the parallel between deontic and alethic modalities is very convenient for natural law theory
sketch out a formal natural law theory that leans on this by having the same modalities be alethic or deontic depending on whether it’s a free being
Here’s a compact “one-operator” natural law theory that exploits the structural parallel between alethic and deontic modality by letting the same modal vocabulary become alethic or deontic depending on whether the subject is a free being.
1) Language
Two sorts of things:
- Entities: \(x,y,\dots\)
- Worlds: \(w,v,\dots\)
Predicates / relations:
- \(\mathrm{Free}(x)\): \(x\) is a free (rational/volitional) agent.
- \(\mathrm{Nat}(x)\): the “nature” (essence/functional organization) of \(x\). (Treat as a parameter supplying constraints.)
- \(\mathrm{End}(x,e)\): \(e\) is a proper end \(telos\) of \(x\).
- \(\mathrm{Means}(e,\varphi)\): \(\varphi\) is a constitutive means for \(e\).
- \(\mathrm{Act}_x(\varphi)\): “\(x\) sees to it that \(\varphi\)” (optional, but helpful for deontic readings).
- \(\mathrm{Flour}(x)\): \(x\) flourishes / realizes its form well.
One modal pair, indexed by subject:
- \(\Box_x \varphi\)
- \(\Diamond_x \varphi := \neg \Box_x \neg \varphi\)
Intended neutral gloss of \(\Box_x\):
“\(\varphi\) holds by the law of \(x\)’s nature.”
The trick is that “law of nature” bites as necessity for non-free beings, and as obligation for free beings.
2) Semantics: one frame family, different constraints
Model:
\[
\mathcal{M}=(W,{R_x}_{x\in E},V,\mathrm{Free},\mathrm{Nat},\dots)
\]
Truth clause:
\[
\mathcal{M},w \vDash \Box_x \varphi \quad\text{iff}\quad \forall v,(wR_x v \rightarrow \mathcal{M},v \vDash \varphi).
\]
Now impose different frame conditions on \(R_x\) depending on freedom:
(A) Non-free beings: alethic (necessity)
If \(\neg \mathrm{Free}(x)\), then \(R_x\) is reflexive (at least), optionally S4/S5-ish if you like strong natural necessity:
- Reflexive: \(\forall w, (wR_x w)\)
validates T: \(\Box_x \varphi \to \varphi\)
Intuition: non-free things “follow” their natures as necessities (acorns become oaks unless impeded; electrons repel, etc.). When the modal says “must,” it’s literally must.
(B) Free beings: deontic (obligation)
If \(\mathrm{Free}(x)\), require \(R_x\) to be serial (at least):
- Serial: \(\forall w,\exists v, (wR_x v)\)
validates D: \(\Box_x \varphi \to \Diamond_x \varphi\)
But do not require reflexivity, so you do not validate T.
That blocks “ought implies is”:
- For free \(x\): \(\Box_x \varphi\) does not entail \(\varphi\) at the actual world.
Intuition: if you can fail, then “bound by nature” shows up as practical necessity (obligation), not as guaranteed fact.
So the same operator \(\Box_x\) yields:
- non-free \(x\): natural necessity
- free \(x\): natural obligation
3) Core “Natural Law” principles
Now add axioms that make this natural-law-shaped rather than just “modal toggling.”
NL1. Teleological grounding
A minimal teleology-to-law schema:
\[
\mathrm{End}(x,e)\ \wedge\ \mathrm{Means}(e,\varphi)\ \to\ \Box_x \varphi.
\]
Reading: if \(\varphi\) is constitutive of realizing a proper end of \(x\), then \(\varphi\) is required by \(x\)’s nature.
- For non-free \(x\), this is an alethic requirement: the unfolding of form.
- For free \(x\), this becomes a deontic requirement: what practical reason must treat as action-guiding.
