks of gratitude to my
teacher Prof. Deepti Sinha, who gave me the golden
opportunity to do this wonderful project on HISTORY OF
LOGIC, and immensely helped and motivated me to
research on the topic. During this I came to know about so
many new things which enriched me with knowledge and
gave me an insight about how human race developed logic
and how we progressed.
Secondly, I would also like to thank my parents to whom I
am greatly indebted who brought me up with so much
encouragement and love and helped me a lot in finalizing
this project within the limited time frame. I have no valuable
words to express my thanks, but my heart is still full of the
favours received from everyone.
ABSTRACT
Aristotle was the first logician to attempt a systematic analysis of
logical syntax, of noun (or term), and of verb. He was the first formal
logician, in that he demonstrated the principles of reasoning by
employing variables to show the underlying logical form of an
argument.
This article discusses the basic events of initiation logic and provides
an overview of its different fields emerged from different parts of the
world from different mathematicians.
Christian and Islamic philosophers such as Boethius (died 524), Ibn
Sina (Avicenna, died 1037) and William of Ockham (died 1347) further
developed Aristotle's logic in the Middle Ages, reaching a high point in
the mid-fourteenth century, with Jean Buridan. The period between the
fourteenth century and the beginning of the nineteenth century saw
largely decline and neglect, and at least one historian of logic regards
this time as barren. Empirical methods ruled the day, as evidenced by
Sir Francis Bacon's Novum Organon of 1620.
Logic began independently in ancient India and continued to develop to
early modern times without any known influence from Greek logic.
Medhatithi Gautama (c. 6th century BC) founded the anviksiki school
of logic. The Mahabharata, around the 5th century BC, refers to the
anviksiki and tarka schools of logic. Pāṇini (c. 5th century BC)
developed a form of logic (to which Boolean logic has some
similarities) for his formulation of Sanskrit grammar. Logic is
described by Chanakya (c. 350-283 BC) in his Arthashastra as an
independent field of inquiry.
Two of the six Indian schools of thought deal with logic: Nyaya and
Vaisheshika. The Nyaya Sutras of Aksapada Gautama (c. 2nd century
AD) constitute the core texts of the Nyaya school, one of the six
orthodox schools of Hindu philosophy. This realist school developed a
rigid five-member schema of inference involving an initial premise, a
reason, an example, an application, and a conclusion. The idealist
Buddhist philosophy became the chief opponent to the Naiyayikas.
Nagarjuna (c. 150-250 AD), the founder of the Madhyamika ("Middle
Way") developed an analysis known as the catuṣkoṭi (Sanskrit), a
"four-cornered" system of argumentation that involves the systematic
examination and rejection of each of the 4 possibilities of a proposition.
INDEX
1. OBJECTIVE AND BACKGROUND
2. HISTORY
2.1 19TH CENTURY
2.2 20TH CENTURY
3. FONDATIONAL THEORIES
4. SET THEORY AND PARADOXES
5. SYMBOLIC LOGIC
6. CONCLUSION
7. REFERENCES
INTRODUCTION
OBJECTIVE AND BACKGROUND
Our main objective is to study about how logic evolved and how
different logicians have taken into different course and developed in
their own creative way.
The modern era saw major changes not only in the external appearance
of logical writings but also in the purposes of logic. Logic for Aristotle
was a theory of ideal human reasoning and inference that also had clear
pedagogical value. Early modern logicians stressed “logic” had come to
mean an elaborate scholastic theory of reasoning that was not always
directed toward improving reasoning. A related goal was to extend the
scope of human reasoning beyond textbook syllogistic theory and to
acknowledge that there were important kinds of valid inference that
could not be formulated in traditional Aristotelian syllogistic.
One such goal was the development of an ideal logical language that
naturally expressed ideal thought and was more precise than natural
languages. Another goal was to develop methods of thinking and
discovery that would accelerate or improve human thought or would
allow its replacement by mechanical devices. . Logic for Aristotle was
a theory of ideal human reasoning and inference that also had clear
pedagogical value.
