Selfextensional Logics

The notion of logic adopted by Abstract Algebraic Logic stems from Tarki’s. By a logic we understand any consequence relation on the algebra of formulas of some propositional language, namely any relation ` between sets of formulas and formulas such that 1. φ ` φ; 2. if Γ ` φ and Γ ⊆ ∆, then ∆ ` φ; 3. if for every ψ ∈ Γ , ∆ ` ψ and Γ ` φ, then ∆ ` ψ; 4. if Γ ` φ, then for any substitution σ of formulas for variables, σΓ ` σφ. A logic S is said to be selfextensional if the relation of mutual consequence (φ a` ψ) is a congruence relation on the algebra of formulas.