NL2. Freedom-sensitive compliance
Make the “is/ought split” explicit:
\[
\neg \mathrm{Free}(x) \to (\Box_x \varphi \to \varphi) \qquad\text{(necessity entails fact)}
\]
\[
\mathrm{Free}(x) \to \neg(\Box_x \varphi \to \varphi) \qquad\text{(obligation needn’t be met)}
\]
(Second line is not a single axiom in normal systems, but you can enforce it by not giving free agents reflexivity.)
NL3. Ought implies can (internalized)
Use the serial/D condition for free agents as the modal version:
\[
\mathrm{Free}(x) \to (\Box_x \varphi \to \Diamond_x \varphi).
\]
This is “ought implies can” in the same modality.
NL4. Practical form: obligations target agency
If you want obligations to be about what the agent does, tie \(\Box_x\) to agency:
\[
\mathrm{Free}(x)\ \wedge\ \Box_x \varphi \to \Box_x,\mathrm{Act}_x(\varphi)
\]
(or instead define obligation as \(\Box_x \mathrm{Act}_x(\varphi)\) and keep \(\varphi\) as outcome propositions.)
4) The natural law “bridge” without an is/ought fallacy
In this setup, the classical natural law move:
“From facts about nature to norms”
is recast as:
“From teleological structure to \(\Box_x\)-claims; whether those are is-like or ought-like depends on \(\mathrm{Free}(x)\).”
So you never infer an “ought” from a bare “is.” You infer a modal claim (“required by nature”), and then the logic itself tells you what kind of requirement it is, because the subject is free or not.
A handy definitional gloss:
- Natural necessity: \(\mathrm{NatNec}(x,\varphi) := \neg\mathrm{Free}(x)\wedge \Box_x\varphi\)
- Natural obligation: \(\mathrm{NatOught}(x,\varphi) := \mathrm{Free}(x)\wedge \Box_x\varphi\)
Same \(\Box\), different mode of governance.
5) Defect and wrongdoing as one notion, two manifestations
Define defect relative to nature:
\[
\mathrm{Defect}(x,\varphi) := \Box_x \varphi \wedge \neg \varphi.
\]
- If \(\neg \mathrm{Free}(x)), (\mathrm{Defect}(x,\varphi)\) is typically impossible unless you also model impediments (injury, interference). Defect is “broken functioning.”
- If \(\mathrm{Free}(x)), (\mathrm{Defect}(x,\varphi)\) is exactly the shape of wrongdoing: an obligation unmet.
This yields a unified natural law idea:
vice is a kind of malfunction in a rational animal.
6) Tiny illustrative example
Let \(a\) be an acorn, \(h\) a human.
- \(\neg \mathrm{Free}(a)\), \(\mathrm{Free}(h)\)
- \(\mathrm{End}(a,\text{oak})\), \(\mathrm{Means}(\text{oak},\text{develop_roots})\)
- \(\mathrm{End}(h,\text{flourishing})\), \(\mathrm{Means}(\text{flourishing},\text{seek_truth})\)
From NL1:
- \(\Box_a,\text{develop_roots}\)
By reflexivity for non-free beings, \(\text{develop_roots}\) follows in the normal case (alethic “must”). - \(\Box_h,\text{seek_truth}\)
But no T for free beings: it does not follow that \(h\) seeks truth. It’s an “ought.”
Same modal skeleton; different upshot because \(h\) is free.
7) What you get “for free” with this design
- A single formal connective \(\Box_x\) that covers both “must” and “ought.”
- A clean explanation of why natural law looks “descriptive” in biology/physics but “normative” in ethics: freedom is the switch.
- A unified account of deviation (impairment vs wrongdoing) as failure to realize natural modal requirements.
If you want to push it further, the next natural move is to add a ranking (a “better-than” ordering) over \(R_x\)-accessible worlds for free agents (à la preference/ideality semantics), so \(\Box_x\) becomes “true in all best \(x\)-realizations of nature,” while for non-free agents it remains “true in all nature-compatible worlds.” That gives you a very natural-law-ish story about prudence, conflict of goods, and exception handling without changing the basic modal parallel.
