As Henkin (1950) demonstrated, weakening the semantics of second-order logic to general models preserves completeness but loses categoricity, so validity in full models and validity relative to θ come apart.

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Reasons For

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Reason for

1.Henkin's general models achieve completeness because they allow non-standard interpretations that validate exactly the theorems provable in SOL.
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2.Categoricity requires full models only; general models necessarily admit multiple non-isomorphic models of the same theory, making categoricity impossible.
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3.The divergence between full and general model validity demonstrates that standard semantics captures intended meaning while Henkin semantics captures proof-theoretic structure.
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Reasons Against

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Reason against

1.Henkin's completeness theorem doesn't 'lose' categoricity—full second-order logic was never categorical for all theories, only for specific axiomatizations like PA.
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2.The distinction between full and general model validity may reflect our incomplete understanding of quantification over properties, not a genuine semantic difference.
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3.Preserving completeness while accepting non-categoricity might be desirable: it trades metaphysical assumptions about properties for proof-theoretic tractability.
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Key Terms

Full models(as the standard against which other models are compared): The strictest, most complete interpretation of the rules in a logical system, where nothing is left out or loosened.
General models(as a weakened version of strict logical rules): A looser, more flexible interpretation of what counts as a valid 'world' or 'situation' in a logical system.
Henkin (1950)(as a historical reference to foundational research): Leon Henkin was a logician who published important work in 1950 about how different logical systems work; this reference points to a specific discovery he made about the rules governing certain types of logic.
Second-order logic(as used in mathematical logic): A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.
categoricity(Joyce's term for the inescapable practical force of moral demands): The property of moral requirements whereby they apply to agents unconditionally, regardless of the agent's contingent desires, goals, or interests
completeness(Used to transfer results from model theory to proof theory): The property of a logical system whereby anything valid (model-theoretically) is deducible (proof-theoretically)
semantics(Distinguished from metasemantics and pragmatics in Kaplan 1989): The domain that concerns the facts about what meanings words or phrases have.
validity(Formal logic; distinguished from syntactic deducibility): The model-theoretic counterpart to deducibility; an argument is valid if its conclusion is true under every interpretation in which its premises are true

Connections

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Proof of definition segments1 linked Truth & Knowledge1 linked

Categoricity requires full models only; general models necessarily admit multipl...Checking the validity of an arbitrary second-order sentence φ can be recursively...

Details

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