Journal of Philosophical Logic. Vol.40 N° 5 Logic in India

Logic in India—Editorial Introduction
Hans van Ditmarsch · Rohit Parikh ·
R. Ramanujam

1 History of Indian Logic

In the words of David B. Zilberman,
The most remarkable feature of Indian formal logic (as it was reflected
by the most advanced system of Indian logic, by Navya–Nyaya) is
clearly a close connection of a logical formalism to a linguistic material
. . . A common characteristic of Indian knowledge on all stages of its
existence was a consistent intentionalism, whereas European logic was
still a predominantly extentional one. Important properties appeared to
be also a utilization of non-quantum formalized expressions, presence
of a complicated theory relations, and a unique theory of multi-level
abstraction. (...) According to Bochenski, Indian logic can be of interest
to Western logicians because it was ‘initiated on different foundations’.
[13, p. 119], [3, p. 517]
Logic arose in ancient India from the art of conducting philosophical debate,
prevalent probably as early as the time of the Buddha in the sixth century BCE
but became more systematic and methodical in the subsequent four hundred
years. By the second century BCE, there were several manuals for formal debates,
perhaps the most systematic of them being Nyaayasutras of Aksapaada
Gautama. Aksapaada defined a method of philosophical argumentation called
the nyaya method. It starts with an initial doubt, as to whether p or not-p is the
case, and ends with a decision that p, or not-p as the case may be. There are
five ‘limbs’ in a structured reasoning: the statement of the thesis, the statement
of reason or evidence, the citation of an example, showing of the thesis as a
case that belongs to the general one and the assertion of the thesis as proven.
The Buddhist logicians argued that the first two or three of these were relevant.
In any case, the discussion was on articulation of inference schemata.
There was a continuing tradition of logic and the Jaina logicians were
concerned with epistemological questions. Perhaps themost important ‘school’
in the long list of logician communities was that of the Navya–Nyaya founded
in the 13th century CE by the philosopher Gange´sa. His Tattvacintamani
(“Thought-Jewel of Reality”) dealt with logic, some set theory, and especially
epistemology. This school developed a sophisticated idiom for analysing inference,
one that has been refined over centuries and is still used by scholars.
The systems of Indian logic are a topic of research and debate to this day,
and a community of scholars undertake studies, meet periodically and discuss
their observations.

2 Logic in India in the Twentieth Century
The widespread influence of the eminent philosopher Sarvepalli Radhakrishnan
on Indian schools of philosophy meant that many modern Indian philosophers
focussed on spiritualism in Indian thought rather than formal logic.
While a few did take up studies on formal semantics, modern developments
in mathematical logic were largely unfluential in Indian studies. Modal logic
and incompleteness phenomena attacted some Indian mathematicians [11, 12]
but only in the last two or three decades of the twentieth century did research
in logic come into its own in India.
From the perspective of philosophical logic, the work of Frege and Quine,
and the role of formalization intrigued many philosophers, especially in relation
to similar notions in Indian systems of logic. The influence of thinkers
such as Wittgenstein was also considerable. Towards the end of the century,
notions from non-classical logics such as non-monotonicity and imprecision in
truth, especially in relation to formal epistemology, attracted the attention of
many researchers [4, 5].
On the other hand, mathematical studies in logic were few. Algebraic
logic, inspired by the work of Helena Rasiowa [9] offered a home for some
mathematicians [2]. However, it was the advent of computer science that gave
a tremendous fillip to logic studies in India. Studies in logics of programs,
programming language semantics, temporal logics and artificial intelligence
Logic in India—Editorial Introduction 559
led to interest in mathematical logic per se, and soon, with the exception of
a handful in Mathematics and Philosophy, logic became a subject of teaching
and research in the computer science departments in India. A newly emergent
and confident theoretical computer science community sought to build bridges
with mathematicians in the areas of combinatorics, graph theory and number
theory, and with logicians in the areas of model theory and proof theory,
bringing algorithmic and complexity theoretic notions into the tools [7, 8, 10]

3 The Assocation for Logic in India
It was in such a background that ALI, the Assocation for Logic in India
(see http://ali.cmi.ac.in/) was formed in 2007, with the basic aim of building
a logic community in India, promoting research and education in logic and its
applications. A foundation for this had been provided by the annual meetings
of the Calcutta Logic Circle (a regular feature for two decades), the first two
editions of the Indian Conference on Logic and its Applications (ICLA) at IIT
Bombay (January 2005 and January 2007) and the International Conference
on Logic, Navya–Nyaya and Applications at Kolkata in January 2007. By
now the Indian Conference on Logic and its Applications (ICLA) is biennial,
taking place in the January of odd years, and the two-week long Indian School
on Logic and its Applications (ISLA) is biennial as well, taking place in the
January of even years.

