Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic

The Poset of All Logics I: Interpretations and Lattice Structure

“…Now, recall that is determined by a matrix , whose set of designated elements is a singleton. By a minor variant of [1, Theorem 8], this implies that is a class of matrices such that F is either empty or a singleton. Then consider a matrix such that is non-trivial.…”

A computational glimpse at the Leibniz and Frege hierarchies

“…In this case ρ(x, y, z) is a set of congruence formulas with parameters for L. Analogously, a logic is called equivalential if there is set of formulas ρ(x, y) in two variables and without parameters which satisfies (1). In this case ρ(x, y, z) is called a set of congruence formulas for L. Protoalgebraic logics can be characterized syntactically as follows [14, Proposition 6.7 finitely regurlarly algebraizable…”

“…The following result will be used later on: 1 On the contrary, it is not straightforward to build a nice Hilbert calculus for L V out of Σ. In particular, the obvious idea of considering the logic axiomatized by the rules α β for all α ≈ β ∈ Σ does not work in general.…”

“…First, every finitary selfextensional logic with a classical deductiondetachment theorem is fully selfextensional [15,Theorem 4.46] (see also [23]). 3 Second, every truth-equational fully selfextensional logic is also fully Fregean [1,Theorem 22].…”

On equational completeness theorems

“…Accordingly, a logic admits an equational completeness theorem precisely when it has an algebraic semantics. 1 For instance, in view of the well-known equational completeness theorem of classical propositional logic CPC with respect to the variety of Boolean algebras, stating that for every set of formulas Γ ∪ {ϕ}, Γ CPC ϕ ⇐⇒ for every Boolean algebra A and a ∈ A, if A γ A ( a) ≈ 1 for all γ ∈ Γ, then A ϕ A ( a) ≈ 1, the variety of Boolean algebras is an algebraic semantic for CPC.…”