1+1=2: A mathematical equation so obvious that we use it metaphorically for an easy task or an apparent logical conclusion. But the nature of mathematics is such that everything apart from an axiom requires valid proof. And our seemingly innocuous equation is no exception either. In Principia Mathematica, Bertrand Russell and Alfred Whitehead take over 360 pages to logically conclude that 1+1=2.
A staggering amount of work to prove something that you learned in Kindergarten. But there’s a difference between knowing addition and understanding addition. The objective of Russell and Whitehead was to establish a firm foundation for mathematics with logic. Unfortunately, their mission hit a dead-end with Godel’s Incompleteness theorems. But we continue to progress in math and science with logical reasoning, which has its beginning in philosophical inquiry.
Logic is the study of methods and principles used to distinguish between correct and incorrect reasoning.1
In Western philosophy, the earliest formal study of logic is found in Aristotle’s treatise known as Organon. Deductive reasoning forms the basis of Aristotle’s logic, with syllogisms as the key logical structures.
Thousands of miles away, Indian philosophers took a different route for their logical theories. It’s not purely deductive but uses ‘evidence’ or ‘examples’ to arrive at a conclusion. It’s undoubtedly rigorous but proceeds in a way different from Aristotle’s syllogisms. But we will get back to this point later in the article. For now, we begin by understanding inference in Nyāya philosophy.
Inference (Anumān, अनुमान)
In the post on Nyāya epistemology, we learned about various valid means to knowledge, perception being the primary source followed by inference.
The Sanskrit word for inference is Anumān (अनुमान), which means “after cognition/knowledge,” or knowledge/cognition that follows some other knowledge.
Perception is antecedent to inference. You first need to view the phenomenon before determining its cause. For example, you see that there's smoke on the mountain. Based on your prior experience, you can infer that there must be fire on the mountain because of the smoke - “The mountain is fiery because it smokes and whatever smokes is fiery.”
A straightforward logical conclusion. Some might even dismiss it as common sense. But it is an inferential argument nonetheless. And just like everything in philosophy, there’s more than what meets the eye. So, let’s proceed by understanding the terminologies of Anumān.
The elements of Anumān
The Naiyāyikas give a five-member form (pañcāvayava-vākya, पञ्चावयव वाक्य) for the inferential argument:
प्रतिज्ञाहेतुदाहरणोपनयनिगमनानि "अवयवा:" ।।१।१।३२।।
The members (of a syllogism) are proposition, reason, example, application, and conclusion. - Complete NyāyaSūtra translation by Sinha and Vidyābhūsana
Let’s take an example to understand the five-member syllogism.
The mountain is fiery (pratijñā, प्रतिज्ञा ) [Proposition]
Because it is smoky/smoke-possessing (hetu, हेतु ) [Reason]
Wherever there’s smoke, there’s fire, like in the kitchen but unlike a lake (udāharana, उदाहरण ) [Examples]
The mountain, since it possesses smoke, has fire (upanaya, उपनय ) [Application]
Therefore, the mountain is fiery (nigamana, निगमन) [Conclusion]
We will understand each of these five members in detail.
साध्यनिर्देशः "प्रतिज्ञा" ।।१।१।३३।।
A proposition is a declaration of what is to be established.
(First Member) The mountain is fiery
The first statement, pratijñā (प्रतिज्ञा), is a proposition — We need to prove it with proper reasoning.
The mountain is the subject of inference, referred to as “paksha (पक्ष),” while the fire is the property that we need to prove, called as “sādhya (साध्य).”
उदाहरणसाधर्म्यात्साध्यसाधनं "हेतु:" ।।१।१।३४।।
The reason is the means for establishing what is to be established through the affirmative character of the example.
(Second Member) Because it is smoky/smoke-possessing
The second statement is called the hetu or ‘reason.’ We see the smoke on the mountain that becomes the reason to proceed with the argument. It’s also called “linga,” or a mark/sign that belongs to the mountain.
साध्यसाधर्म्यातदधम्म्र्भावी द्र्ष्टान्त "उदाहरणम्" ।।१।१।३६।।
An affirmative example is a familiar instance which is known to possess the property to be established, and which implies that this property is invariably contained in the reason given.
(Third Member) Wherever there’s smoke, there’s fire, like in the kitchen but unlike a lake
The third member of the argument essentially differentiates Indian syllogism from its Western counterpart. We can further break it down into two parts - Vyāpti (व्याप्ति) and associated examples.
