{
  "format": "recallweave-deck/1",
  "title": "Boolean logic: from truth values to valid inference",
  "attribution": "Original questions, explanations and transfer prompts authored for RecallWeave by the HAMON estate contribution chatgpt-acd057031fb2, with AI assistance. Conceptual references: OpenStax Contemporary Mathematics, sections 2.2 and 2.4–2.7 (https://openstax.org/books/contemporary-mathematics/pages/2-2-compound-statements). No textbook exercise or figure is reproduced.",
  "license": "Original lesson text: CC BY 4.0. Classical two-valued propositional logic only; these questions do not establish the empirical truth of premises or measured learning benefit.",
  "concepts": ["Truth values and connectives", "Material implication", "Logical equivalence", "Valid inference"],
  "items": [
    {
      "id": "logic-connectives-1",
      "concept": "Truth values and connectives",
      "prerequisites": [],
      "prompt": "A and B are both True. What are the values of (A or B) and (A xor B), in that order?",
      "options": ["True, True", "True, False", "False, True", "False, False"],
      "answer": 1,
      "explanation": "Inclusive or is True when at least one operand is True, including the both-True case. Exclusive or (xor) requires exactly one True operand. With two True operands, the pair is therefore True, False. The two operators agree on the mixed assignments but differ here.",
      "transfer": "Invent two propositions about a fictional display. Describe how an inclusive-or rule and an exclusive-or rule would behave when both propositions are True."
    },
    {
      "id": "logic-connectives-2",
      "concept": "Truth values and connectives",
      "prerequisites": [],
      "prompt": "Which assignment makes A and not (B or C) True?",
      "options": ["A=True, B=True, C=False", "A=True, B=False, C=True", "A=False, B=False, C=False", "A=True, B=False, C=False"],
      "answer": 3,
      "explanation": "The outer and requires A=True and not (B or C)=True. The latter requires B or C=False, which happens only when both B and C are False. An assignment with B=True or C=True makes the parenthesized or True; A=False fails the outer and. Only A=True, B=False, C=False satisfies both parts.",
      "transfer": "Write an expression that is True exactly when A is True and at least one of B or C is True. Compare its accepted assignments with this question's expression."
    },
    {
      "id": "logic-connectives-3",
      "concept": "Truth values and connectives",
      "prerequisites": [],
      "prompt": "Set A=True and B=False. What are the values of ((not A) and B) and not (A and B), in that order?",
      "options": ["False, True", "True, False", "False, False", "True, True"],
      "answer": 0,
      "explanation": "In the first expression, not changes A to False, and False and False is False. In the second, A and B is False, so negating the entire conjunction gives True. The parentheses change what not applies to; negating one operand is different from negating the whole compound expression.",
      "transfer": "Choose another assignment and evaluate both expressions in small steps. Explain why agreement on one assignment would not establish equivalence."
    },
    {
      "id": "logic-implication-1",
      "concept": "Material implication",
      "prerequisites": ["Truth values and connectives"],
      "prompt": "Which assignment makes the material implication A -> B False?",
      "options": ["A=True, B=True", "A=False, B=True", "A=True, B=False", "A=False, B=False"],
      "answer": 2,
      "explanation": "A -> B excludes the case where A is True but B is False. It is True on each of the other three assignments. This is the truth-functional material conditional: its table alone says nothing about a causal, temporal or explanatory connection between the propositions.",
      "transfer": "Invent a fictional if-then rule and identify the one combination of antecedent and consequent that would violate its material-implication table."
    },
    {
      "id": "logic-implication-2",
      "concept": "Material implication",
      "prerequisites": ["Truth values and connectives"],
      "prompt": "A is False and the material implication A -> B is True. What does this tell you about B?",
      "options": ["B can be either True or False.", "B must be False because A is False.", "B must be True because the implication is True.", "The given truth values are inconsistent."],
      "answer": 0,
      "explanation": "The rows (A=False, B=True) and (A=False, B=False) both make A -> B True. The information therefore leaves B undetermined. Inferring not B from not A would add a condition that the original implication does not contain.",
      "transfer": "Compare A -> B with A <-> B on the two rows where A is False. Explain how the biconditional imposes an additional restriction."
    },
    {
      "id": "logic-implication-3",
      "concept": "Material implication",
      "prerequisites": ["Truth values and connectives"],
      "prompt": "Which expression is the contrapositive of (A and B) -> C?",
      "options": ["C -> (A and B)", "not (A and B) -> not C", "not C -> not (A and B)", "not C -> ((not A) and (not B))"],
      "answer": 2,
      "explanation": "The contrapositive of P -> Q is not Q -> not P. Here the complete antecedent P is (A and B), so it must be negated as a unit: not C -> not (A and B). Negating a conjunction means at least one operand is False, not necessarily both. This contrapositive agrees with the original implication on all eight assignments.",
      "transfer": "Form the contrapositive of (A or B) -> C. Keep the antecedent grouped, then use a truth table to check your expression against the original."
