On Tautology and Obligation

Ethical ideas are normative, or at least we take them to be so. This means they must serve as a standard for action. To say they are a standard for action is to say they command us both to imagine how the world could be different, better, than it is, and to try to transform it to align with these imaginings. To demand such a thing, ethical commands must be violable. If the world can’t be otherwise than it is, if we can’t even imagine how it ought to be, there could be no standard for acting differently from how we do act. No action could be right or wrong, it would simply be. If an action cannot be otherwise, that is, if it is logically necessary or an inexorable law of nature, it can’t be normative, just because it can’t be violated. But downstream from this apparent truism lurks an abundance of perplexing consequences that threaten to undermine some of our most immediate ethical intuitions.

Determinacy, natural and logical

Kant saw the problem. He was consumed by the question of how to preserve the possibility of morality in a world increasingly described by deterministic models of nature, most notably, Newtonian mechanics. In the Critique of Practical Reason (CPrR, Bk. I, V. Problem 1), he asserts that if human action is wholly determined by natural causality, then no practical law of freedom can be valid. Likewise, in the Groundwork of the Metaphysics of Morals, he insists that the moral law “must carry with it absolute necessity” (GMM, Preface & §2), a necessity that cannot be grounded in the mere regularities of the empirical world.

Kant’s puzzle is how to make sense of right and wrong in a world that amounts to nothing more than blind physical forces acting on matter, a reductionism that threatens to extirpate the very space in which moral judgment operates. For Kant, autonomy (Freiheit) and its attendant notion of obligation becomes crucial to any defense of ethics. To be autonomous is not to act without constraint, but to act according to principles one recognizes as one’s own. Autonomy, for Kant, is the capacity of rational beings to give themselves the moral law (GMM, §2), i.e., to constrain their actions to fit the demands of their morally rational nature.

In the Lectures on Ethics (Collins), Kant asserts that “moral actions must be contingent if they are to be determined” (Methuen ed., pp. 16). The thought here is that an action can be necessary for a will in two different ways. For imperfect (human) wills, a moral act is not merely objectively necessary (necessitas), it must also be subjectively necessitated (necessitatio/Nötigung).

For a divine being, reason just is the will; the morally necessary act (objective) and the will's own inclination (subjective) coincide perfectly. But the human will must actively constrain its inclination to fit the objective demands of morality. This act of fitting is what Kant calls subjective necessitation. Because the human will can fail to achieve this fit, even when necessitation succeeds, it remains contingent. For this reason, Kant claims that a perfect (holy) will’s moral action is objectively necessary, while from the human standpoint, a moral action can appear both to have a binding character and yet still fail to be carried out by us. This gap is a necessary condition of moral obligation: only those wills that can both succeed and fail to live up to the demands of the moral law are subject to obligation. A perfect will is not obligated to act morally, since it is in its very nature to do so. A diabolical will - a will that makes transgression of the moral law its maxim - could never act morally, and so likewise in this case no obligation arises.[1] From this it follows that an unchosen act cannot be moral, just as an irresistible act cannot be free. Moral obligation entails both the freedom to choose to act in accordance with the moral law and the freedom to violate it. Since our moral obligations need to be chosen, and not simply logically or circumstantially necessary, we must allow for deontic violability, i.e., the circumstantial violation of duties (formalized: O(p) → ◊_c¬p), so that actions can be morally obligatory only if they are not unavoidable. In short, the fact that we can fail to live up to our moral duties is, in part, what makes them duties.

Now, circumstantial (read: “real”) possibility, according to Kant, presupposes alethic (read: “merely logical”) possibility. He affirms this, among other places, in a footnote to the preface of the second edition of the Critique of Pure Reason (Bxxvi):

“I can think whatever I like, as long as I do not contradict myself, i.e., as long as my concept is a possible thought…but in order to ascribe objective validity to such a concept (real possibility, for the first sort of possibility was merely logical) something more is required."

Ascriptions of objective validity require a concept, which must be logically possible (i.e., not self-contradictory), but a logically possible concept does not require objective validity. Similarly, in the Doctrine of the Elements (4. The postulates of empirical thinking in general , A218/B265–266), he lays out the chain of dependence between possibility and actuality:

"1. Whatever agrees with the formal conditions of experience (in accordance with intuition and concepts) is possible. 2. That which is connected with the material conditions of experience (of sensation) is actual. 3. That whose connection with the actual is determined in accordance with general conditions of experience is (exists) necessarily."

The ordering here makes clear that the material conditions of experience must be connected (zusammengehängt) with the formal conditions of experience, but not vice versa. A thought may agree with only the formal conditions, regardless of whether it agrees with the material conditions of experience. Hence, what is taken to be logically possible may be thought independently of what is taken to be really possible, but the converse does not hold.

