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.

References

1. Banerjee, M., & Seth, A. (Eds.) (2011). Logic and its applications—4th Indian Conference,

ICLA 2011. Proceedings (Vol. 6521). LNCS: Springer.

2. Banerjee, M., & Chakraborty, M. K. (1996). Rough sets through algebraic logic. Fundamenta

Informaticae, 28(3–4), 211–221.

3. Bochenski, I. M. (1955). Formale logik. München: K. A. Verlag

4. Chakraborty, M. K. (1995). Graded consequence: Further studies. Journal of Applied Non-

Classical Logics, 5(2), 127–137.

5. Chakraborty, M. K., & Chatterjee, A. (1996). On representation of indeterminate identity via

vague concepts. Journal of Applied Non-Classical Logics, 6(2).

6. Gardner, P., & Yoshida, N. (Eds.) (2004). 15th international Conference on Concurrency

Theory (CONCUR) (Vol. 3170). LNCS: Springer.

7. Lodaya, K., & Pandya, P. K. (2006). A dose of timed logic, in guarded measure. In E. Asarin,

& P. Bouyer (Eds.), FORMATS. Lecture notes in computer science (Vol. 4202, pp. 260–273).

Springer.

8. Lodaya, K., Parikh, R., Ramanujam, R., & Thiagarajan, P. S. (1995). A logical study of

distributed transition systems. Information and Computation, 119(1), 91–118.

9. Rasiowa, H., & Sikorski, R. (1970). The mathematics of metamathematics. Warsaw: Polish

Scientific Publishers.

10. Seth, A. (1992). There is no recursive axiomatization for feasible functionals of type ∼2. In

LICS (pp. 286–295). IEEE Computer Society.

11. Shukla, A. (1967). A note on the axiomatizations of certain modal systems. Notre Dame

Journal of Formal Logic, 8(1–2), 118–120.

12. Shukla, A. (1972). The existence postulate and non-regular systems of modal logic. Notre

Dame Journal of Formal Logic, 13(3), 369–378.

13. Zilberman, D. B. (2006). History of Indian logic. In R. S. Cohen, &H. Gourko (Eds.), Analogy

in Indian and Western philosophical thought. Boston studies in the philosophy of science

(Vol. 243, pp. 110–120). Springer

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