Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    If the NP-completeness of BHP is provable only within sys... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

    Challenges→The Boolean Halting Problem (BHP) is in P only if P equals NP

    If the NP-completeness of BHP is provable only within systems that already assume P≠NP, the conditional 'BHP ∈ P → P=NP' is trivially vacuous rather than informative.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.A conditional is vacuous if its consequent becomes provable only by assuming the antecedent false, making the implication logically uninformative.
      ?

      Think about whether this reason is strong or weak

    • 2.If P≠NP is required to prove BHP∈NP is false, then we cannot derive information about P vs NP from BHP's complexity without circular reasoning.
      ?

      Think about whether this reason is strong or weak

    • 3.Informative theorems should increase knowledge within a fixed framework; conditionals requiring framework-changes provide no substantive insight.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Vacuity requires the conditional to be true in all models; if BHP∈P genuinely implies P=NP in standard models, it's informative regardless of proof methods.
      ?

      Think about whether this reason is strong or weak

    • 2.Many important results are provable only in stronger systems; this doesn't make them vacuous—it reflects genuine logical dependencies, not circularity.
      ?

      Think about whether this reason is strong or weak

    • 3.The conditional's value lies in ruling out a possibility (BHP∈P∧P≠NP), which is meaningful even if the proof strategy presupposes P≠NP's truth.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.

    Connections

    2 topics

    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    A conditional is vacuous if its consequent becomes provable only by assuming the...If P≠NP is required to prove BHP∈NP is false, then we cannot derive information ...Informative theorems should increase knowledge within a fixed framework; conditi...Many important results are provable only in stronger systems; this doesn't make ...
    +3 moreShow less
    The Boolean Halting Problem (BHP) is in P only if P equals NPThe conditional's value lies in ruling out a possibility (BHP∈P∧P≠NP), which is ...Vacuity requires the conditional to be true in all models; if BHP∈P genuinely im...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit