topics: |  philosophy | india | logic | language

to me what is exciting in this discussion is how right from the start, the main question is related to induction - what are the valid mechanisms for arriving at the major premise - the inductive relation or the universal. (major premise: e.g. "all men are mortal", "where there is a smoke, there is fire").

in the western philosophical tradition, "logic" has largely meant formal deductive logic. a typical introductory course on logic will spend nearly all its energy on outlining the methods of proof, and mention induction, if at all, as a almost as a footnote.

even today, if you go to conferences on logic, or take a course in logic, you will get to hear very little about induction. almost the entire discourse will be on formal deductive proof, with various distinctions such as in modal logics, where the inference depends on contexts or possible worlds, on intuitionistic logics, which focuses on provability (true) and refutability (a kind of negation). Within the tradition, the discussion on the validity of the major premise has discussed issues such as justified true belief but this also while side-stepping the minefield of induction.

in contrast, matilal suggests that even western logicians may agree with an interpretation of logic which is more than formal deduction:

	"Logic" I shall here understand to be the systematic study of
	informal inference-patterns, the rules of debate, the identification
	of sound inference vis-a-vis sophistical argument, and similar
	topics.

what matilal is saying without emphasizing it, is that he is talking about
what "logic" means before it gets formal (or algebraic).  Reading between
the lines, we see that this definition is a departure from the formal
traditions in western logic today.  The word "informal" is the key - but
then all ancient philosophy, in this sense, is informal; even if you
consider aristotelian syllotgsm.

from the introduction in Copi

We find considerable overlap between this definition and the following by
Charles Sanders Peirce:

	It will generally be conceded that [the central problem of logic] is
	the classification of arguments, so that all those that are bad are
	thrown into one division, and those which are good into another.

The standard text by Copi quotes the above definition by Peirce, but goes
on to clarify it as:

	The logician is concerned with the correctness of the completed
	process.  His question is always: does the conclusion reached follow
	from the premisses used or assumed?

Which of course, reduces the problem to deduction.   The entire text has
one paragraph on inductive reasoning, on p.3, where it says: 

	Inductive arguments involve the claim only that their premisses
	provide some grounds for their conclusions. Neither the term 'valid'
	nor its opposite 'invalid' is properly applied to inductive
	arguments.

The position in Indian logic however, is that the major premise cannot be 
taken as a given.  It must also be defended via inductive arguments.  

There is some overlap between this position and the famous argument by
C.I. Lewis in 1912 that distinguished the propositions of the type

	1. Either Caesar died, or the moon is made of green cheese. 
	2. Either Matilda does not love me, or I am beloved. 

Here we know that (2) is true without knowing whether the premise is true
or not, but (1) is true only if we know the premise to be true. 
Thus (2) exhibits a 'purely logical or formal character'.   This has led to 
entire branches of logic (modal logics).  

In the Sanskrit inductive tradition, one uses an udAharaNa or example, as
part of the five-fold syllogism, to justify the inducability of the major
premise. 

Deductive logic in practice : Artificial Intelligence

when the field of artificial intelligence began in the 1950s, it was felt
that deductive logic would give us all the answers.  but increasingly we
are finding that there are very few real universals that are invariably
true.  even bachelor(x) may not always equal male(x) & unmarried(x); in
fact, the set of exceptions is huge, probabily unbounded, which gives rise
to what is called the "frame problem" in philosophical logic.

there is a sub-field called inductive logic programming where you try to
induce more compact descriptions given a set of logical descriptions.
things are usually assumed to be static; there are problems with fluents,
and the frame problem remains (if you want to try it out, check out the
Progol system by stephen muggleton).

But mainstream logic remains strongly deductive (and boolean).  perhaps
the philosophers need to learn more from machine learning, where rules of
the kind "All x are y" are practically dead, and one only considers the
conditional probability p(y|x).  Then inferences are equivalent to bayesian
esitmation of probabilities.   Thus, boolean logic has given way to
probabilistic logic.

History of Induction

In the western logical tradition, there isn't much serious discussion on
induction until Hume (see J. R. Milton, Induction before Hume, in
Gabbay and Hartmann's Handbook of the History of Logic: v.10 - Inductive
logic.)

As a student in my first course on symbolic logic (Copi), at one point I
was struck by the absence of any discussion on how we can find more
universals - given that these are extremely strong statements
(conjunctions over the universe).

