Pullbacks: matching two premises to the same world

We are using classical truth regions again. Our original X,Y assignments are A: one-zero, B: one-one, C: zero-one and D: zero-zero. A witness for two premises together must be one world satisfying both. Two different worlds, each satisfying just its own premise, are not a common witness. One means true and zero means false. Recall Modus Ponens. Call the implication region R; it contains B, C and D. Call the X region S; it contains A and B. A pair taking one point from R and one from S need not describe a single world satisfying both premises. A Cartesian product consists of ordered pairs: first an element of R, then an element of S. Our product has six pairs. Their order matters because the first coordinate witnesses the implication premise and the second witnesses X. Both premise regions sit inside the same ambient valuation space. Their inclusion maps keep each point's identity. We now retain only pairs whose two images in that ambient carrier agree. Following either coordinate into the ambient space must reach the same valuation. This is an equality requirement, not a comparison of physical position or observation signature. Different valuations with equal signatures are still different ambient points here. C satisfies implication and A satisfies X, so the pair C, A belongs to the product. But the two inclusions land at different valuations. The pair does not describe one world where both premises hold, and must be rejected. B lies in both premise regions. Its two images agree, so B, B is retained. Checking the other product pairs leaves no additional match. The matching-pair set therefore contains just this one pair. This set of compatible pairs is the pullback of the two inclusions. When the maps are inclusions, agreement means both coordinates are the same point v belonging to the intersection. The surviving pairs are exactly the diagonal pairs v, v from R intersect S. Sending an intersection point to its repeated pair gives a one-to-one correspondence. Taking either coordinate returns the original point. This identifies the matching-pair set with the ordinary intersection, without creating a new valuation or a new inference rule. Composition means applying one function after another. The square commutes because the two paths send each compatible pair to the same ambient point. The equality is guaranteed by the matching condition we used to construct the set. So far we have explained the points and maps. A topological pullback also needs the appropriate topology. The next lecture will identify it with the intersection's induced topology and explain the unique compatible factorization. Form the compatible-pair set for the inclusions of the X and Y truth regions. Which pair survives? Why is A, C not a valid common-world witness even though its two coordinates satisfy the two premises separately? Only B lies in both regions, so only B, B survives. The pair A, C lands at different ambient valuations and fails the equality condition. Individual premise witnesses must coincide to witness their conjunction at one world. The categorical diagram makes the matching requirement explicit. For truth-region inclusions its points are just an intersection in pair notation. Its value here is structural organization, not an extra form of logical consequence beyond truth-set inclusion.