XL can have a strongly complete calculus — Carmelics

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XL can have a strongly complete calculus

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1 reason for

2 reasons against

Reasons For

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Reason for

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1.XL satisfies Compactness and Löwenheim-Skolem
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2.XL has recursive enumerability of validities
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3.These three properties together are sufficient for strong completeness
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Reasons Against

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Reason against 1 of 2

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1.Recursive enumerability of validities is necessary but not sufficient for strong completeness without a complete proof system being explicitly constructible.
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2.XL's many-sorted quantification over heterogeneous domains may generate validity-preservation failures under certain sort-collapsing interpretations that standard enumeration procedures miss.
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3.Lindström's theorem establishes that classical FOL is maximal for compactness plus Löwenheim-Skolem, but XL's sort structure introduces expressive additions that may escape this maximality boundary.
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Reason against 2 of 2

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