WEBVTT

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With X, Y and Z, there are eight complete valuations.

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This is a new labelling scheme, with each three-bit string beside

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its letter.

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A is now one, zero, zero, rather than an extension of every

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earlier two-variable label by the same last bit.

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The first bit is X, the second Y, and the third Z.

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The labels are merely names for complete assignments.

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Any calculation must follow the displayed bit strings, not a remembered alphabetical

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ordering from the four-state board.

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Our first premise is X implies Y.

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It fails when X is true and Y false.

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Z may be either zero or one.

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On the new table those rows are A and E.

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All other rows satisfy it, so its truth region is B, C,

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D, F, G and H.

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Keep that region marked without removing its complement from the displayed carrier.

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We are constructing the premise set before making a joint update.

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A and E remain visible as the reason this implication is not

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a tautology.

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Our second premise is Y implies Z.

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It fails at true Y and false Z.

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X may be either zero or one.

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Those rows are B and D.

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Its satisfying region is A, C, E, F, G and H.

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Each letter has the same bit string as in the first premise

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table.

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Intersect the two premise regions.

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C, F, G and H survive both.

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A and E fail the first premise, while B and D fail

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the second.

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The surviving labels therefore account for every row of the eight-row

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table.

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X implies Z fails at A and D, where X is true

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and Z false.

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Its truth region is B, C, E, F, G and H.

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Every common premise survivor belongs to it.

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The inclusion proves hypothetical syllogism: the two implication premises entail the third.

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It is proper because B and E satisfy the conclusion but not

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both premises.

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Validity requires inclusion, not equality of the two regions.

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Assume both premise implications hold at one valuation.

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If X is false, the conclusion implication is true.

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If X is true, the first premise forces Y true, and then

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the second forces Z true.

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In either case the conclusion holds.

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This chain is a truth-table argument, not a promise that X

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causes Y or that events occur in that temporal order.

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All three variables have fixed values at one complete valuation throughout the

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reasoning.

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Pick B or E, a valuation allowed by the conclusion but excluded

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by the joint premises.

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Identify which premise fails there.

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Does this extra conclusion world invalidate the inference?

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At B, X is false and Z false, so X implies Z

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is true.

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But Y is true and Z false, making the second premise false.

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This extra conclusion world does not invalidate the inference: no premise survivor

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falsifies the conclusion.

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The method scales from four worlds to eight without changing the validity

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criterion.

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What must be rebuilt is the labelled valuation table.

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Keeping the old letters without their new bit strings would make a

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correct set calculation impossible to audit.