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    Syntactic consistency cannot be equivalent to satisfiabil... — Carmelics
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    Challenges→Each consistent set of many-sorted formulas has a model, making syntactic consistency and semantic satisfiability equivalent

    Syntactic consistency cannot be equivalent to satisfiability when 'satisfiability' picks out different model classes depending on unresolved foundational choices about sort ontology.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
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    • 1.Model-theoretic semantics requires fixed sort ontologies; without resolving what counts as a domain element, 'satisfiability' remains ambiguous.
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    • 2.Syntactic consistency is proof-theoretically determinate regardless of ontological commitments, making it independent of model-class variation.
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    • 3.Different foundational frameworks (constructivism vs. classical set theory) yield incompatible model classes for identical syntactic theories.
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    Reasons Against

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    Reason against
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    • 1.Completeness theorems show syntactic consistency reliably tracks satisfiability across standard model classes, regardless of foundational metaphysics.
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    • 2.Ontological indeterminacy about sorts doesn't make satisfiability meaningless—it just relativizes it to explicit framework choices we can specify.
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    • 3.Syntactic consistency itself presupposes logical rules whose legitimacy depends on the same foundational choices as model-class selection does.
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    Key Terms

    Foundational choices(in explaining why different ethical theories have unavoidable trade-offs)
    Basic starting assumptions a theory makes that everything else builds on—like whether God exists or whether morality comes from reason alone.
    Model classes(in logic and philosophy of language)
    Different categories of possible worlds or setups that we use to test whether statements can be true; think of them as different types of scenarios you might imagine to check if something works.
    Sort ontology(in metaphysics and logic)
    Basic choices about what kinds of 'things' exist and how to classify them—like deciding whether numbers, abstract ideas, and physical objects are all the same type of thing or fundamentally different types.
    Syntactic consistency(logic)
    Whether a set of rules or statements follows all the formal grammatical and logical rules, without contradicting the structure of the system itself.
    satisfiability(classical first-order logic)
    A formula φ of language L is satisfiable if and only if there exists a Σ-structure A and assignment s on it such that A,s⊨φ holds.

    Connections

    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Completeness theorems show syntactic consistency reliably tracks satisfiability ...Different foundational frameworks (constructivism vs. classical set theory) yiel...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Each consistent set of many-sorted formulas has a model, making syntactic consis...
    Model-theoretic semantics requires fixed sort ontologies; without resolving what...
    +3 moreShow less
    Ontological indeterminacy about sorts doesn't make satisfiability meaningless—it...Syntactic consistency is proof-theoretically determinate regardless of ontologic...Syntactic consistency itself presupposes logical rules whose legitimacy depends ...