#### Supplement to Supervenience

## Appendix: List of Definitions

For ease of reference, the technical definitions discussed in earlier
sections are collected in this section. ‘*A*’ and
‘*B*’ should be understood as variables ranging
over non-empty sets of properties (though the quantification is
typically implicit).

**Individual Supervenience**

Weak Individual SuperveniencePossible-Worlds Formulation:Aweakly supervenes onBiff for any possible worldwand any individualsxandyinw, ifxandyareB-indiscernible inw, then they areA-indiscernible inw.

Modal Operator Formulation:Aweakly supervenes onBiff necessarily, if anythingxhas some propertyFinA, then there is at least one propertyGinBsuch thatxhasG, and everything that hasGhasF.

□∀x∀F∈A[Fx→ ∃G∈B(Gx& ∀y(Gy→Fy))]Strong Individual SuperveniencePossible-Worlds Formulation:Astrongly supervenes onBiff for any possible worldsw_{1}and w_{2}and any individualsxinw_{1}andyinw_{2}, ifxinw_{1}isB-indiscernible fromyinw_{2}, thenxinw_{1}isA-indiscernible fromyinw_{2}.□∀

Modal Operator Formulation:Astrongly supervenes onBiff necessarily, if anythingxhas some propertyFinA, then there is at least one propertyGinBsuchxhasG, and necessarily everything that hasGhasF.x∀F∈A[Fx→ ∃G∈B(Gx& □∀y(Gy→Fy))]

**Regional Supervenience****Weak**:*A*weak-regionally supervenes on*B*iff for any possible world*w*and any space-time regions*r*_{1}and*r*_{2}in*w*, if r_{1}and*r*_{2}are exactly alike in every intrinsic*B*-respect in*w*, then they are exactly alike in every intrinsic*A*-respect in*w*.**Strong**:*A*strong-regionally supervenes on*B*iff for any possible worlds*w*_{1}and*w*_{2}and any space-time regions*r*_{1}in*w*_{1}and spacetime region*r*_{2}in*w*_{2}, if*r*_{1}in*w*_{1}is exactly like*r*_{2}in*w*_{2}in every intrinsic*B*-respect, then*r*_{1}in*w*_{1}is exactly like*r*_{2}in*w*_{2}in every intrinsic*A*-respect.**Similarity-Based Supervenience****Weak Similarity-Based**:*A*weakly supervenes on*B*iff for any world*w*, and for any*x*and*y*in*w*, if*x*and*y*are not largely different with respect to*B*-properties, then they are not largely different with respect to*A*-properties.**Strong Similarity-Based**:*A*strongly supervenes on*B*iff for any worlds*w*_{1}and*w*_{2}, and for any*x*in*w*_{1}and*y*in*w*_{2}, if*x*in*w*_{1}is not largely different from*y*in*w*_{2}with respect to*B*-properties, then*x*in*w*_{1}is not largely different from*y*in*w*_{2}with respect to*A*-properties.**Global Supervenience****Generic Global**:*A*globally supervenes on*B*iff for any worlds*w*_{1}and*w*_{2}, if*w*_{1}and*w*_{2}have exactly the same world-wide pattern of distribution of*B*-properties, then they have exactly the same world-wide pattern of distribution of*A*-properties.**Weak Global**:*A*weakly globally supervenes on*B*iff for any worlds*w*_{1}and*w*_{2}, if there is a*B*-preserving isomorphism between*w*_{1}and*w*_{2}, then there is an*A*-preserving isomorphism between them.**Intermediate Global**:*A*intermediately globally supervenes on*B*iff for any worlds*w*_{1}and*w*_{2}, if there is a*B*-preserving isomorphism between*w*_{1}and*w*_{2}, then there is an*A*-and-*B*-preserving isomorphism between*w*_{1}and*w*_{2}.**Strong Global**:*A*strongly globally supervenes on*B*iff for any worlds*w*_{1}and*w*_{2}, every*B*-preserving isomorphism between*w*_{1}and*w*_{2}is an*A*-preserving isomorphism between them.**Multiple Domain Supervenience**-
**Weak Multiple Domain**:*A*weakly supervenes on*B*relative to relation*R*just in for any world*w*and any*x*and*y*in*w*, if*R*|*x*in*w*is*B*-indiscernible from*R*|*y*in*w*, then*x*and*y*are*A*-indiscernible in*w*.**Strong Multiple Domain**:*A*strongly supervenes on*B*relative to relation*R*just in case for any worlds*w*_{1}and*w*_{2}, any*x*in*w*_{1}and*y*in*w*_{2}, if*R*|*x*in*w*_{1}is*B*-indiscernible from*R*|*y*in*w*_{2},*x*in*w*_{1}is*A*-indiscernible from*y*in*w*_{2}.**Weak Coincident-Friendly**: For any world*w*and any*x*and*y*in*w*, if*x*in*w*is B-indiscernible from*y*in*w*, then for each thing*x** in*w*to which*x*is*R*-related, there is something*y** in*w*to which*y*is*R*-related, and which is*A*-indiscernible from*x**.**Strong Coincident-Friendly**: For all*x*and*y*, and all worlds*w*_{1}and*w*_{2}, if*x*in*w*_{1}is*B*-indiscernible from*y*in*w*_{2}, then for each thing*x** in*w*_{1}to which*x*is*R*-related, there is something*y** in*w*_{2}to which*y*is*R*-related, and which is*A*-indiscernible from*x**.