The simplest protoalgebraic logic

The logic I is the sentential logic defined in the language with just implication → by the axiom of reflexivity or identity “φ→ φ” and the rule of Modus Ponens “from φ and φ→ ψ to infer ψ”. The theorems of this logic are exactly all formulas of the form φ→ φ. I argue that this is the simplest protoalgebraic logic, and that in it every set of assumptions encodes in itself not only all its consequences but also their proofs. In this paper I study this logic from the point of view of abstract algebraic logic, and in particular I use it as a relatively natural counterexample to settle some open problems in this theory. It appears that this logic has almost no properties: it is neither equivalential nor weakly algebraizable; it does not have an algebraic semantics; it does not satisfy any form of the Deduction Theorem, other than the most general parameterized and local one that all protoalgebraic logics satisfy; it is not filter-distributive; and so on. It satisfies some forms of the interpolation property but in a rather trivial way. Very few things are known about its algebraic counterpart, save that its “intrinsic variety” is the class of all algebras of the similarity type.