ICLA The biennial Indian Conference on Logic and its Applications (ICLA)
is a forum for bringing together researchers from a wide variety of fields that
formal logic plays a significant role in, along with mathematicians, philosophers
and logicians studying foundations of formal logic in itself. The fourth
conference was held at Delhi University, in January 2011, and the proceedings
published as LNCS 6521 in the FoLLI series [1]. It had as a special feature the
inclusion of studies in systems of logic in the Indian tradition, and historical
research on logic.

ISLA The Indian School on Logic and Applications (ISLA) is a biennial
event as well. The previous editions of the school were held in IIT Bombay
(2006), IIT Kanpur (2008), and University of Hyderabad (2010), and an
upcoming ISLA is at Manipal University (2012). The objective is to present
before graduate students and researchers in India some basics as well as active
research areas in logic. The School typically attracts students and teachers
from mathematics, philosophy and computer science departments. The school
adopts a dual format: the mornings will consist of introductory courses on
fundamental aspects of logic, by eminent researchers in the area. The afternoons
have workshops, which can be of the nature of advanced tutorials, or
presentations on research areas, in different aspects of logic and applications.

4 Contents
This special issue on Logic in India aims to provide a sampler of work from
both traditions, that of Indian logic, as well as work from logicians active in
mathematics and computer science in India.
‘Possible Ideas of Necessity in Indian Logic’ by Sundar Sarukkai is a contribution
motivated by the history of Indian logic, on the conception of necessity.
Logical necessity is presumably absent in Indian logic, where the structure of
the logical argument in Indian logic is often given as a reason for this claim.
In Indian logic, the analysis of ‘invariable concomitance’ (vyapti) is of crucial
importance and its definitions are very complex. The author argues how vyapti
can be understood in terms of contingent necessity in the Leibnizian sense and
also how the complex definitions can be interpreted as an attempt to define
contingent necessity in terms of logical necessity.
‘Fine-grained concurrency with separation logic’ by Kalpesh Kapoor, Kamal
Lodaya and Uday Reddy is a contribution in the area of computer science,
on reasoning about concurrent programs. Such reasoning involves ensuring
that concurrent processes manipulate disjoint portions of memory but the
division of memory between processes is in general not static. The implied
ownership of memory cells may be dynamic and shared, allowing concurrent
access. Concurrent Separation Logic with Permissions, developed byO’Hearn,
Bornat and others (see [6] for various contributions), is able to represent
sophisticated transfer of ownership and permissions between processes. The
authors demonstrate how these ideas can be used to reason about fine-grained
concurrent programs.
‘Context-sensitivity in Jain Philosophy. A Dialogical Study of Siddharsigani’s
Commentary On The Handbook of Logic’ by Nicolas Clerbout, Marie-Hélène
Gorisse, and Shahid Rahman is a contribution on the history of Indian logic. In
classical India, Jain philosophers developed a theory of viewpoints (naya-vada)
according to which any statement is always performed within and dependent
upon a given epistemic perspective or viewpoint. The Jainas furnished this
epistemology with an (epistemic) theory of disputation that takes into account
the viewpoint in which the main thesis has been stated. The paper delves
into the Jain notion of viewpoint contextualisation and develops a suitable
logical system that offers a reconstruction of the Jainas’ epistemic theory of
disputation.
‘A Logic for Multiple-source Approximation Systems with Distributed
Knowledge Base’ by Mohua Banerjee and Aquil Khan is a contribution in
the area of mathematics and computer science, focussing on rough sets, which
are approximations of sets. The primitive notion is that of an approximation
space, which is a pair consisting of a domain of discourse (the knowledge base)
and an equivalence relation on that domain (the granularity of information
about objects in the domain). The authors focus on the situation where
information is obtained from different sources. The notion of approximation
space is extended to define a multiple-source approximation system with
distributed knowledge base, that can reflect how individual sources perceive
the same domain differently (depending on what information the group /
individual source has about the domain). The same concept may then have
approximations that differ with individuals or groups.
It is hoped that this issue will generate interest in Logic in India within the
wider international community of logicians and philosophers.

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