HISTORY
Mathematical logic emerged in the mid-19th
century as a subfield of mathematics,
reflecting the confluence of two traditions:
formal philosophical logic and mathematics
(Ferreirós 2001, p. 443). "Mathematical
logic, also called 'logistic', 'symbolic logic',
the 'algebra of logic', and, more recently,
simply 'formal logic', is the set of logical
theories elaborated in the course of the last
[nineteenth] century with the aid of an
artificial notation and a rigorously deductive
method." The first half of the 20th century
saw an explosion of fundamental results, accompanied by vigorous
debate over the foundations of mathematics.
19th century
In the middle of the nineteenth century, George Boole and then
Augustus De Morgan presented systematic mathematical treatments
of logic. Their work, building on work by algebraists such as George
Peacock, extended the traditional Aristotelian doctrine of logic into a
sufficient framework for the study of foundations of mathematics
(Katz 1998, p. 686). Charles Sanders Peirce later built upon the work
of Boole to develop a logical system for relations and quantifiers,
which he published in several papers from 1870 to 1885.
Gottlob Frege presented an independent development of logic with
quantifiers in his Begriffsschrift, published in 1879, a work generally
considered as marking a turning point in the history of logic. Frege's
work remained obscure, however, until Bertrand Russell began to
promote it near the turn of the century. The two-dimensional notation
Frege developed was never widely adopted and is unused in
contemporary texts.
From 1890 to 1905, Ernst Schröder published Vorlesungen über die
Algebra der Logik in three volumes. This work summarized and
extended the work of Boole, De Morgan, and Peirce, and was a
comprehensive reference to symbolic logic as it was understood at
the end of the 19th century.
20th century
In the early decades of the 20th
century, the main areas of study
were set theory and formal logic.
The discovery of paradoxes in
informal set theory caused some to
wonder whether mathematics itself
is inconsistent, and to look for
proofs of consistency.
In 1900, Hilbert posed a famous list
of 23 problems for the next century.
The first two of these were to
resolve the continuum hypothesis and prove the consistency of
elementary arithmetic, respectively; the tenth was to produce a method
that could decide whether a multivariate polynomial equation over the
integers has a solution. Subsequent work to resolve these problems
shaped the direction of mathematical logic, as did the effort to resolve
Hilbert's Entscheidungsproblem, posed in 1928. This problem asked for
a procedure that would decide, given a formalized mathematical
statement, whether the statement is true or false.
Foundational theories
Concerns that mathematics had not
been built on a proper foundation
led to the development of
axiomatic systems for fundamental
areas of mathematics such as
arithmetic, analysis, and geometry.
In logic, the term arithmetic refers
to the theory of the natural
numbers. Giuseppe Peano (1889)
published a set of axioms for
arithmetic that came to bear his
name (Peano axioms), using a variation of the logical system of Boole
and Schröder but adding quantifiers. Peano was unaware of Frege's
work at the time. Around the same time Richard Dedekind showed that
the natural numbers are uniquely characterized by their induction
properties. Dedekind (1888) proposed a different characterization,
which lacked the formal logical character of Peano's axioms.
Dedekind's work, however, proved theorems inaccessible in Peano's
system, including the uniqueness of the set of natural numbers (up to
isomorphism) and the recursive definitions of addition and
multiplication from the successor function and mathematical induction.
In the mid-19th century, flaws in Euclid's axioms for geometry became
known (Katz 1998, p. 774). In addition to the independence of the
parallel postulate, established by Nikolai Lobachevsky in 1826
(Lobachevsky 1840), mathematicians discovered that certain theorems
taken for granted by Euclid were not in fact provable from his axioms.
Among these is the theorem that a line contains at least two points, or
that circles of the same radius whose centers are separated by that
radius must intersect. Hilbert (1899) developed a complete set of
axioms for geometry, building on previous work by Pasch (1882). The
success in axiomatizing geometry motivated Hilbert to seek complete
axiomatizations of other areas of mathematics, such as the natural
numbers and the real line. This would prove to be a major area of
research in the first half of the 20th century.