This establishes that

1. obligation presupposes the circumstantial possibility of violation and

2. circumstantial possibility presupposes logical (alethic) possibility.

It follows that the obligation to “p” entails the logical (alethic) possibility of “¬p”: (O(p) → ◊¬p). An action can be a duty only if it can possibly fail to be the case, i.e., is not a logical tautology.

Building a modal bridge

Schwarz (2024) details some significant problems that arise in formalizing models that bridge alethic logics (what necessarily or, at least possibly, is the case) and deontic logics (what ought to be the case, what is permissible).[2] He provides an analysis that can profitably be extended to the case of deontic violation in a Kantian context. In modal logics defined in terms of Kripkean possible worlds semantics, universal (☐-type) modal operators are closed under logical consequence (monotonicity). For deontic logics this means that for any action that is obligatory and logically implies another action, the consequent action is obligatory as well.

Now, tautologies are entailed by every proposition. That is, any material conditional is true just in case its antecedent term is false or its consequent term is true. Since tautologies are always true, every conditional statement with a tautology in the consequent position is true, so if we axiomatize monotonicity for obligation it follows that in a bridged alethic-deontic model with at least one obligation, any tautology will be obligatory.

(O(ψ) ∧ (ψ → ☐Φ)) ⊧ O(☐Φ)

This is not good for the principle of deontic violability. It should be noted that we can’t altogether dispense with monotonicity for deontic logics without losing basic features of deontic reasoning. Doing so would, for instance, invalidate the inference from the duty to pay taxes to the duty to file taxes.

Perhaps we could take issue with the definition of the conditional here. Kant does not concern himself with material implication, but rather with a conception of conditionality defined in terms of a “ground-and-consequent” relationship between propositions in a hypothetical judgment (A 73/B98). In this conception, a conditional statement (i.e., hypothetical judgment) is true only if there is a relevant and necessary relation between the antecedent and consequent propositions. Discussing the role of inference in the hypothetical judgment at A303/B360 he states:

“In every inference there is a proposition that serves as a ground, and another, namely the conclusion, that is drawn from the former, and finally the inference (consequence) ac­cording to which the truth of the conclusion is connected unfailingly with the truth of the first proposition“ (emphasis mine).

Now, given this constraint on implication and Kant’s conviction that only contingent propositions can be obligatory, a path out of the bind emerges. Kant needs to prohibit hypothetical judgments that bring together contingent antecedents and necessary consequents. Doing so would be in keeping with the ancient/scholastic syllogistic principle of peiorem sequitur semper conclusio partem (the conclusion always follows the weaker part), which holds that conclusions must not exceed the strength of the weakest premise. Expressed in conditional form with modal operators, this principle would rule out the implication from a merely possible antecedent to a necessary consequent.

Kant’s remarks on modality complicate the issue to a degree. In the Critique (A73–76/B98–101), he doesn't treat the antecedent (Grund) and consequent (Folge) as each carrying their own modal value. Instead, he holds that the antecedent and consequent, taken as component propositions of a hypothetical judgment, are merely problematic (possible), while the consequence (the "if/then" relation) between them is assertoric (asserted as true). As noted in Kovač (2026), this is a "purely intensional" theory of modality.[3] Kant ostensibly treats modal statements in terms of the strength of assertion of a judgment: the problematic being weakest (a free assumption), the assertoric being stronger (actual acceptance as true), the apodictic being strongest (grounded in logical necessity).

Accordingly, what is asserted is the hypothetical relation itself, not the propositional content of the antecedent and consequent clauses. Further, whatever necessity (”apodicticity”) may be at play here is confined to the nature of the inference from the antecedent to the consequent (if the ground is true, the consequent necessarily follows; in Kant’s jargon, the propositions are “connected unfailingly”, see above). So, a hypothetical judgment is never formulated as an implication from a merely possible antecedent to a necessary consequent. Given these constraints, a hypothetical judgment with a contingent obligatory proposition in the antecedent position cannot imply a tautology, meaning no obligatory tautologies occur by monotonicity. At the same time, deontic monotonicity may be preserved between antecedents and consequents that are both contingent.

Symbolization

This may be formalized as follows. The Kantian hypothetical judgment (“ψ is the ground of φ”) is expressed by the binary predicate “H(ψ, φ).” It should be noted that this predicate is not arbitrary. Rather, it has a structured intensional character with three important constraints: asymmetry (H(ψ, φ) → ¬H(φ, ψ)), a restriction on the relative modal strength of the terms (modality(φ) ≤ modality(ψ)), and relevance by shared syntactic elements between terms, defined as follows:

S(ψ, φ) := Σ(ψ) ∩ Σ(φ) ≠ ∅

where Σ = atomic formulae for propositional logic / variables for predicate logic

The predicate “H” applies to ordered pairs of propositions, and its truth stands or falls on the semantic relations between the contents of these propositions.

