Philosophy Index

Set membership

When a logical object, whether it's a propositional formula, constant (in first-order logic) or a possible world (in modal logic), belongs in a group of other objects, for whatever reason, we say that the object is a member, or an element, of that set.

We use the symbol ∈ to denote membership in a set. So, if we say φ ∈ Γ, then the constant φ is a member of the set Γ.

The symbol ∉ is sometimes used to clarify that something is not a member of a set, so φ ∉ Γ means that φ is not a member of the set Γ.

Set equality

Two sets are equal when they contain identical members. So, if set Γ = { ¬α, (β ∧ γ) } and set Δ = { ¬α, (β ∧ γ) }, then Γ = Δ.