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It is not the case that Any account of the Trinity that treats the Persons as numerically identical faces a serious problem.
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Reasons For
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Reason for
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1.
It is self-evident that if any x and any y are numerically identical, then x and y can never differ.
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2.
Any account of the Trinity must allow that the Persons differ at least in respect of origin: only the Son is begotten, only the Spirit proceeds, and only the Father begets the Son.
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Reasons Against
2 perspectives
Reason against 1 of 2
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1.
Leibniz's Law holds that numerically identical entities share all properties, including relational and modal ones.
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2.
The Father has the property 'being unbegotten' while the Son lacks it, constituting a genuine difference in intrinsic properties.
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3.
No coherent identity relation can obtain between entities that differ in intrinsic properties, even under revisionary theology.
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Reason against 2 of 2
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1.
Relative identity solutions (per Geach) require distinct sortal concepts under which entities are 'the same F', presupposing non-identity at the absolute level.
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2.
If the Persons were numerically identical absolutely, relative identity distinctions would collapse into mere linguistic variance, not real Trinitarian distinction.
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