Question 6244: Typed MP Goals and Minimum Depths

Original question and givens

Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).

• G001 — Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).

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-proof

Step-by-step derivation

n=4;given facts=[0];original indexed rules=[[0, 0, 1], [1, 1, 2]]

use only given facts and single-premise MP;not-derived does not mean False

phase=base;given facts=[0]

phase=base;active_rules=[0,1]

phase=base;given fact=0

phase=base;derive atom=0;depth=0;basis=given

base MP depth for atom1 := sum(0, 1)

base MP depth for atom1 = 1

phase=base;rule[0]=0→1;antecedent depth=0

phase=base;derive atom=1;depth=1;from=0;rule=0

base MP depth for atom2 := sum(1, 1)

base MP depth for atom2 = 2

phase=base;rule[1]=1→2;antecedent depth=1

phase=base;derive atom=2;depth=2;from=1;rule=1

phase=base;edge closure/depth audits=[[0, 0, 1, 0, 1, Yes], [1, 1, 2, 1, 2, Yes]]

phase=base;depths=[0,1,2,null]

query atom=0;inside Fin(4)=Yes

derived=Yes;depth/status=0

query atom=101;inside Fin(4)=No

derived=No;depth/status=type mismatch

query atom=2;inside Fin(4)=Yes

derived=Yes;depth/status=2

Reverse checks

• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute base MP depth for atom2 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute base MP depth for atom1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([Yes, No, Yes], [Yes, No, Yes], [0, ill-typed, 2])

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.