SET THEORY AND PARADOXES
Ernst Zermelo (1904) gave a proof that
every set could be well-ordered, a result
Georg Cantor had been unable to obtain. To
achieve the proof, Zermelo introduced the
axiom of choice, which drew heated debate
and research among mathematicians and the
pioneers of set theory. The immediate
criticism of the method led Zermelo to
publish a second exposition of his result,
directly addressing criticisms of his proof
(Zermelo 1908a). This paper led to the
general acceptance of the axiom of choice
in the mathematics community.
Zermelo (1908b) provided the first set of axioms for set theory. These
axioms, together with the additional axiom of replacement proposed by
Abraham Fraenkel, are now called Zermelo–Fraenkel set theory (ZF).
Zermelo's axioms incorporated the principle of limitation of size to
avoid Russell's paradox.
In 1910, the first volume of Principia Mathematica by Russell and
Alfred North Whitehead was published. This seminal work developed
the theory of functions and cardinality in a completely formal
framework of type theory, which Russell and Whitehead developed in
an effort to avoid the paradoxes. Principia Mathematica is considered
one of the most influential works of the 20th century, although the
framework of type theory did not prove popular as a foundational
theory for mathematics (Ferreirós 2001, p. 445).
Fraenkel (1922) proved that the axiom of choice cannot be proved from
the axioms of Zermelo's set theory with urelements. Later work by Paul
Cohen (1966) showed that the addition of urelements is not needed, and
the axiom of choice is unprovable in ZF. Cohen's proof developed the
method of forcing, which is now an important tool for establishing
independence results in set theory.
Symbolic logic
Leopold Löwenheim (1915) and Thoralf Skolem (1920) obtained the
Löwenheim–Skolem theorem, which says that first-order logic cannot
control the cardinalities of infinite structures. Skolem realized that this
theorem would apply to first-order formalizations of set theory, and that
it implies any such formalization has a countable model. This
counterintuitive fact became known as Skolem's paradox.
In 1931, Gödel published On Formally Undecidable Propositions of
Principia Mathematica and Related Systems, which proved the
incompleteness (in a different meaning of the word) of all sufficiently
strong, effective first-order theories. This result, known as Gödel's
incompleteness theorem, establishes severe limitations on axiomatic
foundations for mathematics, striking a strong blow to Hilbert's
program. It showed the impossibility of providing a consistency proof
of arithmetic within any formal theory of arithmetic. Hilbert, however,
did not acknowledge the importance of the incompleteness theorem for
some time.
Gödel's theorem shows that a consistency proof of any sufficiently
strong, effective axiom system cannot be obtained in the system itself,
if the system is consistent, nor in any weaker system. This leaves open
the possibility of consistency proofs that cannot be formalized within
the system they consider. Gentzen (1936) proved the consistency of
arithmetic using a finitistic system together with a principle of
transfinite induction. Gentzen's result introduced the ideas of cut
elimination and proof-theoretic ordinals, which became key tools in
proof theory. Gödel (1958) gave a different consistency proof, which
reduces the consistency of classical arithmetic to that of intuitionistic
arithmetic in higher types.
The first textbook on symbolic logic for the layman was written by
Lewis Carroll, author of Alice in Wonderland, in 1896.
CONCLUSION
The history of logic deals with the study of the development of the
science of valid inference (logic). Formal logics developed in ancient
times in India, China, and Greece. Greek methods, particularly
Aristotelian logic (or term logic) as found in the Organon, found wide
application and acceptance in Western science and mathematics for
millennia. The Stoics, especially Chrysippus, began the development of
predicate logic.
The article will discuss the mathematicians who initiated the logic and
an overview of its acceptance from different parts of the world.
Logic revived in the mid-nineteenth century, at the beginning of a
revolutionary period when the subject developed into a rigorous and
formal discipline which took as its exemplar the exact method of proof
used in mathematics, a hearkening back to the Greek tradition. The
development of the modern ""mathematical" logic during this period by
the likes of Boole, Frege, Russell, and Peano is the most significant in
the two-thousand-year history of logic, and is arguably one of the most
important and remarkable events in human intellectual history.
REFRENCES
https://en.wikipedia.org/wiki/
https://plato.stanford.edu/entries/aristotle-logic/
https://www.britannica.com/topic/logic/Logical-systems
https://www.youtube.com/
