Learning a premise: restricting the possible worlds

Compare two meanings of acquiring information. For this example, X means dot and Y means stripe. A has only a dot, B both, C only a stripe, and D neither. Their two-bit annotations follow X,Y order. These are the same four cards at the start of both branches. Both branches begin with all four valuations and only X selected. The opens are empty, A and B, and the whole carrier. Reset both branches before changing anything; otherwise their effects cannot be compared fairly. In the first branch we add Y to the observation menu. In the second we accept Y as a premise. The first changes which events can be certified; the second changes which valuations remain possible. Adding the available test Y retains A, B, C and D. It produces the six-open positive topology. The fact that we can now ask for a Y certificate does not say the actual valuation satisfies Y. Do not remove the Y-false cards in this branch. Availability is a feature of the interface, whereas a learned premise asserts a condition on possible worlds. Confusing the two would make the comparison meaningless. Return the other branch to the original one-test menu X. Accept Y as true. The possible carrier becomes B and C, the intersection of the old possibility region with the truth region of Y. Restrict every old open to this new carrier. The old dot region meets B and C at B alone. The whole old carrier restricts to B and C. Together with the empty set, these are the induced subspace opens. For a surviving point, intersect its old smallest neighborhood with the new possibility region. Under X alone, B's old neighborhood was A and B; after learning Y it becomes B. This is restriction, not a newly invented topology. Learning a second premise intersects again. The final possibility region is unchanged if we regroup the intersections, reverse the order of the premises, or repeat one premise. These properties are called associativity, commutativity, and idempotence. They concern accepted classical premises, not every real-world learning protocol. After learning Y, accepting X retains only B. On a singleton carrier the induced topology has the empty set and that singleton. This does not retroactively mean X was learned when it was merely added as an available test. Inconsistent premises leave no surviving valuation. The empty subspace is a legitimate mathematical outcome. It does not identify an impossible state card or assign truth values to a newly drawn empty-world point. Reset to all four states with Y as the sole test. Compare adding X as an available test with learning X as a premise. Give the carrier in each branch and the induced opens in the premise branch. Adding X keeps the full carrier and gives the positive X,Y topology. Learning X instead keeps only A and B while retaining the menu Y. Restricting Y's region gives B, so the premise branch has opens empty, B and the carrier A and B. State the menu and carrier after every change. Refinement changes expressibility on fixed points. Premise update changes the current points and retains the induced observation structure. Later inference will ask whether all surviving worlds satisfy a conclusion.