## A Brief History *of* **Lambda Calculus**

A historical perspective on the threads that lead to the developments of Lambda Calculus

A good way to orient oneself to study a subject is to understand the prior art that has went into the development of the subject. To clarify the standing of Lambda Calculus and its stature within logic, let us sketch out the historical threads in the Western tradition that lead up to its creation. We will observe how it was influenced by and exerted influence on related fields of inquiry.

One has to be careful when studying history through the lens of modern concepts for they act as retroactive ideas when understanding the past. People in the past had an entirely different epistemology (way of understanding ideas about the world) compared to us. What we now understand to be terms like for instance, modal logic, during earlier times, were known by a combination of different terms. And for some concepts, they could be non-existent or could remain tacit in the conceptual structure of early periods. So when ascribing that a certain person invented a modern idea, we are usually referring to something that is identifiable in the form we understand it today, though its historical roots could be fused together with an entirely disconnected piece of work in a remote domain in modern parlance.

### I. Antiquity

The tradition of logic as we know it today can be considered to have taken shape with the philosophical discourses of Pre-Socratic philosophers. The roots quite likely extend further back in history at least as far as Egyptian and Babylonian times. For the current context of understanding the evolution of Lambda Calculus, we will begin our journey by describing the main schools of logic in ancient Greece.

A great personality to start the discussion of history of logic with is Aristotle. Aristotle was a student of Plato. Logical argumentation was a significant theme in the Greek intellectual discourses. Aristotle was a key figure who systematized the structure of these logical investigations. He wrote extensively on Prior and Posterian Analytics. In Prior Analytics, he rigorously examined the kinds of structures possessed by logical arguments. Much of the evolutionary threads of Lambda Calculus conversed with his canon and borrowed ideas from it for their subsequent developments.

In antiquity logic and rhetoric were seen as two parts of a whole. Modern logic is derived from the study of rhetorical arguments which was given an important cultural value in Ancient Greece.

Aristotle established his work in the late 4th century. It emerged from the ambient dialectical milieu where debates and discussions, that employed logic among peers were a part and parcel of the intellectual hertiage. Aristotle created his canon by closely examining the internal form of language constructs, and logical arguments. With this insight he built a framework that categorized the internal structure of arguments of these discourses. This close connection between rhetoric and logic has to be kept in mind when reading about ancient works. This is because if a concept feels incomplete in the logical doctrine, it may be complemented in the rhetorical doctrine. The resulting logic he built is commonly known as *term logic* or *syllogistic logic*.

The building blocks of these logic are terms which can be thought of as categories or classes treated as subjects, which are then linked to something called a predicate using a construct for relating them called copula. These statements can then be *converted* to various other kinds of representations and related to one another resulting in further classifications known as *moods* and *figures*.

Aristotle’s logic can be thought of as a term logic. His work also contained within them the seeds of modal logic whereby the possibilities and necessities of arguments are analyzed.

This work became the canon and remained the common touchpoint for evolution of logic up until late 19th century and in its original form still continues in certain practices to this day.

After this, his students Theophrastus and Eudemus developed the subject into late antiquity (400-100 B.C.) One interesting work could be the Poryphry tree which is a categorization format that resembles the knowledge graphs of today.

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