WEBVTT

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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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One means true and zero false.

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To test an inference, every world satisfying all its premises must also

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satisfy its conclusion.

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All those premises refer to the same world.

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Material implication fails only when X is true and Y false.

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Its truth region is B, C and D.

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We will intersect that region with a second premise, keeping both premises

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evaluated at the same valuation.

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Modus Ponens adds X.

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The common survivors must satisfy implication and X together.

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B belongs to both regions; A fails implication, and C and D

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fail X.

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The common premise region is B.

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The conclusion Y is true at B and C, so every common

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survivor satisfies Y.

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This proves the finite truth-region instance of Modus Ponens.

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More generally, an implication can be false only at true antecedent and

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false consequent.

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Once its antecedent is true and the implication is assumed true, that

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falsifying combination is excluded.

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The consequent must be true.

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Reset the premise selection.

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Modus Tollens adds not Y instead of X.

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Intersect the implication region B, C and D with the no-Y

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region A and D.

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Only D survives.

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D falsifies X and therefore satisfies not X.

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Every survivor belongs to the conclusion region C and D.

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We have changed which premise is supplied while retaining the same implication

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and carrier.

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If Y is false but implication remains true, X cannot be true.

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Otherwise we would have exactly the implication's falsifying combination.

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This explains the rule independently of our particular A–D names.

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These are classical formulas evaluated at complete valuations.

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Later open-set implication and negation will impose observability through interior.

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We should not silently substitute those operations into a calculation that has

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used classical complement all along.

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If implication and Y are premises, the common region contains B and

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C.

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B satisfies X, but C does not.

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C is an explicit counterexample to the proposed conclusion X.

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A true consequent does not force a true antecedent.

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The rule's name is less important than locating this valuation: it

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satisfies every premise and falsifies the conclusion.

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The same inspection procedure distinguishes a valid rule from a plausible-looking

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reversal.

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Use a new implication: if X or Y, then X.

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Treat X or Y as its whole antecedent.

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If that implication and its antecedent are both premises, do they entail

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X?

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Compute the common truth region before checking the answer.

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The implication is false only at C.

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Its region is A, B and D.

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We have evaluated this compound implication before intersecting it with the other

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premise.

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The antecedent premise X or Y has region A, B and C.

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Keep both input regions visible and require a valuation to satisfy both

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at once.

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Their intersection leaves A and B.

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This is exactly X's region, so the inference is valid.

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The named rule applies to this compound antecedent through the same truth-

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value argument.

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Modus Ponens and Modus Tollens work with compound formulas too.

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Their validity comes from the same common-survivor inclusion criterion.

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A full table or set calculation should make that reason visible rather

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than relying on the authority of a rule name.