Hypothetical syllogism and eight possible worlds

With X, Y and Z, there are eight complete valuations. This is a new labelling scheme, with each three-bit string beside its letter. A is now one, zero, zero, rather than an extension of every earlier two-variable label by the same last bit. The first bit is X, the second Y, and the third Z. The labels are merely names for complete assignments. Any calculation must follow the displayed bit strings, not a remembered alphabetical ordering from the four-state board. Our first premise is X implies Y. It fails when X is true and Y false. Z may be either zero or one. On the new table those rows are A and E. All other rows satisfy it, so its truth region is B, C, D, F, G and H. Keep that region marked without removing its complement from the displayed carrier. We are constructing the premise set before making a joint update. A and E remain visible as the reason this implication is not a tautology. Our second premise is Y implies Z. It fails at true Y and false Z. X may be either zero or one. Those rows are B and D. Its satisfying region is A, C, E, F, G and H. Each letter has the same bit string as in the first premise table. Intersect the two premise regions. C, F, G and H survive both. A and E fail the first premise, while B and D fail the second. The surviving labels therefore account for every row of the eight-row table. X implies Z fails at A and D, where X is true and Z false. Its truth region is B, C, E, F, G and H. Every common premise survivor belongs to it. The inclusion proves hypothetical syllogism: the two implication premises entail the third. It is proper because B and E satisfy the conclusion but not both premises. Validity requires inclusion, not equality of the two regions. Assume both premise implications hold at one valuation. If X is false, the conclusion implication is true. If X is true, the first premise forces Y true, and then the second forces Z true. In either case the conclusion holds. This chain is a truth-table argument, not a promise that X causes Y or that events occur in that temporal order. All three variables have fixed values at one complete valuation throughout the reasoning. Pick B or E, a valuation allowed by the conclusion but excluded by the joint premises. Identify which premise fails there. Does this extra conclusion world invalidate the inference? At B, X is false and Z false, so X implies Z is true. But Y is true and Z false, making the second premise false. This extra conclusion world does not invalidate the inference: no premise survivor falsifies the conclusion. The method scales from four worlds to eight without changing the validity criterion. What must be rebuilt is the labelled valuation table. Keeping the old letters without their new bit strings would make a correct set calculation impossible to audit.