Ineffable Inconsistencies

For any given consistent tarskian logic it is possible to find another non-trivial logic that allows for an inconsistent model yet completely coincides with the initial given logic from the point of view of their associated single-conclusion consequence relations. A paradox? This short note shows you how to do it. This may be read as the description of an expedition into unexplored regions of abstract logic, the theory of valuations and paraconsistency. 1 INCONSISTENT CLASSICAL LOGIC Plus on voit ce monde, et plus on le voit plein de contradictions et d’inconséquences. —Voltaire, Dictionnaire Philosophique, XVIII century. Take your preferred presentation of classical propositional logic. More concretely, take some denumerable set At of atomic sentences and some non-empty functionally complete set of logical constants C. As usual, the set S of classical formulas will be inductively built as the free algebra generated by C over At. Let V be a set of truth-values, D ⊆ V a set of designated values (shades of truth) and U ⊆ V a set of undesignated values (shades of falsehood), where D∪U = V and D∩U = ∅. Semantically, a classical state of the world will be simulated by an assignment Asg : At → V , where both D and U are required to be non-empty —usually, they are taken to be singletons, symbolizing ‘the true’ and ‘the false’, if you like. Yes, if you have a boolean mind, you will probably be expecting each such assignment Asg to be uniquely extendable into a valuation § : S → V , according to the truthfunctional interpretation of each connective in C. Indeed, say you are talking about disjunction and negation, ∨ and ∼. In that case you are probably expecting their semantical interpretations to be induced by the set Sem of all valuations § : S → V such that: §(α ∨ β) ∈ D iff §(α) ∈ D or §(β) ∈ D §(∼α) ∈ D iff §(α) ∈ U Because classical logic has a truth-functional semantics and because this semantics was formulated above in order to display the dependence of each complex classical formula on its immediate subformulas, and only on them, each of the §-clauses regulating the set Sem could be written with an ‘iff’ and have a very specific format, indicating the similarity between the algebra of classical formulas and the classical (boolean) algebra of truth-values.