Artificial Intelligence 1943

A Logical Calculus of the Ideas Immanent in Nervous Activity

Warren S. McCulloch & Walter Pitts

Treat a neuron as all-or-none, and a network of them becomes a machine that computes logic.

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In depth · the introduction

In 1943 a brain scientist and a self-taught runaway logician wrote down the first equation for a single nerve cell — and, almost by accident, drew the first blueprint for artificial intelligence.

The idea, unpacked

A nerve cell, McCulloch and Pitts started from, is all-or-none: it either fires or it doesn't, like a light switch rather than a dimmer. So they asked a daring question — if a single neuron is just a switch, what can a whole network of switches do?

Their answer was that, wired the right way, such networks compute logic. Give a neuron a rule — "fire only if at least two of your inputs are on" — and it behaves as a logical AND. Change the rule and you get OR, or NOT. And since all of logic, and so all of computation, can be built from those few pieces, a network of these tiny yes/no decisions can in principle compute anything a computer can. The brain, on this view, is a vast machine made of switches.

Where it came from

The pair were an unlikely match. Warren McCulloch was a philosophically minded neurophysiologist in his forties, haunted by the question of how the wet matter of the brain could possibly embody logic. Walter Pitts was barely out of his teens — a runaway who had taught himself formal logic, written to Bertrand Russell as a boy, and studied under the logician Rudolf Carnap before McCulloch took him into his home.

Fusing neurophysiology with the formal logic of Whitehead and Russell, and with Alan Turing's brand-new idea of a universal computing machine, they produced a paper so densely symbolic it was almost unreadable. Yet the right person read it: John von Neumann, who two years later built their "logical neuron" into his design for the modern stored-program computer.

Why it mattered

It was the first time anyone gave a precise, mathematical answer to an ancient question: could a machine, in principle, think? By showing that the brain's own hardware could be described as computation, the paper launched two fields in one stroke. It told engineers that networks of simple switches could compute anything — a promise that runs straight to today's neural networks. And it told brain scientists that the mind might be understood as information being processed. Every artificial neural network since, and the AI you may be using right now, descends from this idea.

A way to picture it

Think of a bouncer deciding whether to let the night go ahead. He'll allow it only if enough approved guests have shown up — that's a threshold. But there's one banned name on the list, and if that person walks in, the night is off, no matter how big the crowd. That "enough guests, unless the banned one appears" rule is exactly the McCulloch–Pitts neuron: count the excitatory inputs against a threshold, but let any inhibition cancel everything. Try it yourself below.

Where it sits

This paper is a hinge in the history of ideas. Behind it stand George Boole, who in 1854 turned logic into algebra, and Alan Turing, who in the 1930s turned computation into something a machine could carry out. Ahead of it stands the whole of modern AI: Frank Rosenblatt's perceptron (1958) gave the neuron the power to learn; decades later AlexNet (2012) and the Transformer (2017) stacked millions of these units into the systems we now simply call AI. McCulloch and Pitts are where that long line begins.

The original document

Original source text

Warren S. McCulloch & Walter Pitts · Bulletin of Mathematical Biophysics, Vol. 5, pp. 115–133 · 1943

Abstract

Because of the "all-or-none" character of nervous activity, neural events and the relations among them can be treated by means of propositional logic.

It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes.

The five physical assumptions

From the known physiology of the neuron the authors abstract five idealizing assumptions. They are stated plainly, and everything in the paper is built on them:

1. The activity of the neuron is an "all-or-none" process.

2. A certain fixed number of synapses must be excited within the period of latent addition in order to excite a neuron at any time, and this number is independent of previous activity and position on the neuron.

3. The only significant delay within the nervous system is synaptic delay.

4. The activity of any inhibitory synapse absolutely prevents excitation of the neuron at that time.

5. The structure of the net does not change with time.

From neurons to propositions

On these assumptions each neuron becomes a two-valued device, firing or not firing at each tick of a discrete clock set by the synaptic delay. Its firing at one instant is read as a proposition asserting that the conditions for it held an instant before. The paper then shows that nets without closed loops realize exactly the expressions of propositional logic — conjunction, disjunction, negation and their combinations — while nets that contain loops ("circles") acquire memory, letting a fact reverberate and so depend on the whole past.

[ … ]

The reach of the result

Building on Turing's 1936 notion of computability, the authors argue that, apart from the provision of an unlimited tape, such nets can compute whatever is effectively computable — that the logic of the idealized brain and the logic of the universal machine are, in this precise sense, the same. The closing sections turn to consequences for psychology and the theory of knowledge. The notation, borrowed from Carnap and Russell, is famously forbidding; the full nineteen pages, with every theorem and proof, are at the source below.

Chicago · 1943