To complete the solution, we add a contingency constraint on obligation:

∀φ (☐(φ) → ¬Oφ)

(and “∀φ (Oφ → ∇(φ))” by contraposition)

We then add in a monotonicity rule that requires H(ψ, φ) to transfer obligation:

[(H(ψ, φ) ∧ Oψ) → Oφ].

We thus rule out obligatory tautologies, while preventing the implausible global prohibition on deontic monotonicity. Order restored!

However, an even bigger problem for deontic violability looms on the horizon. Consider the duality of obligation and permissibility. An action (𝜑) is not obligatory (¬O) just in case it is permissible (P) not to do it (¬(𝜑)). Conversely, we are obligated not to perform any impermissible action. This can be symbolized as “¬O(𝜑) ≡ P¬(𝜑)” and “O¬(𝜑) ≡ ¬P(𝜑)”, respectively. If we claim that no tautologies are obligatory this immediately implies, by duality, that all logical impossibilities are permissible.

∀φ(□φ → ¬Oφ) ⟹ ∀φ(□¬φ → Pφ)[4]

What should we, or Kant, make of this? Despite the fact that the first formulation of the categorical imperative (CI-1) is an explicit injunction against rational inconsistency, expressed in terms of the non-universalizability of certain maxims for action, perhaps Kant still has some wiggle room. After all, CI-1 does not claim that rational inconsistency never occurs; people act in inconsistent, non-universalizable ways all the time in the actual world, by lying, for instance. Lying as an occasional matter is logically possible because it happens against the backdrop of truth-telling as an overwhelmingly more common and normatively correct phenomenon. But universalized, internal to a world (w_L) in which everyone always lies, it produces performative contradictions of self-reference, as can be seen in sentences like “everyone in this world (w_L) always lies.”

Now, the problem with axiomatizing the duality of obligation and permissibility in Kantian deontic logic is not that it makes lying possible – universalizability be damned! – but that the prohibition on obligatory tautologies guarantees the permissibility of lying iff it is universalized, since it is only as a universalized maxim that it forms a contradiction. On this model, contra Kant, universalizing the maxim “always lie” makes it permissible!

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Notes

[1] In fact, there is reason to doubt that the “diabolical will” is even a coherent concept in Kant's framework. His account suggests that freedom presupposes at least implicit recognition of the moral law's authority. This can be seen in the Wille/Willkür distinction: whereas Wille (legislative reason) just is the source of the law and can't itself be "evil," Willkür (the power of choice) is that faculty that can adopt bad maxims. On this reading, a being that made rebellion against the law itself its fundamental maxim might not even count as a rational agent bound by the moral law at all.

[2] Schwarz, Modal Logic (2024), ch. 6, Deontic Logic

[3] Kovač, Immanuel Kant: Logic (2026), §4.c., The Internet Encyclopedia of Philosophy

[4] We make the following assumptions:

1. Premise: ∀φ(□φ → ¬Oφ); “No tautologies are obligatory.”

2. Deontic duality: (Pφ  ≡  ¬O¬φ); “φ is permissible iff its negation is not obligatory.”

3. Classical logic\De Morgan:   ¬(A ∨ B) ≡ (¬A ∧ ¬B)

4. □ := logical truth\tautology: “□ψ” is instantiated by a tautology such as “(p ∨ ¬p)”; “¬ψ” corresponds to the contradiction “¬(p ∨ ¬p)”; “□¬ψ” recovers the tautologous form “□¬(p ∨ ¬p).”

and derive:

i. From 1., by universal instantiation with φ := ¬ψ : □¬ψ  →  ¬O¬ψ.

ii. But by 2., ¬O¬ψ ≡ Pψ. So, from the instantiation of 1. we get: □¬ψ  →  Pψ

iii. Generalizing over ψ to recover the universal form yields: ∀ψ(□¬ψ → Pψ)

Conclusion: if no tautologies are obligatory, all contradictions are permissible. Making this explicit, we take a concrete tautology instance “(p ∨ ¬p)” and derive “□(p ∨ ¬p) → ¬O(p ∨ ¬p)”: tautologies are not obligatory. Duality of obligation yields:

¬(p ∨ ¬p) ≡ (¬p ∧ ¬¬p) ≡ (¬p ∧ p), by De Morgan

To sum up: the universal instantiation with ¬ψ in i) picks out those ψ that correspond to contradiction; ii) makes those contradictions yield Pψ (“ψ is permissible”) via the duality ¬O¬ψ ≡ Pψ. Negating the “non-obligatory tautology” gives a contradiction (by De Morgan) and the duality turns the non-obligation of the negated tautology into the permissibility of the contradiction.