Disjunctive syllogism: A OR B, NOT A, therefore B

Disjunctive syllogism: from A OR B and NOT A, conclude B. The validity check holds on all four rows; the premises are jointly true only on A = FALSE, B = TRUE, where B is indeed true.

Either A or B; not A; therefore B. Eliminating one branch of an OR leaves the other, which is the reasoning behind every process of elimination. The model has the premises p1, A OR B, and p2, NOT A, and the conclusion B; the engine's reading of the conclusion is “B is true”. On the four-row table p1 is false only on row 0, where both inputs are false, and p2 is true on rows 0 and 1, where A is false.

The validity check holds: no admissible assignment makes every premise true and the conclusion false. The only row where both premises are true is row 1, A = FALSE, B = TRUE, and the conclusion is true there. The premise-consistency check holds with that same witness, A = FALSE, B = TRUE, so the argument is valid for a substantive reason rather than vacuously. Because the OR is inclusive, the row A = TRUE, B = TRUE satisfies p1 but not p2, and it plays no role in the verdict.

This form appears whenever a fault tree or a diagnostic flow narrows the cause: the failure is in the network or in the database; the network checks out; so look at the database. It is also the logic of a fallback branch in code. The thing to watch is the first premise: the argument is only as good as the claim that the two options are exhaustive. If a third cause exists, A OR B is false on the row that matters and the conclusion is unsupported, which the table cannot see because the missing option is not a variable.

p1 — A OR B · premisep2 — NOT A · premiseconclusion — B · conclusion

Exact checks

Truth table (4 rows)

Plain-language reading

Logic diagram

The common slip

Eliminating the wrong way: from A OR B and A concluding NOT B. The OR is inclusive, so the row A = TRUE, B = TRUE makes both premises true and NOT B false; that argument is invalid. Disjunctive syllogism needs the negation of one branch, not its affirmation.

Questions

Would the argument still be valid with an exclusive or?

Yes: A XOR B is false on row 3 as well as row 0, which only removes a row where p2 was already false. The row A = FALSE, B = TRUE remains the only one with both premises true, and the conclusion holds there. The affirming version, from A to NOT B, becomes valid only under XOR.

Related

Guides

• Rules of inference

Operators and gates

• NOT gate (inverter): truth table, symbol and the NOT operator
• OR gate: truth table for 2, 3 and 4 inputs, inclusive or explained

Tools

• Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
• Argument validity checker: premises, conclusion, counterexample or proof
• Karnaugh map solver: 2 to 4 variables, groups, SOP and POS, don’t-cares from assumptions

See also

• Modus tollens: A → B, NOT B, therefore NOT A (valid)
• Constructive dilemma: (A → B) AND (C → D), A OR C, therefore B OR D
• OR gate: truth table for 2, 3 and 4 inputs, inclusive or explained
• Inclusive vs exclusive or (OR vs XOR), with truth table
• Argument validity checker: premises, conclusion, counterexample or proof
• Premise in logic: meaning and examples

Worked examples

• Entry rule: (member OR invited) AND NOT banned, checked every way
• Entry rule slip: member OR (invited AND NOT banned) and the case that differs
• Majority of three (2-out-of-3 voting): truth table, K-map and NAND form
• Odd parity of three inputs: A XOR B XOR C

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.