Binary Logic

Binary logic studies two possibilities, such as true and false. Test each possibility against the stated rules.

Choose one truth assignment, propagate it, and reject the entire case if any statement conflicts. A truth table is a short proof, not guesswork.

Exactly one of two statements is true: A says “The box is red”; B says “The box is not red”. What follows? 1. If the box is red, A is true and B is false. 2. If it is not red, A is false and B is true. 3. Both cases fit, so the colour cannot be determined.

The table checks the two possible truth assignments. Whether the box is red or not red, exactly one of A and B is true. The added condition therefore supplies no distinguishing information. A correct conclusion must hold across every surviving case, not only the case tried first.

Exactly one of A,B,C tells the truth. A says "B is truthful"; B says "C is truthful"; C says "A and B are liars." If A were truthful, B would be truthful too, impossible. If B were truthful, C would be truthful too, impossible. Thus C alone is truthful. Verify: A's claim about B is false, B's claim about C is true, a contradiction! Therefore the supplied conditions have no solution.

Distinguish a consistent possibility from a unique answer. Always verify the surviving case against every speaker. Elimination alone can leave a case that also fails.

If P then Q rules out P with not-Q. It does not imply that Q requires P. The contrapositive, if not-Q then not-P, is valid.

If a file is approved, it is signed. An unsigned file cannot be approved. A signed file may or may not be approved, because signing could happen for other reasons.

Choose one truth assignment, propagate it, and reject the entire case if any statement conflicts. A truth table is a short proof, not guesswork.

Exactly one of A,B,C tells the truth. A says "B is truthful"; B says "C is truthful"; C says "A and B are liars." If A were truthful, B would be truthful too, impossible. If B were truthful, C would be truthful too, impossible. Thus C alone is truthful. Verify: A's claim about B is false, B's claim about C is true, a contradiction! Therefore the supplied conditions have no solution.

Always verify the surviving case against every speaker. Elimination alone can leave a case that also fails.

A truth-teller makes true statements; a liar makes false statements. If an alternator is defined to alternate truth values between consecutive statements, test both possible starting truth values. Always use the definitions supplied in the puzzle.

Exactly one of A,B,C is a truth-teller, one a liar and one an alternator. Their two statements in order are: A: “I am not the liar.” “C is the liar.” B: “I am the liar.” “A is the truth-teller.” C: “I am the alternator.” “B is the liar.” B cannot be truthful, because “I am the liar” would be false, and cannot be a liar, because that statement would be true. Thus B alternates false then true, establishing A as truthful and C as the liar. A's statements are both true; C's are both false. This checks all three roles. Conditional logic “If P, then Q” rules out P together with not-Q and is equivalent to “if not-Q, then not-P”. It does not imply Q→P. “P only if Q” means P→Q; “P if Q” means Q→P. Inclusive “P or Q” permits both unless exactly one is specified.

Do not infer an alternator's starting truth value from its role alone. A single statement about a liar needs the complete role constraints before it determines who is speaking.