Contrary-to-duties as conjunctions via weaker semantics

In natural language, it seems that we can paraphrase a contrary-to-duty obligation simply by saying forbidden(A) & forbidden(A & ~B), so that whoever fails the contrary-to-duty obligation to B has violated two prohibitions (the one against A and the one against A&~B) instead of just one. However this is well-known to fail in the usual deontic logic semantics, which has motivated dyadic deontics and so on. With no disparagement of those alternative projects, consider whether a change in the semantics of the usual (monadic) deontic logic can allow for this strategy.

Great question. The short version is: yes—if you tweak the semantics (not the syntax) of ordinary, monadic deontic logic so that it pays attention to violation-minimization with specificity, then the paraphrase

CTD “If A, ought B” ≈ forbidden(A) ∧ forbidden(A ∧ ¬B)

starts behaving exactly the way we want. Below is a compact way to do it.

1) Why the usual (Kripke/KD) semantics makes the paraphrase collapse

In standard SDL (serial Kripke frames), \(F\varphi\) is \(O\neg\varphi\), true iff \(\neg\varphi\) holds at all ideal worlds. So from \(F(A)\) we already know all ideal worlds have \(\neg A\); hence \(F(A\wedge \neg B)\) becomes vacuously true (those worlds never see \(A\) anyway). That’s why you don’t get the “two violations vs one” distinction in \(A\)-worlds.

2) A ranked, specificity-sensitive semantics (monadic language, new models)

Keep the monadic language (just \(O,F\)). Change models:

Worlds and valuation. As usual, a set \(W\) with classical valuation \(V\).

Norm base. At a world \(w\), let \(\Pi_w\) be the set of prohibited propositions that are in force there: \[ \Pi_w={\chi : (M,w)\vDash F(\chi)}. \] Intuitively, \(\Pi_w\) is “the list of prohibitions that hold at \(w\).”

Specificity order on prohibitions. Write \(\chi \succ \psi\) if \(\chi \models \psi\) but not conversely (so \(A\wedge \neg B \succ A\)). Treat logically equivalent formulas as the same prohibition.

Violation profile and lexicographic betterness. For each world \(u\), define its violation vector relative to \(\Pi_w\):

• level 0 counts violations of the most specific prohibitions in \(\Pi_w\),
• level 1 counts violations of next-most-specific prohibitions, and so on (finite if \(\Pi_w\) is finite; otherwise use a well-founded refinement of \(\succ\)).

Compare worlds by lexicographic minimization of these vectors. Write \(u \preceq_w v\) if \(u\)’s vector is lexicographically \(\le\) \(v\)’s.

Truth clause for \(O/F\). Now evaluate obligations against these “best” worlds: \[ (M,w)\vDash O\varphi \quad\text{iff}\quad \varphi \text{ holds at every } u \text{ that is } \preceq_w\text{-minimal in } W. \] (As usual, \(F\chi\) is \(O\neg\chi\).)

What this buys you for CTD

Let \(\Pi_w\) contain exactly \(A\) and \(A\wedge \neg B\). Specificity gives \[ A\wedge \neg B \succ A. \]

• Globally best worlds. The \(\preceq_w\)-minimal worlds are the ones violating neither prohibition, i.e., \(\neg A\) worlds. So \(F(A)\) holds.

• Among \(A\)-worlds. All \(A\)-worlds already violate \(A\). The next comparison tier is the more specific prohibition \(A\wedge \neg B\).

  • Worlds with \(A\wedge B\) violate one prohibition (the general \(A\)).
  • Worlds with \(A\wedge \neg B\) violate two (both \(A\) and \(A\wedge \neg B\)). Hence the best \(A\)-worlds are exactly the \(A\wedge B\) worlds.

Consequently, your monadic pair \[ F(A) \; \& \; F(A\wedge \neg B) \] entails the right CTD behavior: in any situation where \(A\) (wrongly) holds, the violation-minimizing way to comply is to make \(B\) true; failing \(B\) under \(A\) counts as an extra violation. In other words, the dyadic reading “if \(A\), ought \(B\)” is recovered semantically from those two monadic prohibitions.

3) Logical fingerprints (and what you give up)

This semantics is non-normal (neighborhood-style rather than Kripke-normal):

• It blocks Ross’s paradox and the monotonicity rule \(O\varphi \Rightarrow O(\varphi\vee\psi)\). Good: \(F(A)\) no longer trivializes \(F(A\wedge \neg B)\).
• It validates a deontic consistency axiom \(D\) (no world can make both \(O\varphi\) and \(O\neg\varphi\) true, since “best” worlds are classically consistent).
• Aggregation \(O\varphi \wedge O\psi \Rightarrow O(\varphi\wedge\psi)\) can fail in general (as it arguably should in presence of independent, possibly conflicting prohibitions).
• The system is hyperintensional enough to distinguish \(F(A\wedge \neg B)\) from \(F(\neg A\rightarrow B)\): logical equivalence of matrices does not force equivalence of deontic claims, because specificity depends on informational strength, not mere truth-conditions.

4) Two quick variants (same spirit)

• Ranked/penalty semantics without explicit specificity: assign each prohibition a cost; make costs higher for more specific constraints (e.g., by default, cost \(= \) formula strength). Order worlds by the total cost of violated constraints, lexicographically by cost levels.
• Layer-wise evaluation clause: require that \(F(\chi)\) be true not only at globally best worlds, but also at the best worlds consistent with the positive part of \(\chi\). For \(\chi= A\wedge \neg B\), that forces \(\neg B\) to be avoided at the best \(A\)-worlds—again yielding \(A \Rightarrow B\) at the violation layer.

5) Bottom line

If you keep the monadic language but switch to a specificity- and violation-minimizing semantics, the natural-language paraphrase

\[ \text{“CTD }(A \Rightarrow B)\text{” } \quad\leadsto\quad F(A)  \; \& \; F(A\wedge \neg B) \]

works exactly as intended: it distinguishes one-violation \(A\wedge B\) from two-violation \(A\wedge \neg B\), and it guides action inside \(A\)-scenarios without introducing a dyadic operator. The price is leaving normal Kripke semantics (and with it, rules like RM) — but that’s precisely what prevents the collapse that originally motivated dyadic systems.