Imperative Logic

Origin. Jørgen Jørgensen, "Imperatives and logic" (Erkenntnis, 1937–38), which posed the dilemma the field is named for; Alf Ross, "Imperatives and logic" (1944), which gave it its standard counterexample; Hare's neustic/phrastic analysis (1949, 1952); Segerberg (1990), Hansen (2008), and Vranas (2008, 2011) for the modern treatments.

Models. "Shut the door" is neither true nor false. Logical consequence is standardly truth preservation. Yet imperatives plainly figure in inferences that look valid — from "Post the letter" it seems to follow that you may "Post the letter or burn it", and from "Do all of these" that you should "Do this one". Either there is a logic here that is not about truth, or the appearance is illusory and the inferences are really about something else.

Formalism.

Jørgensen's dilemma:

Logical inference requires relata with truth values.
Imperatives have no truth values.
There are valid-seeming inferences among imperatives. Any two of these deny the third. Every position in the field is a choice of which to give up.

Ross's paradox (1944): !p ⊨ !(p ∨ q)? "Post the letter" ⊨ "Post the letter or burn it" The inference is valid if imperatives inherit ∨-introduction from their contents. It is plainly not something one may infer. So the inheritance fails — or ∨ in imperatives is not ∨.

The reductive route (Hofstadter–McKinsey 1939, Åqvist): !p is Op or "bring it about that p". Logic of imperatives = deontic logic. Cost: the free-choice and Ross problems reappear as deontic ones, unsolved.

Hare's neustic/phrastic (1949): Every sentence: a phrastic (the content) plus a neustic (the force). "Shut the door" = ⊢! (the door's being shut) "The door is shut" = ⊢. (the door's being shut) Logic operates on phrastics, which are shared. The neustic rides along. Cost: the neustic's behaviour under the connectives is exactly what is at issue.

Satisfaction semantics (Segerberg, Hansen): An imperative is satisfied or violated, never true or false. Consequence: !φ ⊨ !ψ iff every world satisfying !φ satisfies !ψ. This validates Ross's paradox and so needs a further constraint.

Support-based (Vranas 2008, 2011): A prescription has a context (the condition) and a satisfaction/violation pair. Γ ⊨ φ iff every world in which Γ is satisfied and unviolated supports φ. Ross's inference fails: satisfying "post it" does not support "post-or-burn", because the disjunction is violated by burning and the premise does not exclude it.

Free choice permission: P(p ∨ q) seems to give P(p) ∧ P(q). Invalid in every standard deontic logic. The imperative version — "Take an apple or a pear" — is where the imperative and deontic literatures share a problem and neither has an answer.

Symbols.

Symbol	Unicode	Name	Meaning
!	—	Imperative mood	!p: let it be that p
⊢! / ⊢.	—	Neustics	Hare's imperative / assertoric force
O	—	Obligation	The deontic reduction's target
P	—	Permission	Where free choice bites
⊨	U+22A8	Consequence	Not truth preservation here

Metatheory. Jørgensen's dilemma has no consensus solution after eighty-five years, and the field is organized by which horn is taken: the reductivists deny (2) by finding a proposition behind the imperative, the expressivists deny (3) by relocating the inferences to pragmatics, and the revisionists deny (1) by defining consequence over satisfaction rather than truth. Vranas's support semantics is the most developed of the third kind and blocks Ross while keeping a compositional treatment, which no earlier system managed. The connection to deontic logic runs both ways and is unresolved: SDL was built as an imperative logic in disguise, and the paradoxes it cannot solve — Ross, free choice, contrary-to-duty — are all inherited from the imperative case.

Applies to. Deontic logic's foundations, where the reduction question is live. Speech-act theory and the semantics of mood. Norm-governed multi-agent systems and BDI architectures, where instructions must compose. Legal drafting, where the same disjunction problems arise with consequences.

Limitations. No standard system: the field is a set of competing proposals with a shared problem list rather than a discipline with a core. The satisfaction-based approaches must stipulate away Ross's paradox rather than derive its failure, except on Vranas's account, whose primitives (support, context) are doing work that is not independently motivated. And the whole subject presupposes that the valid-seeming inferences are valid, which the expressivist denies and no argument settles — so the field may be a logic of an illusion.