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    Two functions can be co-extensional over all inputs while... — Carmelics
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    Challenges→A function f(x) is in FP if and only if it is definable by a Σ^B₁-formula relative to which it is provably total in V¹

    Two functions can be co-extensional over all inputs while differing in their Σ^B₁-definability status under alternative but equally valid formalization choices, undermining the biconditional's necessity claim.

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    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').
    Co-extensional(describing when two things match up but aren't necessarily connected in a meaningful way)
    Having exactly the same members or applying to exactly the same things, even if for completely different reasons.
    Formalization(describing what Frege did with existence)
    The process of taking an idea and expressing it precisely using logical symbols and strict rules, like translating messy everyday language into mathematical logic.
    Function(Boorse's account of biological function, grounding his theory of disease)
    A species-typical contribution to survival and reproduction

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    definability(combinatory logic)
    A combinator can be defined by a term whether or not that term is in normal form; additionally, a combinator may be defined by two or more distinct terms in normal form.
    necessity claim(in logic and philosophy)
    An argument that something *must* be true in all cases, not just sometimes or usually.
    Σ^B₁(in symbolic logic notation)
    A technical classification system in mathematical logic that groups statements by how complex they are to describe—this is an advanced formal category.

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    Truth & Knowledge1 linkedModality & Possibility1 linked

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    A function f(x) is in FP if and only if it is definable by a Σ^B₁-formula relati...

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