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 it 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 the incompatibility between logical or natural necessity and moral obligation is upstream from an abundance of perplexing consequences, that threaten to undermine some of our most immediate ethical intuitions. In what follows, I hope to reveal some of these difficulties by way of an examination of alethic and deontic modal logics, to consider what dangers lurk in our attempts to bridge this chasm.

Kant saw the problem. He was consumed by the question of how to preserve the possibility of ethics in a world increasingly described by deterministic models of the natural world, most notably, Newtonian mechanics. In the Critique of Practical Reason (1788), he warns that if human action is wholly determined by natural causality, then “no practical law of freedom can be valid” (KpV, 5:99). Likewise, in the Groundwork of the Metaphysics of Morals, he insists that the moral law “must carry with it absolute necessity” (GMM, 4:389), a necessity that cannot be grounded in the mere regularities of empirical nature.

Kant’s puzzle is how to make sense of right and wrong, good and evil if the world amounts to nothing more than blind physical forces acting on matter. Later thinkers like Darwin and Nietzsche would radicalize this challenge to morality by recasting human behavior as the expression of blind biological processes – the will for survival, “the will to power” – by conceiving ethics as nothing more than a chance adaptation that might (or not) aid in the survival of a species, or a deeply capricious, even self-destructive, psychological tendency. For Kant, however, such reductionism extirpates the very space in which moral judgment operates.

For Kant, autonomy (Freiheit) and its attendant notion of duty becomes crucial to any defense of ethics. When Kant speaks of autonomy, he does not mean freedom from all law; rather, he means obedience to a law that reason itself legislates. 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, to act “not merely in accordance with the law, but from respect for the law” (GMM, 4:400). It is autonomy that distinguishes the realm of freedom from that of nature.

When we speak of an act as ethical or unethical, we presuppose the possibility of choice. As Kant observes, “an action can be called morally good only if it is done from duty” (GMM, 4:397) – and duty, in turn, implies that the agent could have acted otherwise. “If man could not resist the moral law, then there would be no moral worth in obedience to it. Hence the possibility of transgression is contained in the very idea of morality” (Lectures on Ethics (Collins), 27:284–285). An unchosen act cannot be moral, just as a merely mechanical one cannot be free. And this freedom “reveals itself through the possibility of acting contrary to the [moral] law” (KpV, 5:99–100). Freedom to choose entails the freedom to violate our duties.

Kant thinks our moral obligations need to be chosen, not simply logically or circumstantially necessary. This means that we must allow for deontic violability, i.e., the circumstantial violation of duties (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 (i.e., physical) possibility, according to Kant, presupposes logical possibility. He affirms this, among many other places, in the Jäsche Logic (9:65): “logical possibility is the condition of all other kinds of possibility; for that which involves contradiction cannot be thought at all, much less be possible in nature.” So 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.

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).[1] 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 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 moral and legal 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. In this conception, a conditional statement (i.e., hypothetical judgment) is true only if there is a relevant and necessary connection between the antecedent and consequent propositions. Kant writes, “in the hypothetical judgment, the relation of the antecedent to the consequent is…the relation of ground and consequence (of reason and what follows from it)” (A73/B98). This relation operates “according to the rule of identity, which requires that if the ground is posited, the consequence must also be posited” (A304/B361).

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. And in fact, he does just that. Remember, in a hypothetical judgment, understood as a rule of dependence (“If A, then B”), the consequent is grounded in the antecedent. But for Kant, the modal strength of the grounded proposition cannot exceed that of its ground, because the consequent’s necessity would then be independent of the antecedent, which contradicts the very notion of a ground. Kant argues this explicitly in §36 of the Prolegomena: “if the consequent is necessary, then the ground must also be necessary, for otherwise the necessity of the consequent would be without sufficient ground.” (cf. Prolegomena §36, note, and Critique of Pure Reason A289/B345–A291/B347 on the relation of grounds and modality)

Given these constraints, if the antecedent proposition is ever obligatory it is also contingent and as such cannot imply a tautology, so no obligatory tautologies occur by monotonicity. At the same time, deontic monotonicity may be preserved between antecedents and consequents that are both contingent.

We may formalize this 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)[2]

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, in a world (w_L) in which everyone always lies, it produces contradictions 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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[1] Schwarz, Modal Logic (2024), ch. 6, Deontic Logic

[2] 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)”; the negation of “□¬ψ” corresponds to the contradiction “¬(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: x□¬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 exactly those ψ whose necessity corresponds to contradiction; ii) makes those contradictions yield Pψ (“ψ is permissible”) via the duality ¬O¬ψ ≡ Pψ. In short, 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.