 Returning to the former case: as this use of the term ‘consequences’ reminds us, a closure operation on the set of formulas of some language is conventionally called a consequence operation, and perhaps to use the notation Cn(⋅) rather than simply C(⋅), though it is often more convenient to talk in terms of the corresponding consequence relation, holding between a set of formulas Γ and an individual formula φ—notated in some such way as “Γ ⊩ φ”, “Γ ⊢ φ”, etc.—when φ ∈ Cn(Γ); when the relation has been defined semantically it is not uncommon to press “⊨” into service in this role. (This will occur once or twice below.) Certain shortcuts are usually taken with this notation, so that one writes, for example “Γ, Δ, φ, ψ ⊢ χ” and “⊢ φ” to mean “Γ ∪ Δ ∪ {φ,ψ} ⊢ χ” and “∅ ⊢ φ”, respectively. (For the moment, capital Greek letters range over sets of formulas.) The conditions given earlier for C’s being a closure operation can be recast as conditions definitive of a consequence relation, conditions we do not give explicitly here. A consequence relation ⊢ on some language is finitary if whenever Γ ⊢ φ then we have Γ0 ⊢ φ for some finite Γ0 ⊆ Γ; to be substitution-invariant whenever Γ ⊢ φ then s(Γ) ⊢ s(φ) where s is any substitution, i.e., any endomorphism of the algebra of formulas (and s(Γ) is {s(ψ) | ψ ∈ Γ}); and, finally, to be congruential when φ ⊣⊢ ψ implies χ(φ) ⊣⊢ χ(ψ), in which the notation is to be understood as follows. “φ ⊣⊢ ψ” abbreviates “φ ⊢ ψ and ψ⊢ φ” (and if we are decorating the turnstile “⊢” with superscripts or subscripts, these are written only on the ⊢ and not also on the ⊣ part of the composite notation); χ is some formula constructed from propositional variables including at least one, pk, say—to indicate which we may sometimes write χ = χ(pk)—and χ(φ), for a formula φ, is the result of substituting φ for all occurrences of pk in χ. (So χ(φ) is s(χ) for the unique substitution s for which s(pi) = pi for i ≠ k and s(pk) = φ.) The χ(⋅) notation amounts to thinking of χ as a (sentential) context, and putting φ into this context amounts to producing the formula χ(φ); more generally, we can think of an n-ary context as a formula containing n distinguished propositional variables pk1,…, pkn, say—and perhaps additional variables  too.[3]  If there are no additional variables, then the formula χ(pk1,…, pkn) can be thought of as a derived or composite connective of the language under consideration. Returning to the 1-ary case, when φ ⊣⊢ ψ we describe φ and ψ as equivalent according to ⊢, and, following Smiley (1962), when for all χ(⋅), the formulas χ(φ) and χ(ψ) are equivalent according to ⊢, we describe φ and ψ as synonymous (according to ⊢). Thus a congruential consequence relation is one for which any equivalent formulas are synonymous.