    },
    {
      "id": "logic-equivalence-1",
      "concept": "Logical equivalence",
      "prerequisites": ["Truth values and connectives", "Material implication"],
      "prompt": "Which expression has the same truth value as not (A or B) for every assignment?",
      "options": ["(not A) or B", "(not A) or (not B)", "not (A and B)", "(not A) and (not B)"],
      "answer": 3,
      "explanation": "For A or B to be False, A and B must both be False. Therefore not (A or B) is True exactly when not A and not B are both True. This is one of De Morgan's laws. Replacing and with or would also admit mixed assignments, so it would change the truth table.",
      "transfer": "Derive the corresponding expression for not (A and B). Find a mixed assignment that distinguishes the two De Morgan laws."
    },
    {
      "id": "logic-equivalence-2",
      "concept": "Logical equivalence",
      "prerequisites": ["Truth values and connectives", "Material implication"],
      "prompt": "Which listed case is a counterexample to the claim that A -> B and B -> A are logically equivalent?",
      "options": ["A=True, B=True", "A=False, B=True", "A=False, B=False", "No truth assignment can distinguish them."],
      "answer": 1,
      "explanation": "At A=False, B=True, the first implication is True but B -> A is False. A single differing assignment refutes equivalence. The both-True and both-False rows give matching outputs, so neither listed matching row is a counterexample. The converse reverses direction; the contrapositive reverses and negates both sides.",
      "transfer": "Locate the other differing assignment for these two implications. Then replace the converse with the contrapositive and check all four rows."
    },
    {
      "id": "logic-equivalence-3",
      "concept": "Logical equivalence",
      "prerequisites": ["Truth values and connectives", "Material implication"],
      "prompt": "What must a complete truth table show to establish that A -> B and (not A) or B are logically equivalent?",
      "options": ["They use the same variable letters.", "Their outputs agree when A and B are both True.", "Their outputs agree for every assignment of A and B.", "Each expression has at least one True row."],
      "answer": 2,
      "explanation": "Equivalence requires matching truth values for every assignment, not just a selected example. These expressions both have the column True, False, True, True when the assignments are (True,True), (True,False), (False,True), (False,False). Using the same variables or merely having some True rows is insufficient.",
      "transfer": "Give two expressions that agree on one assignment but are not equivalent. Explain what additional rows are needed to decide the question completely."
    },
    {
      "id": "logic-inference-1",
      "concept": "Valid inference",
      "prerequisites": ["Material implication", "Logical equivalence"],
      "prompt": "The premises A -> B and A are both True. Which listed conclusion must be True?",
      "options": ["not B", "B", "not A", "A and not B"],
      "answer": 1,
      "explanation": "A is True by the second premise. If B were False, the first premise A -> B would be False, contrary to the given premises. Hence B must be True. This valid inference pattern is called modus ponens; it preserves truth when its premises are true.",
      "transfer": "Write the single conditional ((A -> B) and A) -> B and examine every row. Explain how its being always True corresponds to the inference pattern."
    },
    {
      "id": "logic-inference-2",
      "concept": "Valid inference",
      "prerequisites": ["Material implication", "Logical equivalence"],
      "prompt": "An argument has premises A -> B and B, with conclusion A. Which assignment makes both premises True and the conclusion False?",
      "options": ["A=False, B=True", "A=True, B=False", "A=True, B=True", "A=False, B=False"],
      "answer": 0,
      "explanation": "With A=False and B=True, A -> B is True and B is True, yet A is False. This is a counterexample to the argument's validity. The invalid pattern is called affirming the consequent. B might be true in a case where A is false; the original one-way implication does not rule that out.",
      "transfer": "Create a fictional example of affirming the consequent. Show a possible situation in which the two premises hold but the proposed conclusion does not."
    },
    {
      "id": "logic-inference-3",
      "concept": "Valid inference",
      "prerequisites": ["Material implication", "Logical equivalence"],
      "prompt": "A complete truth table has no row where all an argument's premises are True and its conclusion is False. What has been established?",
      "options": ["Every premise is true in the actual world.", "The conclusion is True under every assignment, even when premises are False.", "The premises describe a cause of the conclusion.", "The argument's form is valid; whether its premises are actually true is a separate question."],
      "answer": 3,
      "explanation": "The absence of a counterexample establishes the truth-preserving form of the argument. It does not verify the actual premises, make the conclusion a tautology, or establish causation. A sound argument additionally has true premises; a truth table of the form alone cannot supply that empirical information.",
      "transfer": "Give a valid argument with a premise whose real-world truth would need evidence. Describe what the logical test establishes and what remains to be checked."
    }
  ]
}