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-20260921-proof-foundations · 2026-09-21 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is divisible by 6” and Q mean “the integer n is divisible by 3”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is a multiple of 9” and Q mean “the integer n is a multiple of 3”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is prime and greater than 2” and Q mean “the integer n is odd”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the triangle has three equal sides” and Q mean “the triangle has at least two equal sides”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the quadrilateral is a square” and Q mean “the quadrilateral has four right angles”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is a multiple of 10” and Q mean “the integer n is a multiple of 5”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is a perfect square” and Q mean “there exists an integer k such that n=k²”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the rational number r equals 0” and Q mean “the absolute value of r equals 0”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “set A is empty” and Q mean “A has no elements”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “function f is constant” and Q mean “any two inputs have the same output”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the angle measure is 90°” and Q mean “the angle is a right angle”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the angle measure is 180°” and Q mean “the angle is a straight angle”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is divisible by 8” and Q mean “the integer n is divisible by 2”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is a multiple of 15” and Q mean “the integer n is divisible by 5”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the number x is greater than 7” and Q mean “x is greater than 0”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the number x is less than -3” and Q mean “x is less than 0”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the polygon is a pentagon” and Q mean “the polygon has five sides”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the rectangle has equal length and width” and Q mean “the rectangle is a square”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is even” and Q mean “n² is even”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the denominator b of a/b is nonzero” and Q mean “a/b is a defined fractional representation”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “set A is a subset of B” and Q mean “every element of A is also in B”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “propositions R and S both hold” and Q mean “proposition R holds”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?
Can
The premises are P→Q and P, exactly the form of modus ponens, so Q follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-ponens · candidate
Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is divisible by 6” and Q mean “the integer n is divisible by 3”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is a multiple of 9” and Q mean “the integer n is a multiple of 3”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is prime and greater than 2” and Q mean “the integer n is odd”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the triangle has three equal sides” and Q mean “the triangle has at least two equal sides”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the quadrilateral is a square” and Q mean “the quadrilateral has four right angles”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is a multiple of 10” and Q mean “the integer n is a multiple of 5”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is a perfect square” and Q mean “there exists an integer k such that n=k²”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the rational number r equals 0” and Q mean “the absolute value of r equals 0”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “set A is empty” and Q mean “A has no elements”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “function f is constant” and Q mean “any two inputs have the same output”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the angle measure is 90°” and Q mean “the angle is a right angle”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the angle measure is 180°” and Q mean “the angle is a straight angle”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is divisible by 8” and Q mean “the integer n is divisible by 2”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is a multiple of 15” and Q mean “the integer n is divisible by 5”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the number x is greater than 7” and Q mean “x is greater than 0”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the number x is less than -3” and Q mean “x is less than 0”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the polygon is a pentagon” and Q mean “the polygon has five sides”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the rectangle has equal length and width” and Q mean “the rectangle is a square”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is even” and Q mean “n² is even”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the denominator b of a/b is nonzero” and Q mean “a/b is a defined fractional representation”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “set A is a subset of B” and Q mean “every element of A is also in B”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “propositions R and S both hold” and Q mean “proposition R holds”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?
Can
The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-modus-tollens · candidate
Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is divisible by 6” and Q mean “the integer n is divisible by 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is a multiple of 9” and Q mean “the integer n is a multiple of 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is prime and greater than 2” and Q mean “the integer n is odd”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the triangle has three equal sides” and Q mean “the triangle has at least two equal sides”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the quadrilateral is a square” and Q mean “the quadrilateral has four right angles”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is a multiple of 10” and Q mean “the integer n is a multiple of 5”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is a perfect square” and Q mean “there exists an integer k such that n=k²”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the rational number r equals 0” and Q mean “the absolute value of r equals 0”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “set A is empty” and Q mean “A has no elements”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “function f is constant” and Q mean “any two inputs have the same output”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the angle measure is 90°” and Q mean “the angle is a right angle”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the angle measure is 180°” and Q mean “the angle is a straight angle”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is divisible by 8” and Q mean “the integer n is divisible by 2”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is a multiple of 15” and Q mean “the integer n is divisible by 5”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the number x is greater than 7” and Q mean “x is greater than 0”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the number x is less than -3” and Q mean “x is less than 0”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the polygon is a pentagon” and Q mean “the polygon has five sides”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the rectangle has equal length and width” and Q mean “the rectangle is a square”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is even” and Q mean “n² is even”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “the denominator b of a/b is nonzero” and Q mean “a/b is a defined fractional representation”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “set A is a subset of B” and Q mean “every element of A is also in B”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
Let P mean “propositions R and S both hold” and Q mean “proposition R holds”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?
Cannot
Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-invalid-converse · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=1. Does this witness refute that universal proposition?
Yes
n=1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 0.” and the stated witness n=0. Does this witness refute that universal proposition?
Yes
n=0 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than or equal to 0.” and the stated witness n=-1. Does this witness refute that universal proposition?
Yes
n=-1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 3.” and the stated witness n=1. Does this witness refute that universal proposition?
Yes
n=1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 5.” and the stated witness n=2. Does this witness refute that universal proposition?
Yes
n=2 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer equals 0.” and the stated witness n=1. Does this witness refute that universal proposition?
Yes
n=1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 5.” and the stated witness n=5. Does this witness refute that universal proposition?
Yes
n=5 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is less than 10.” and the stated witness n=10. Does this witness refute that universal proposition?
Yes
n=10 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=-3. Does this witness refute that universal proposition?
Yes
n=-3 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 0.” and the stated witness n=-1. Does this witness refute that universal proposition?
Yes
n=-1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 7.” and the stated witness n=3. Does this witness refute that universal proposition?
Yes
n=3 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer equals 0.” and the stated witness n=-2. Does this witness refute that universal proposition?
Yes
n=-2 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than -4.” and the stated witness n=-4. Does this witness refute that universal proposition?
Yes
n=-4 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is less than 0.” and the stated witness n=0. Does this witness refute that universal proposition?
Yes
n=0 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 2.” and the stated witness n=5. Does this witness refute that universal proposition?
Yes
n=5 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than or equal to 0.” and the stated witness n=-9. Does this witness refute that universal proposition?
Yes
n=-9 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 11.” and the stated witness n=4. Does this witness refute that universal proposition?
Yes
n=4 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?
Yes
n=9 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 0.” and the stated witness n=-8. Does this witness refute that universal proposition?
Yes
n=-8 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 100.” and the stated witness n=100. Does this witness refute that universal proposition?
Yes
n=100 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is less than -2.” and the stated witness n=-2. Does this witness refute that universal proposition?
Yes
n=-2 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 4.” and the stated witness n=6. Does this witness refute that universal proposition?
Yes
n=6 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer equals 0.” and the stated witness n=12. Does this witness refute that universal proposition?
Yes
n=12 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than or equal to 0.” and the stated witness n=-12. Does this witness refute that universal proposition?
Yes
n=-12 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 9.” and the stated witness n=10. Does this witness refute that universal proposition?
Yes
n=10 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.
Reviewed step-by-step derivation
Mathematical proof · qf-proof-counterexample-universal · candidate
Canonical question JSONL · English JSONL · Validation report