A Logical Calculus of the Ideas Immanent in Nervous Activity

Why?

McCulloch & Pitts built the first rigorous bridge between neuro-physiology and mathematical logic.

They idealised a biological neuron as a binary device that either fires (1) or stays silent (0) once every small time-step. Showing that suitably interconnected neurons can realise every formula of propositional logic—and in fact any computable function.

What?

The paper opens with five axioms. Re-phrased with symbols I can understand better:

1. Activity for a neuron is either 1 or 0: $x_i(t)\in{0,1}$.
2. If total excitation ≥ fixed threshold (θ) the neuron fires in the next instant: $x_i(t+1)=H!\bigl(\sum_j w_{ij}\,x_j(t)-θ_i\bigr)$ where $H$ is the unit step function (MP treat inhibition as a veto; I’m modelling that as –∞ for convenience).
3. Delay is negligible and identical for every synapse.
4. Synapses are either excitatory (positive weight) or inhibitory (sufficient to block firing regardless of excitation), i.e. an inhibitory input sets the argument of H to $-\infty$.
5. The network structure (weights, connections) is fixed. No learning considered; this is a logic circuit.

With this, we can easily build logic gates with a single neuron:

Because AND and NOT form a functionally complete set, any propositional formula can be implemented by a finite network.

They prove that every finite truth-table can be translated into a finite MP-net whose output reproduces the table row-by-row in one tick:

1. Disjunctive normal form – Any Boolean function $f(x_1,\dots,x_n)$ can be written $f=\bigvee_k (l_{k1}\land l_{k2}\land\dots)$ where each $l_{kj}$ is a literal $x_i$ or ¬$x_i$.
2. One neuron per term – Implement each parenthesised AND term with a neuron whose threshold equals the number of positive literals and which receives an inhibitory input for every negated literal.
3. Final OR neuron – Collect excitatory links from every term-neuron and set threshold = 1.

(I was very confused by just reading this but the code made it a lot clearer, see below)

Code

NINF = -float("inf") # veto value


class FormalNeuron:
    def __init__(self, weights, threshold):
        self.weights = weights
        self.threshold = threshold

    def output(self, inputs):
        total = 0
        for name, w in self.weights.items():
            v = inputs.get(name)
            if w == NINF:               # inhibitory line
                if v == 1:              # veto active → immediate silence
                    return 0
                else:                   # veto line silent → skip
                    continue
            total += w * v              # regular excitatory weight

        return 1 if total >= self.threshold else 0


# Three‑neuron XOR network
n1  = FormalNeuron({'x': 1, 'y': NINF}, threshold=1)  # x AND (NOT y)
n2  = FormalNeuron({'x': NINF, 'y': 1}, threshold=1)  # (NOT x) AND y
xor = FormalNeuron({'n1': 1, 'n2': 1}, threshold=1)   # OR of hidden nodes

# Truth‑table 
print("x y | n1 n2 | xor")
print("-" * 17)
for x in (0, 1):
    for y in (0, 1):
        inputs = {'x': x, 'y': y}
        inputs['n1'] = n1.output(inputs)
        inputs['n2'] = n2.output(inputs)
        inputs['xor'] = xor.output(inputs)
        print(f"{x} {y} |  {inputs['n1']}  {inputs['n2']} |  {inputs['xor']}")

Which gets you:

x y | n1 n2 | xor
-----------------
0 0 |  0  0 |  0
0 1 |  0  1 |  1
1 0 |  1  0 |  1
1 1 |  0  0 |  0

Random thoughts

• I’ve used o3 extensively to actually understand what I was reading. I did expect this. It is actually delightful and way faster to learn something. I’ve spent Sunday morning on this, I think without AI help it would’ve taken me at least three times that, or I just wouldn’t have done it.
• Coding has been the most fun part, also the easiest. I’ve tried to avoid using AI there for a change.
• Even if reading the paper and trying to understand has been painful, it’s also fun in other ways. I do wish sometimes I had stayed in Uni and done a PhD.
• Biology concepts are very far back on my brain.
• I did not expect math notation to have changed so much since the 40s? At least the notation that I was taught in school, which admittedly was Engineering and not Mathematics. I find modern notation way easier to understand.
• {AND, NOT} is already functionally complete—OR can be written ¬(¬x ∧ ¬y) but I still use an explicit OR neuron in examples for clarity.