ProLT Observation Topologies

From cards to observations

In the home-page example, A records colour (0 = teal, 1 = gold) and B records shape (0 = circle, 1 = square). The state 01 is a teal square. Recording colour alone groups it with the teal circle; recording shape as well separates them. The cards stay fixed while the observer gains a distinction.

This page takes the next step: it compares a complete record of yes/no answers with what can be established using positive answers alone. The first gives groups of matching answers; the second gives a topology.

The mathematical tool

In a Propositional Logical Topology (ProLT), possible truth assignments are points. A chosen family of formulas acts as observation tests. States with the same observation signature cannot be distinguished by those tests, even if they are distinct in the underlying state space.

From the selected observations we can build a finite topology, a specialization order, and distances on observation signatures. This makes the observer’s available distinctions explicit.

A four-state example

Start with the four states 00, 01, 10, and 11, where the first digit is A (colour) and the second is B (shape). Test A asks “Is it gold?”; test B asks “Is it square?” If an observer records both yes and no answers to A, it sees two groups: {00, 01} and {10, 11}. Recording B as well gives each state a different two-bit signature.

For the positive-observation construction, an open set is a region that can be described using positive test results, combined with “and” and “or,” together with the empty region and the whole space. Taking the complement of a region is not automatically allowed: “not gold” is not itself a positive result of the test “Is it gold?”

ProLT also asks what follows when a test contributes only its positive truth region. With A alone, the positive region is {10, 11}; the topology generated by this one region has just the open sets ∅, {10, 11}, and the full four-state space. Add B, whose positive region is {01, 11}, and the generated topology gains their intersection {11} and further unions. The two-sided signature partition now has four singletons, but {00} is still not an open set of this positive-observation topology. That difference is why the paper keeps a full yes/no record separate from positive observation.

A small general fact. Adding tests can split a two-sided signature group but cannot merge two groups that were already distinct: if states disagree on an old test, they still disagree after more tests are added. This follows immediately by restricting each new signature to the old tests. It is an illustrative construction, not a claim of a new theorem.

What the current manuscript establishes

The ProLT paper brings together finite Boolean, order-topological, observational, and distance constructions. It carefully separates a positive observation from a complete two-sided record, and adding observations from restricting premises. The paper identifies itself as an expository synthesis rather than a claim to a new class of spaces.

Why it matters to The MIND

ProLTs provide a concrete starting point for questions about self and non-self: what can this system tell apart, given the tests it can perform? Hierarchical concepts ask how limited resolution groups states; observation refinement asks what changes when a new test becomes available.

Current boundary

Acquirable Predictive Models adds an operational question: can permitted tests obtain a class that predicts outputs under known actions? It separates background distinguishability, the supplied query library, and an executable experiment. A complete signature in the mathematical description is not a free observation of the hidden state.

The points in this model are logical valuations. Treating them as mental states would require an additional, justified model of a cognitive system. The topology alone does not establish self-awareness.

Continue the example

Next: Observation Refinement distinguishes changes to the recorded answers, the observed states, and their relationships. For the formal definitions and proofs, read the ProLT manuscript online.

Mathematical background

The current manuscript situates its finite constructions alongside Stone’s representation theory for Boolean algebras, Stong’s work on finite topological spaces, and Vickers’s Topology via Logic. These are background references, not claims that ProLT introduced their underlying results.