Logisticon

Summary

Any logical function of n arguments can be represented by a single Venn diagram, each of whose spaces contains a 0 or 1 in Boolean fashion. Using 1 for true and 0 for false they constitute the truth table of the function. The laws by which such functions operate upon each other will be simply stated, and the theory will be extended to cover probabilistic logic by the introduction of p’s in the places normally restricted to 0 or 1, where 0 ≤ p ≤ 1, thus producing a way of evaluating all such functions on simple digital calculators. Every such Venn function can be realized by a formal neuron at a fixed threshold which merely adds afferents. By changing the threshold in unitary steps, any such neuron may be made to compute a dissimilar function for every step, thus producing 2ⁿ + 1 functions ranging from contradiction to tautology. It is necessary and sufficient to assure excitation at the neuron, and inhibition of two kinds, one of which raises the threshold of the neuron while the other prevents an impulse over an afferent fiber from reaching the neuron. The method of construction and minimization of the required constructions for such diagrams will be given.

Combinations of these diagrams will be used for the construction of infallible nets of fallible components for n > 2, and they will be examined to show the limits of that infallibility under perturbation of threshold, amplitude of signals, synapsis, and even scattered loss of neurons. We will show how nets can be constructed which are logically stable under common shift of threshold which alters the function computed by every neuron but not the input-output function of the net. In passing, we will show the construction of polyphecks, that is, all of those Venn functions which, when given n arguments, can produce all functions.

Finally, we will examine the flexibility of minimal nets, consisting of n + 1 neurons with n inputs for n = 2 and n = 3, and show that the former can compute 15 out of the 16 logical functions, and the latter 253 out of 256.

In connection with his Summary the author wrote to us in a letter of April 20, 1960: “My title, however, is taken from the Pythagoreans, for I shall deal with the brain only as a device for handling arithmetic. Hence ‘Logisticon,’ and I will send you a copy of what I have proposed originally on this score to Colin Cherry.” Because Dr. McCulloch’s oral paper “Logisticon,” summarized above, is so similar in its conclusions to what he presented at the Ninth Alfred Korzybski Memorial Lecture of March 12, 1960, the substance of that lecture—not heretofore released in book form-is now quoted in extenso with the kind permissions of both the author and the Institute of General Semantics, which had offset the lecture in Nos. 26/27 of its Bulletin under the title “What Is a Number, that a Man May Know It, and a Man that He May Know a Number?”—Ed.

The work of the Research Laboratory of Electronics, of which the author is a staff member, is supported in part by the U.S. Army (Signal Corps), the U.S. Air Force (Office of Scientific Research, Air Research and Development Command), and the U.S. Navy (Office of Naval Research). The work of Dr. McCulloch’s group also receives support from National Institutes of Health and Teagle Foundation, Incorporated.