This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20260916-logical-connectives · 2026-09-16 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∧ Q true?
No
A conjunction is true exactly when both operands are true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∧ Q true?
Yes
A conjunction is true exactly when both operands are true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-conjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∨ Q true?
No
An inclusive disjunction is true when at least one operand is true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∨ Q true?
No
An inclusive disjunction is true when at least one operand is true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∨ Q true?
No
An inclusive disjunction is true when at least one operand is true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∨ Q true?
No
An inclusive disjunction is true when at least one operand is true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∨ Q true?
No
An inclusive disjunction is true when at least one operand is true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and false respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are false and false respectively. Is P ∨ Q true?
No
An inclusive disjunction is true when at least one operand is true, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P ∨ Q true?
Yes
An inclusive disjunction is true when at least one operand is true, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-inclusive-disjunction-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and false respectively. Is P → Q true?
No
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and false respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and false respectively. Is P → Q true?
No
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and false respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and false respectively. Is P → Q true?
No
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and false respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and false respectively. Is P → Q true?
No
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and false respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and false respectively. Is P → Q true?
No
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and false respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and false respectively. Is P → Q true?
No
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are false and false respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P → Q true?
Yes
Material implication is false only when P is true and Q is false; it is not a causal claim. This value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-material-implication-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and true respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and false respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and false respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and true respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and false respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and false respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and true respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and false respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and false respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and true respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and false respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and false respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and true respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and false respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and false respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and true respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and false respectively. Is P ↔ Q true?
No
A biconditional is true exactly when both directions have the same truth value, so this value is false.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are false and false respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
The truth values of P and Q are true and true respectively. Is P ↔ Q true?
Yes
A biconditional is true exactly when both directions have the same truth value, so this value is true.
Reviewed step-by-step derivation
Logical connectives · qf-logic-biconditional-truth · candidate
Canonical question JSONL · English JSONL · Validation report