Discussive Logic (D2)

Origin. Stanisław Jaśkowski, "Propositional calculus for contradictory deductive systems" (Studia Societatis Scientiarum Torunensis, 1948; English 1969) — the first paraconsistent logic, published the year Priest was born and fifteen years before da Costa. Jaśkowski was answering Łukasiewicz's 1910 challenge to construct a system in which the law of non-contradiction fails without triviality. A second paper (1949) added the discussive conjunction, which the first had done without. Kotas (1974) gave the first axiomatization; da Costa and Dubikajtis (1977) worked it into Jaśkowski's own discussive language.

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 by that ◇: a thesis needs only one participant, and two theses need not share one. Adjunction is the first casualty — 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. Atoms are left alone; the outer ◇ below does the work. p* = p for atomic p (¬A)* = ¬A* (A ∨ B)* = A* ∨ B* (A ∧ B)* = A* ∧ B* (A →_d B)* = ◇A* → B* (discussive implication)

The consequence relation: ⊢_D2 A iff ⊢_S5 ◇A* A₁, …, Aₙ ⊢_D2 B iff ⊢_S5 ◇A₁* → (⋯ → (◇Aₙ* → ◇B*)⋯) A discussive thesis is an S5-possibility, and discussive consequence relates possibilities.

Why adjunction fails: ◇A and ◇B can hold together where ◇(A ∧ B) does not — take B = ¬A. One participant asserts A, another ¬A, and no one asserts the conjunction. So A, B ⊬_D2 A ∧ B.

Why explosion fails: Two routes, both blocked. Via adjunction: A, ¬A ⊢ A ∧ ¬A ⊢ B — the first step is unavailable. Via disjunctive syllogism: ∨ translates plainly, so ◇A ⊨ ◇(A ∨ B) and disjunction introduction survives; but the participant asserting A ∨ B may be the one asserting A while ¬A comes from another, so ◇(A ∨ B), ◇¬A ⊭ ◇B and the syllogism fails. Negation is classical at every step. D2 is paraconsistent for a reason that has nothing to do with negation.

Discussive conjunction: The second paper recovers a usable conjunction: A ∧_d B =df ◇A ∧ B. Adjunction holds for it, since S5 ⊨ ◇A ∧ ◇B → ◇(◇A ∧ B), so A, B ⊢_D2 A ∧_d B. Explosion still fails: (A ∧_d ¬A) →_d B is not a thesis. D2 is therefore non-adjunctive in ∧ and adjunctive in ∧_d — the ◇, not the failure of adjunction, is what carries the paraconsistency. ∧_d is asymmetric, which is the price. Read as a clause of the translation rather than a definition in the modal language it yields a different logic, Ciuciura's D2*.

The family: Non-adjunctive paraconsistency: Jaśkowski, Rescher–Manor's approach, Schotch–Jennings preservationism. Subvaluationism is the same shape on another frame: ◇ over precisifications rather than participants. Distinct from the many-valued family (LP, FDE), the relevant family, and da Costa's C-systems.

Symbols.

Metatheory. D2 is defined by translation into S5 and inherits its metatheory wholesale: decidability, the finite model property, and completeness are the host's rather than results that had to be established for D2. That is its real distinction — LP is decidable and completely axiomatized too, but that had to be proved of LP. S5 is sufficient and not necessary: Furmanowski, Perzanowski, and Nasieniewski–Pietruszczak identified much weaker normal and regular logics with the same ◇-theses, and any of them defines D2. Axiomatizing D2 directly in the discussive language proved harder than the modal definition suggests, which is why Kotas's result was revisited by Omori and Alama (2018). Jaśkowski's diagnosis is that inconsistency in a body of assertions is a failure of aggregation, not of negation, and D2 shows the diagnosis coherent: a logic can tolerate A and ¬A with every connective classical 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. Nor does adjunction failure explain as much as the textbook billing suggests — the negation-free fragments are outright classical, and the ◇ kills disjunctive syllogism in a language with no conjunction at all, so non-adjunctivity is a symptom of the translation rather than the mechanism behind it. 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.

© 2026 Lingenic LLC