1. From Truth Tables to Recursion Tables

When Ludwig Wittgenstein introduced truth tables, he gave the abstract laws of propositional logic a visible form. Each connective could be understood at a glance by inspecting all possible combinations of truth values. What had to be inferred could now be seen.

A truth table does something very simple: it makes completeness explicit. Every possible case is listed. Once the table is filled, nothing remains hidden. The behavior of a logical connective is no longer something hidden in the interplay of rules and axioms. It is something that can be inspected.

This small device had a large effect on how logic could be taught. It replaced part of the burden of reasoning with a form of visual survey. Instead of asking whether a definition covers all cases, one could simply look.

The idea behind what we will call recursion tables is to carry this move into a different domain.

In place of truth values, we now consider values constructed from a small set of rules. Instead of asking how a connective behaves on fixed inputs, we ask how a function behaves on values that are built step by step. The question is no longer about truth, but about construction.

Where truth tables enumerate combinations of values, recursion tables enumerate combinations of constructors. Each row and column corresponds to a way in which an input can be formed. Each cell records what the function does in that case.

The aim is similar. We want to make visible that a definition is complete, that it handles every possible input, and that its recursive structure leads toward a result.

The shift, however, is significant. Truth tables display static outcomes. Recursion tables display processes. Their entries do not merely state what is the case; they indicate how a result is obtained, often by referring back to smaller cases.

This difference reflects a broader change in perspective. In propositional logic, correctness is determined by truth values. In the constructive setting we are moving toward, correctness is determined by how something is built.

Recursion tables are a modest attempt to make that constructive discipline visible. They do not replace formal reasoning. But like truth tables before them, they can give the mind something to hold on to.