Vyāpti is the invariable relation between hetu and sādhya, which refers to smoke and fire in our example. The validity of the argument depends on the fact that whatever smokes is fiery. We establish the universal relation between vyāpaka (pervader) and vyāpya (pervaded) with a positive and negative example.
The hearth in the kitchen is a positive example (sapaksha, सपक्ष) of the relation between smoke(hetu) and fire(sadhya). In contrast, the lake with an absence of fire is a negative example (vipaksha, विपक्ष).
A point worth noting is that hetu (reason) is only valid when:
It is present in the subject of inference (paksha) and related positive examples (sapaksha)
It is not present in related negative examples (vipaksha). The presence of smoke without fire indicates contradiction and invalidates it as a valid reason for inference.
The use of examples is a unique aspect of Nyāya's logic.
Being realists, the Naiyāyikas believe in using evidence or examples to establish an argument. While Aristotlean logic uses relations between major, middle, and minor terms in syllogisms, the Nyāya philosophers rely on universal relationships and evidence.
उदाहरणपेक्षस्तथेत्यूपसंहारो न तथेति वा साध्यस्य "उपनय:" ।।१।१।३८।।
Application is winding up, with reference to the example, of what is to be established as being so or not so.
(Fourth Member)The mountain, since it possesses smoke, has fire
The fourth member of the argument builds upon the previous two to establish that the mountain is fiery as it has smoke. Uddyotakara, a Nyāya philosopher, refers to the fourth step as Liṅgaparāmarśa (लिङ्गपरामर्श) — the relationship between hetu in paksha and vyāpti — the mountain is a case of smoke pervaded by fire (vyāpti).
Thus, Liṅgaparāmarśa is the essential and immediate cause (caramkarana, चरमकारण) of inference for Uddyotakara. According to him, we cannot conclude by merely knowing that the mountain has smoke or that whatever smokes is fiery. The perceived fact and the universal relation have to come together for successful inference.
हेत्वप्देशात्प्रतिज्ञाया: पुनर्वचनं "निगमनम्" ।।१।१।३९।।
Conclusion is the restating of the proposition, after the reason has been mentioned.
(Fifth Member) Therefore, the mountain is fiery
The last member in the five-member syllogism is the concluding statement that proves the initial proposition.
One might object that the first and fifth members are repetitive, and thus both are not required in the syllogism. However, the proposition ascribes sādhya (fire) to paksha (mountain), whereas the conclusion asserts that paksha has sādhya. 2
The concluding statement is akin to the Latin phrase “quod erat demonstrandum (QED),” which is often used at the end of mathematical proofs. It indicates that the proof or the argument is complete and hence is used with the meaning "thus it has been demonstrated"3
We cannot have a one-to-one comparison between Indian and Aristotlean syllogisms. As Karl Potter points out, a more fruitful comparison can perhaps be found with Mill’s canons of inductive reasoning. 4
According to Roy Perrett,
“The focus of concern in Indian logic was the ascertainment of the truth of the universal proposition of an inference and hence the establishment of the validity of the given inference.” 5
As we have seen in the five-member syllogism, the universal relationship of vyāpti is fundamental to Indian syllogism. But we have a small problem. How do we know that vyāpti is universal in nature?
The induction problem of Vyāpti
How can we ascertain that whatever smokes is always fiery? There’s no way that we can view every single instance of smoke and fire. Hence, induction by simple enumeration is not a possibility. Instead, we require a perception of the entire class of smokes as related to fire. But can we have such a perception? Well, we’ve seen the answer previously in one of the previous posts on extraordinary perception. Remember that the Naiyāyikas consider Sāmānyalakṣaṇa (सामान्यलक्षण) as the perception of universals (सामान्य/ जाति).
In Nyāya philosophy, the existence of universals is presupposed in inference. We can only know the universal relation (vyāpti) of smoke and fire through Sāmānyalakṣaṇa, which is an extraordinary perception. Roy Perrett elaborates:
“When we perceive particular smokes and fires, we also perceive the universal smokiness and fieriness inhering in them. Through this sense of contact with smokiness and fieriness, which are generic properties equally shared by all cases of smoke and fire, we can, in turn, perceive all cases of smoke and fire. Thus, the concomitance of smoke and fire is established through an extraordinary perception of the whole class of smoke-possessing things as related to fire.”6
Is the appeal to extraordinary perception “logically sound”? Many would deny it. But here’s a counterpoint — Can we solve the problem of induction through deduction or formal logic? F. C. S. Schiller believes that we cannot reach a universal conclusion with formal logic. It turns out that the problem of induction is one of the most famous problems in philosophy. And many philosophers believe that there’s no definite solution to it.
Roy Bhaskar, a well-known philosopher of science, says that the problem of induction only arises when we deny the inherent nature of something. For example, all coal is black because of its chemical properties; all men are mortal because of cellular aging. But does this work for every inductive argument? Well, we really don’t have a conclusive answer for the problem of induction. The problem has grown larger in recent years, even taking us into statistics and probability. But we will not take that path in this article.
For now, let’s consider another aspect of Nyāya’s theory of inference.
Causal relations and universal inferences
Akṣapāda Gautama gave three classifications for inference in NyāyaSūtra —
Pūrvavat (पूर्ववत्), Śeṣavat (शेषवत्), and Sāmānyatodṛṣṭa(सामान्यतोदृष्ट)
अय तत्पूर्वकं “त्रिविधमनुमानं” पुर्ववच्शेषवत्सामान्यतो दृष्टं च ।।१।१।५।।
Inference is knowledge which is preceded by perception, and is of three kinds — Pūrvavat (पूर्ववत्), Śeṣavat (शेषवत्), and Sāmānyatodṛṣṭa(सामान्यतोदृष्ट).
However, there’s no consensus amongst Nyāya scholars on the correct meaning of these classifications. According to Vātsyāyana, two of them are causal relationships, while the third refers to the general co-existence of things.
Pūrvavat is an inference from cause to effect. For example, we can infer rain from dark clouds.
Śeṣavat is an inference from effect to cause — the rise in the water level of the river or muddy roads indicates heavy rainfall.
Sāmānyatodṛṣṭa is an inference from general correlation.
“In Sāmānyatodṛṣṭa, the inference depends not on a causal connection but certain observed points of similarity between different objects of experience. “- Satischandra Chatterjee
For instance, we infer that the moon orbits earth because we see its position changing in the sky. (The example is given by Upayahrdaya. However, it has a flaw. By analogical reasoning, we can also conclude that the sun revolves around the earth.)
In yet another philosophical interpretation, Uddyotakara believes that the three classifications refer to the universally positive, negative, and positive-negative inferences. Consider the following inference -
All knowable objects are nameable
This pot is knowable
Therefore, it is nameable.7
Such universal positive inferences are known as “Kevalānvayi (केवलान्वयि)”.
On the contrary, there can also be universal negative examples:
No non-soul is animate
All living beings are animate
Therefore, all living beings have souls.8
This is an example of a universal negative inference , Kēvalavyatirēkī (केवलव्यतिरेकी).
However, our example of smoke and fire has both positive and negative examples.
Such inferences are known as “Anvaya-vyatireka(अन्वयव्यतिरेक)”.
Sage Gautama gave no descriptions for the Pūrvavat, Śeṣavat, and Sāmānyatodṛṣṭa in NyāyaSūtra. This led to varied interpretations from different Nyāya scholars. But that’s also the beauty of philosophy — ambiguity of language creating new terrains of thought.
The article covered some of the essential aspects of inference in Nyāya philosophy. But the landscape is too vast. We will need many more articles to cover the immense amount of work done by Naiyāyikas over centuries. But let’s take it slowly. We are here to grasp rather than gobble. Later issues will cover many interesting topics like logical fallacies, empty terms, and double negation. Slowly but surely, we will scale the mountains of Indian logic. But be wary of the smoke. Because wherever there’s smoke, there’s fire.
Introduction to Logic - Copi
Encyclopedia of Indian Philosophies - Volume 2 by Karl Potter
Wikipedia article on QED - https://en.wikipedia.org/wiki/Q.E.D.
Encyclopedia of Indian Philosophies - Volume 2 by Karl Potter
The problem of induction in Indian Philosophy - Roy Perrett
The problem of induction in Indian Philosophy - Roy Perrett
The Nyāya theory of knowledge - Satishchandra Chatterjee
The Nyāya theory of knowledge - Satishchandra Chatterjee