# Question 33040: Truth of a Conjunction

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0065-33040 · q-20260916-logic-conjunction-truth-0015



## Original question and givens

The truth values of P and Q are true and false respectively. Is P ∧ Q true?


- G001 — The truth values of P and Q are true and false respectively. Is P ∧ Q true?





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-logical-connectives](https://ocme.evemisslab.com/en/mko/mko-logical-connectives/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-logic-conjunction-truth, BASE-no-required-operation, BASE-no-required-domain, []]

After

derived = [[SOURCE_YesNo_STRING]]

Conditions: none

MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.

Step identifiers and dependenciesRule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001





S002 · impliesBefore

source roles = [Yes, No, ∧, "P15 \u8207 Q15"]

After

derived = [[[No, No, No], [No, Yes, No], [Yes, No, No], [Yes, Yes, Yes]]]

Conditions: none

P/Q are GIVEN EXACT Booleans;do not infer truth from context.Show all four defining cases.OR is INCLUSIVE;material implication fails ONLY at P true/Q false and is not causation;check BOTH directions for iff.

Step identifiers and dependenciesRule: Logic_EXACT_Boolean_givens_FULL_connective_truth_definition · Uses: G001, S001





S003 · impliesBefore

source roles = [Yes, No, ∧]

After

derived = [No, Yes, No, Yes, No]

Conditions: none

Substitute THIS P/Q to evaluate the connective;iff needs both directions true.A false-premise implication may be true without establishing causation or proving the underlying propositions.

Step identifiers and dependenciesRule: Logic_INSTANCE_forward_backward_and_result_not_underlying_proof · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
- S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
- S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.





## Result

No

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 451081bc2aae1c70aaabee440d45365677a4ffc787ca60bc37a450d4dd56b632
- candidate record SHA-256: 532eed5159e3eb53d79d21487923364a1b7009fbd73c3cf927f5630f3c8d4b6b
- semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
- question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260916-logic-conjunction-truth-0015.json)