It is much later that I discovered that in Indian logic, the validity of
the inductive inference by which one reaches this universal (vyApti) has
been the central concern right from the beginning, as we see in diMnAga's
"wheel" described below.

Excerpts

"Logic" I shall here understand to be the systematic study of informal
inference-patterns, the rules of debate, the identification of 
sound inference vis-a-vis sophistical argument, and similar topics. 
							- [opening line]

One may feel somewhat apologetic today to use the term "logic" in the
context of classical Indian philosophy, for "logic" has acquired a very
specific connotation in modern philosophical parlance. 

Nevertheless, the list supplied in the opening sentence is, I believe, a
legitimate usage of the term, especially when its older senses are taken
into account. 

[other authors have used "indian logic" to refer to concerns, such as the
ideas of pramANas (accredited means of knowing) with a possible emphasis
on anumAna or inference as a means of knowing.  These issues are closer
to epistemology and have misled many non-Sanskritists.
(e.g. S.C. Vidyabhusana's monumental, but by now dated, work "A History of
Indian Logic" (1921) or scholars such as H. N. Randle and
T. Stcherbatsky...)

I shall take "logic" in its extended and older sense in order to carve out
a way for my own investigation. I shall use the traditional shAstras and try
to explain their significance and relevance to our modern discussion of the
area sometimes called "philosophical logic." p.1

I shall include much else besides, as the initial list shows, but will try
to remain faithful to the topic of logic, debate, and the study of
inference. 

The art of conducting a philosophical debate was prevalent probably as early
as the time of the Buddha and the Mahāvira (Jina), but it became more
systematic and methodical a few hundred years later. By the second century
BC, the intellectual climate in India was bristling with controversy and criticism.

At the center of controversy were certain dominant religious and ethical
issues. Nothing was too sacred for criticism. 
Some of the major concerns were
     - "Is there a soul different from body?", 
     - "Is the world ilokd) eternal?", 
     - "What is the meaning, goal, or purpose of life?",
     - "Is renunciation preferable to enjoyment?", etc.

While teachers and thinkers argued about such matters, there arose a gradual
awareness of the characteristics or patterns of correct -— that is,
acceptable and sound -— reasoning, and concern about how it differs from the
kind of reasoning that is unacceptable.

vAdavidyA: the art of debate

logic developed in ancient india from the tradition of vAdavidyA
('वादविद्या' or 'तर्कविद्या'), a discipline dealing with the categories of
debate.

around the beginning of the christian era, several vāda 'वाद' manuals had
become available.  these were meant for students who wanted to learn how to
conduct debates successfully, what tricks to learn, how to find loopholes in
the opponent's position, and what pitfalls to be wary of. we will examine
some of these manuals in chapters 2 and 3. of these manuals, the nyāyasütras
of akSapāda gautama (अक्षपाद गौतम, c. 150 AD) is comparatively more systematic
than others. we shall hence follow it in this introductory exposition.

Debates, in AkShapAda's view, can be
    i. honest debate (vAda), where both sides are seeking the truth,
   ii. tricky-debate (jalpa जल्प), goal to win by fair means or foul
  iii. desctructive debate (vitaNDA वितण्डा), defeat and demolish opponent
     		by any means

Debates, e.g. of second type, between e.g. teachers from different schools,
would take place before a board called the madhyastha, often a king or
other sponsor.  In the third type, the winner may not even have a position,
but must defeat the opponent using only logical arguments, explicitly
destructive and negative.

the first type of debate

Apart from a theory of evidence (pramANa) and argument (tarka) needed
for the first type of debate, a number of situations are listed for
concluding the debate and declaring one side as defeated
(nigraha-sthAna = defeat-situation).  Nyayasutras - 22 such situations
e.g.
   a) if opponent cannot understand your argument,
   b) is confused,
   c) cannot reply within reasonable time, etc.

In addition, manuals list several standard "false" rejoinders or jAti
(24 such listed), as well as some underhand tricks (chala), like
equivocation and confusion of a metaphor for the literal.

METHOD:
akShapAda defines a method of philosophical argumentation. Seven
categories  constituting the "prior" stage of a nyAya,
starts with an initial doubt, whether p or not-p, and ends with a
decision: 1. doubt, 2. purpose, 3. example, 4. basic tenets,
5. the "basic limbs" of formal reasoning, 6. Supportive argument
(tarka), 7. Decision.

