WEBVTT

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We are using classical truth regions again.

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Our original X,Y assignments are A: one-zero, B: one-one,

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C: zero-one and D: zero-zero.

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A witness for two premises together must be one world satisfying both.

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Two different worlds, each satisfying just its own premise, are not a

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common witness.

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One means true and zero means false.

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Recall Modus Ponens.

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Call the implication region R; it contains B, C and D.

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Call the X region S; it contains A and B.

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A pair taking one point from R and one from S need

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not describe a single world satisfying both premises.

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A Cartesian product consists of ordered pairs: first an element of R,

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then an element of S.

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Our product has six pairs.

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Their order matters because the first coordinate witnesses the implication premise and

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the second witnesses X.

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Both premise regions sit inside the same ambient valuation space.

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Their inclusion maps keep each point's identity.

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We now retain only pairs whose two images in that ambient carrier

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agree.

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Following either coordinate into the ambient space must reach the same valuation.

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This is an equality requirement, not a comparison of physical position or

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observation signature.

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Different valuations with equal signatures are still different ambient points here.

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C satisfies implication and A satisfies X, so the pair C, A

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belongs to the product.

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But the two inclusions land at different valuations.

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The pair does not describe one world where both premises hold, and

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must be rejected.

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B lies in both premise regions.

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Its two images agree, so B, B is retained.

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Checking the other product pairs leaves no additional match.

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The matching-pair set therefore contains just this one pair.

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This set of compatible pairs is the pullback of the two inclusions.

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When the maps are inclusions, agreement means both coordinates are the same

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point v belonging to the intersection.

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The surviving pairs are exactly the diagonal pairs v, v from R

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intersect S.

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Sending an intersection point to its repeated pair gives a one-to-

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one correspondence.

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Taking either coordinate returns the original point.

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This identifies the matching-pair set with the ordinary intersection, without creating

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a new valuation or a new inference rule.

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Composition means applying one function after another.

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The square commutes because the two paths send each compatible pair to

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the same ambient point.

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The equality is guaranteed by the matching condition we used to construct

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the set.

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So far we have explained the points and maps.

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A topological pullback also needs the appropriate topology.

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The next lecture will identify it with the intersection's induced topology

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and explain the unique compatible factorization.

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Form the compatible-pair set for the inclusions of the X and

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Y truth regions.

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Which pair survives?

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Why is A, C not a valid common-world witness even though

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its two coordinates satisfy the two premises separately?

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Only B lies in both regions, so only B, B survives.

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The pair A, C lands at different ambient valuations and fails the

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equality condition.

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Individual premise witnesses must coincide to witness their conjunction at one world.

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The categorical diagram makes the matching requirement explicit.

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For truth-region inclusions its points are just an intersection in pair

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notation.

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Its value here is structural organization, not an extra form of logical

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consequence beyond truth-set inclusion.