Modus Ponens and Modus Tollens, one state at a time

Our original X,Y assignments are A: one-zero, B: one-one, C: zero-one and D: zero-zero. One means true and zero false. To test an inference, every world satisfying all its premises must also satisfy its conclusion. All those premises refer to the same world. Material implication fails only when X is true and Y false. Its truth region is B, C and D. We will intersect that region with a second premise, keeping both premises evaluated at the same valuation. Modus Ponens adds X. The common survivors must satisfy implication and X together. B belongs to both regions; A fails implication, and C and D fail X. The common premise region is B. The conclusion Y is true at B and C, so every common survivor satisfies Y. This proves the finite truth-region instance of Modus Ponens. More generally, an implication can be false only at true antecedent and false consequent. Once its antecedent is true and the implication is assumed true, that falsifying combination is excluded. The consequent must be true. Reset the premise selection. Modus Tollens adds not Y instead of X. Intersect the implication region B, C and D with the no-Y region A and D. Only D survives. D falsifies X and therefore satisfies not X. Every survivor belongs to the conclusion region C and D. We have changed which premise is supplied while retaining the same implication and carrier. If Y is false but implication remains true, X cannot be true. Otherwise we would have exactly the implication's falsifying combination. This explains the rule independently of our particular A–D names. These are classical formulas evaluated at complete valuations. Later open-set implication and negation will impose observability through interior. We should not silently substitute those operations into a calculation that has used classical complement all along. If implication and Y are premises, the common region contains B and C. B satisfies X, but C does not. C is an explicit counterexample to the proposed conclusion X. A true consequent does not force a true antecedent. The rule's name is less important than locating this valuation: it satisfies every premise and falsifies the conclusion. The same inspection procedure distinguishes a valid rule from a plausible-looking reversal. Use a new implication: if X or Y, then X. Treat X or Y as its whole antecedent. If that implication and its antecedent are both premises, do they entail X? Compute the common truth region before checking the answer. The implication is false only at C. Its region is A, B and D. We have evaluated this compound implication before intersecting it with the other premise. The antecedent premise X or Y has region A, B and C. Keep both input regions visible and require a valuation to satisfy both at once. Their intersection leaves A and B. This is exactly X's region, so the inference is valid. The named rule applies to this compound antecedent through the same truth-value argument. Modus Ponens and Modus Tollens work with compound formulas too. Their validity comes from the same common-survivor inclusion criterion. A full table or set calculation should make that reason visible rather than relying on the authority of a rule name.