Discussive Logic (D2)

Origin. Stanisław Jaśkowski, "Propositional calculus for contradictory deductive systems" (Studia Societatis Scientiarum Torunensis, 1948; English 1969) — the first paraconsistent logic, four years before Priest was born and fifteen before da Costa. Jaśkowski was answering Łukasiewicz's 1910 challenge to construct a system in which the law of non-contradiction fails without triviality. Da Costa and Dubikajtis (1977) axiomatized it.

Models. A discussion. Each participant asserts consistently; the discussion as a whole contains A and ¬A because different participants said them. A thesis of the discussion is what some participant maintains — so "A holds in the discussion" is ◇A, and the discussion is inconsistent without anyone being. Explosion is blocked not by a glut value, not by variable sharing, not by a controlled negation, but because adjunction fails: A and B are each in the discussion and A ∧ B need not be.

Formalism.

The translation: For a formula A, let A* be its discussive translation. p* = ◇p for atomic p (A ∧_d B)* = ◇A* ∧ B* (discussive conjunction) (A →_d B)* = ◇A* → B* (discussive implication) (A ∨ B)* = A* ∨ B* (¬A)* = ¬A*

The consequence relation: ⊢_D2 A iff ⊢_S5 ◇A* D2 is the S5-diamond fragment: a discussive thesis is an S5-possibility.

Why adjunction fails: ◇A and ◇B are S5-theorems; ◇(A ∧ B) need not be. So A, B ⊬_D2 A ∧ B. That is the whole mechanism: the discussion holds both without holding their conjunction.

Why explosion fails: From A and ¬A, classical logic derives A ∧ ¬A and then everything. Without adjunction the first step is unavailable, and the derivation stops before negation is used. D2 is paraconsistent for a reason that has nothing to do with negation.

Discussive connectives: Plain ∧ would restore adjunction, so Jaśkowski defines ∧_d and →_d with the ◇ built in. A ∧_d B reads "A is discussable and B holds" — asymmetric, which is the price.

The family: Non-adjunctive paraconsistency: Jaśkowski, Rescher–Manor's approach, Schotch–Jennings preservationism. Distinct from the many-valued family (LP, FDE), the relevant family, and da Costa's C-systems.

Symbols.

Metatheory. D2 is a fragment of S5 and is therefore decidable, complete, and as well behaved as its host — which makes it the only paraconsistent logic in this collection that costs nothing metatheoretically, and the reason is that it never touches negation. Jaśkowski's diagnosis is that inconsistency in a body of assertions is a failure of aggregation, not of negation, and D2 is the theorem that this diagnosis is coherent: a logic can tolerate A and ¬A while keeping classical negation, classical disjunction, and a decidable consequence relation. That this was done in 1948 and ignored until the 1970s is why the paraconsistency literature's standard history starts in the wrong place.

Applies to. Belief merging and the aggregation of inconsistent sources. Databases with conflicting records, where the conflict is between rows rather than within one. Multi-agent settings where the group holds what no agent does. The classification of paraconsistent approaches, for which non-adjunctivity is the first and least-known branch.

Limitations. Losing adjunction is expensive in a way losing explosion is not: almost all ordinary reasoning conjoins premises, and D2's ∧_d is asymmetric, so the repair is not a repair so much as a different connective wearing the symbol. The system is paraconsistent about the discussion and not about anything a single agent believes, so it does not touch the dialetheist's cases — the Liar is not a discussion. And the ◇-fragment reading makes D2 a way of talking about S5 rather than a rival to classical logic, which is either its honesty or its irrelevance depending on what one wanted paraconsistency for.

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