Inference as inclusion of truth regions

A truth region collects the worlds where a formula is true. AND corresponds to intersection, inclusive OR to union, and classical NOT to complement inside the current carrier. Our cards use X for dot and Y for stripe: A has only a dot, B both, C only a stripe, and D neither. Premises must be evaluated together at one world. Collect the valuations satisfying all premises. A conclusion follows classically when every one of those surviving valuations also satisfies the conclusion. This is a universal claim, so one failing survivor is enough to disprove it. In set language, the common premise region must be included in the conclusion's truth region. Inclusion permits equality but does not require it. A stronger group of premises may select fewer valuations than its consequence selects. Take X and Y together as premises. Their common truth region is B alone. The conclusion X is true at A and B. Every premise survivor is therefore in the conclusion region. The inclusion is proper, yet the inference is valid. The conclusion allows an additional world A, which does not satisfy both premises. A consequence can be weaker than the information supplied by the premises. Now ask whether X alone entails Y. The premise region contains A and B. A satisfies X and falsifies Y, so A is a surviving counterexample. B's success cannot rescue a universally quantified claim that fails at A. In general, compute the premise region with the conclusion region removed. A nonempty remainder consists of counterexamples. An empty remainder is exactly inclusion, so the same finite procedure tests validity and identifies where invalidity occurs. If each formula entails the other, the two truth regions are included in one another and therefore equal. This is semantic equivalence. It is stronger than sharing some true rows or one-way entailment. If the premises are inconsistent, their common region is empty. There is no surviving valuation that could satisfy all premises and falsify the conclusion. Classical semantic consequence therefore holds vacuously: the universal claim has no counterexample. This does not make the premises satisfiable or provide an actual world witnessing the conclusion. Choosing fewer or more observations changes which events are certifiable. It does not change the classical truth regions of the fixed formulas. A valid inclusion remains valid even when the available interface cannot certify its conclusion at every satisfying point. Conjoining premises intersects subsets of valuations. Intersecting two topologies instead intersects collections of subsets on one carrier. Those are different types of operation, and the latter is not how we compute a common premise region. Do X or Y together with not X entail Y? Find the common premise region, compare it with Y's region, and explain why no counterexample remains. X or Y allows A, B and C. Not X allows C and D. Their intersection is C, which lies inside Y's region B and C. The inference is valid by inclusion, whether or not we attach a traditional rule name. The method is stable: evaluate formulas, intersect premise regions, then test inclusion. Observational topology adds structure to these regions and their maps; it does not introduce a new criterion of classical logical validity.