Limbs or members (propositions) [avayava]

The limbs were the most important formulation.  Five limbs in a
structured reasoning, which are to be articulated in language:

   1. There is fire on the hill. [statement of the thesis; signified]
   2. For there is smoke [reason or evidence; sign]
   3. (Wherever there is smoke, there is fire), as in the
       kitchen. [example, particular, well-recognized]
   4. This is such a case (smoke on the hill). [showing of present case as
       belonging to general case.]
   5. assertion of thesis again, as proved.

Buddhists argued that this was too elaborate.  NyayA school: five
steps are needed to satisfy the expectation (AkAMkShA).

The supportive argument (tarka, method #6) is needed when doubts are
raised about the implication of the middle part of the inference
schema.
[the middle part is the inductive relation, or the major premise]

pseudo-evidence is similar to evidence or reason - "impostor."  Five
types:

1. Deviatory: Since there can be fire without smoke (e.g. red hot iron
 	   ring), someone claiming smoke in the kitchen because of
	   fire there.
2. Contradictory: Where contrary evidence exists, e.g. Pool of water.
3. Unestablished: Where the evidence-reason must be proven to exist, if
	   not it is a pseudo-sign.
4. Counter-balanced: A purported evidence may be countered by a
	   counter-evidence showing the opposite possibility.
5. Untimely: as soon as the thesis is stated, the evidence is no longer
	   evidence (gives reference to Matilal 85).

the sign and the signified

All this implicitly states a theory of what consitutes an adequate
sign.  Sign - _liMga_ लिङ्ग, and what it represents - _liMgin_.  Fully
articulated in the writings of the Buddhist logician DinnAga
(400-480AD) in his theory of the triple-character reason.

Dinnāga formulated the following three conditions, which, he claimed,
a logical sign must fulfill:

	1. It should be present in the case (object) under consideration.
	2. It should be present in a similar case or a homologue.
	3. It should not be present in any dissimilar case, any
		heterologue.

An adequate sign is what should be non-deviating, i.e. it should not be
present in any location when the signified is absent.  If it is, it would
be deviating.

The property A (signified) is implied by sign B in location or context
S.  The context of an argument is the pakSha.  Situations where the
signified occurs are the sapakSha, and the vipakSha is where the
signified is not present.  If the sign is present in a vipakSha it is
deviating.

diMnAga's wheel

Assuming the sign and signified co-occur in the pakSha, DiMnAga
constructs a wheel of reason (3x3 matrix), where sapakSha and vipakSha
may be +, -, or ± :

	 ,-----------------------------------------------.
	 |       \   V=   +            -           ±     |
	 |   S=   \--------------------------------------|
	 |      +  |     +V,+S   |   -V,+S   |   ±V,+S   |
	 |      -  |     +V,-S   |   -V,-S   |   ±V,-S   |
	 |      ±  |     +V,±S   |   -V,±S   |   ±V,±S   |
	 '---------'-------------------------------------'

only the squares [-V,+S] and [-V,±S] satisfy the conditions.

the middle square, (-V,-S), argues diMnAga, is not a case of
sound inference; this sign is a pseudo-sign. For although it satisfies
the two conditions 1 and 3 above, it does not satisfy condition 2.

1,3,7,9 - are pseudosigns - where the sign occurs in some vipaksha or
	the other.

Uddyotkara: DiMnAga did not consider at least two further alternatives -
    the cases where sapakSha or vipAkshA are null. -> now we get sixteen
    cases.

e.g. What is knowable is namable (as in Nyaya).
However, (according to Nyaya) everything can be named (expressed) in
language, so there is no vipakSha to this:
		This is nameable, because it is knowable.

Here knowability is the sign, and is adequate for showing the
nameability of an entity.

Uddyotkara captured another type of adequate reason or logical sign -
formulated in terms of a counterfactual.

     The living body cannot be without a soul, for if it were it would
     have been without life.

This is the generalized inference called "universal negative" -
kevalavyatirekin केवलव्यतिरेकि - The subject which has unique property B
cannot be without A, for then it would be without B, since the presence
of A and B mutually imply each other.  Since B is an unique property,
there is no sapakSha.

In this instance, since A is present uniquely in S and nowhere else,
we cannot derive the pervasion between A and B from example. p.9