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
    The reduction presupposes that θ faithfully captures full... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

    Challenges→Checking the validity of an arbitrary second-order sentence φ can be recursively reduced to checking the validity of a Σ¹₁-sentence.

    The reduction presupposes that θ faithfully captures full second-order semantics, but Henkin models satisfying θ need not validate full comprehension, undermining the biconditional in P3.

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

    No one has weighed in yet. Be the first to share reasons for or against this statement.

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

    Key Terms

    Biconditional(in formal logic)
    A logical statement that says two things are true if and only if each other is true; it's a two-way relationship (like saying 'you can vote if and only if you're 18').
    Full comprehension(a feature of the logical system)
    A principle that says: whenever you can describe a property, there exists a set of all things that have that property.
    Henkin models(Completeness Theorem for second-order logic)
    Models for second-order logic in which the second-order quantifiers range over some, but not necessarily all, subsets of the domain; full Henkin models are those where all subsets are included
    P3(as referenced in formal logic or philosophical argumentation)
    A specific rule or principle (probably the third one in a list) being discussed in the larger argument.

    Next step

    Based on where you are in your exploration

    Explore a random proposition
    Start fresh with something unrelated.
    Second-order semantics(in logic and philosophy of language)
    A way of talking about language where you can make statements not just about things in the world, but about the statements themselves—like commenting on the rules of the language rather than just using it.
    reduction(Lambda calculus or term-rewriting systems)
    A computational process analogous to computing the value of a function, proceeding through a series of discrete calculation steps applied to a term
    θ (theta)(in formal logic notation)
    A variable (placeholder) used to represent any logical statement or formula, like how 'x' represents an unknown number in algebra.

    Connections

    2 topics

    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    Checking the validity of an arbitrary second-order sentence φ can be recursively...

    Details

    Type
    claim
    Perspectives
    0 (0 for, 0 against)
    Edits
    1 edit

    Open for perspectives

    This idea is waiting for its first supporting or challenging perspective.

    Share the first perspective