History of logic


The history of logic deals with the study of the development of the science of valid inference. Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian 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.
Christian and Islamic philosophers such as Boethius, Avicenna, Thomas Aquinas and William of Ockham 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 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 "symbolic" or "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.
Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.

Logic in India

Hindu logic

Origin

The Nasadiya Sukta of the Rigveda contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuskoti: "A", "not A", "A and 'not A, and "not A and not not A".
Logic began independently in ancient India and continued to develop to early modern times without any known influence from Greek logic.

Before Gautama

Though the origins in India of public debate, one form of rational inquiry, are not clear, we know that public debates were common in preclassical India, for they are frequently alluded to in various Upaniṣads and in the early Buddhist literature. Public debate is not the only form of public deliberations in preclassical India. Assemblies of various sorts, comprising relevant experts, were regularly convened to deliberate on a variety of matters, including administrative, legal and religious matters.

Dattatreya

A philosopher named Dattatreya is stated in the Bhagavata Purana to have taught Anviksiki to Aiarka, Prahlada and others. It appears from the Markandeya purana that the Anviksiki-vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy. Dattatreya expounded the philosophical side of Anviksiki and not its logical aspect.

Medhatithi Gautama

While the teachers mentioned before dealt with some particular topics of Anviksiki, the credit of founding the Anviksiki in its special sense of a science is to be attributed to Medhatithi Gautama. Guatama founded the anviksiki school of logic. The Mahabharata, around the 5th century BC, refers to the anviksiki and tarka schools of logic.

Panini

developed a form of logic for his formulation of Sanskrit grammar. Logic is described by Chanakya in his Arthashastra as an independent field of inquiry.

Nyaya-Vaisheshika

Two of the six Indian schools of thought deal with logic: Nyaya and Vaisheshika. The Nyāya Sūtras of Aksapada Gautama 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.

Jain logic

made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning the nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable.
The Jains have doctrines of relativity used for logic and reasoning:
  • Anekāntavāda – the theory of relative pluralism or manifoldness;
  • Syādvāda – the theory of conditioned predication and;
  • Nayavāda – The theory of partial standpoints.
These concepts in Jain philosophy made important contributions to the thought, especially in the areas of skepticism and relativity.

Buddhist logic

Nagarjuna

, the founder of the Madhyamaka developed an analysis known as the catuṣkoṭi, a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the four possibilities of a proposition, P:
  1. P; that is, being.
  2. not P; that is, not being.
  3. File:Eight Patriarchs of the Shingon Sect of Buddhism Nagarjuna Cropped.jpg|thumb|Painting of Nāgārjuna from the Shingon Hassozō, a series of scrolls authored by the Shingon school of Buddhism. Japan, Kamakura period P and not P; that is, being and not being.
  4. not ; that is, neither being nor not being.Under propositional logic, De Morgan's laws would imply that the fourth case is equivalent to the third case, and would be therefore superfluous, with only 3 actual cases to consider.

    Dignaga

However, Dignāga is sometimes said to have developed a formal syllogism, and it was through him and his successor, Dharmakirti, that Buddhist logic reached its height; it is contested whether their analysis actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference-warranting relation, "vyapti", also known as invariable concomitance or pervasion. To this end, a doctrine known as "apoha" or differentiation was developed. This involved what might be called inclusion and exclusion of defining properties.
Dignāga's famous "wheel of reason" is a method of indicating when one thing can be taken as an invariable sign of another thing, but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive.

Logic in China

In China, a contemporary of Confucius, Mozi, "Master Mo", is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Due to the harsh rule of Legalism in the subsequent Qin dynasty, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists.

Logic in the ancient Mediterranean

Prehistory of logic

Valid reasoning has been employed in all periods of human history. However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement". The ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid. Ancient Babylon was also skilled in mathematics. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axioms and assumptions, while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science.

Ancient Greece before Aristotle

While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seemed aware of geometric methods.
Fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The proofs of Euclid of Alexandria are a paradigm of Greek geometry. The three basic principles of geometry are as follows:
  • Certain propositions must be accepted as true without demonstration; such a proposition is known as an axiom of geometry.
  • Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a proof or a "derivation" of the proposition.
  • The proof must be formal; that is, the derivation of the proposition must be independent of the particular subject matter in question.
Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi, probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity. In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the Rhetoricians or Orators and the Sophists, who used arguments to defend or attack a thesis, both in legal and political contexts.

Thales

It is said Thales, most widely regarded as the first philosopher in the Greek tradition, measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering Thales' theorem just as Pythagoras had the Pythagorean theorem.
Thales is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed. Indian and Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon.