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The difference is essentially whether correlations between base and supervenient properties hold within actual worlds only, or across possible worlds. For example, if psychological properties strongly locally supervene on physical properties, then any two people physically the same, in any two worlds, will also be psychologically the same. On the other hand, if psychological properties only weakly locally supervene on physical properties, then those correlations between base and supervenient properties that hold in virtue of the supervenience

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