Aristotle, Formal Logic, and the Shape of “Essence”
1. Introduction
I already wrote about the philosophical journey from Plato to Aristotle to Aquinas. This is of interest to me because the theology of the Catholic Church remains in thrall to Aristotelian categories - there seems little appetite for updating the church's 'physics ontology' with the latest thinking from QFT or GR. Perhaps that is wise.
Anyway, in the previous post - on reflection - I didn't really explore the subtle distinctions between, for example, form and essence. But now, with GPT5.2 just introduced, perhaps we have the cognitive horsepower to make all things clear.
Aristotle would, one hopes, be pleased. So, over to GPT5.2.
2. Aristotle’s Problem (Why This Formalisation Is Needed)
Plato’s theory is pulled in opposite directions:
1) We need genuine universals: “horse” is not merely a convenient noise. There are objective joints in reality.
2) But if universals live in a separate realm (Plato), you owe a non-mystical account of how that realm explains what happens here. (And you risk regress headaches like the Third Man.)
Aristotle’s move is to keep universals real enough to explain, but to make them immanent: the explanatory principle is in the thing. That is the metaphysical brief. Now we give it a modern logical spine.
3. The Modern Apparatus
3.1 Worlds
Let W be a non-empty set of possible worlds. Let w0 ∈ W be the actual world.
3.2 Individuals (Primary Substances)
Let D be a domain of individuals. These are the candidates for Aristotle’s primary substances.
Example individual constant: b ∈ D, intended as Bucephalus (Alexander's horse).
3.3 Existence
E(x, w): “x exists in world w”.
Example: E(b, w0) says Bucephalus exists in the actual world.
3.4 Kinds (Secondary Substances) as First-Class Objects
Up to now our domain D has contained individuals — Aristotle’s primary substances (this horse, this man). But Aristotle also treats species and genera as “secondary substances”: horse, human, animal. Formally, the clean way to represent that without smuggling in Plato is to move to a two-sorted (many-sorted) logic.
We therefore extend the model so that it has two distinct domains:
D = the domain of individuals (primary substances).
K = the domain of kinds (species/genera; secondary substances). And no, this is not Plato's Forms smuggled back in!
So the model has the shape:
M = ⟨ W, w0, D, K, I ⟩
where W is the set of possible worlds, w0 is the actual world, and I is the interpretation function assigning meanings to our symbols.
Variables are typed accordingly:
x, y range over D (individuals).
k, g range over K (kinds).
Example of an individual constant: b ∈ D, intended as Bucephalus.
Example of a kind constant: kH ∈ K, intended as the kind Horse (a species).
We connect individuals to kinds with a typed “instantiation” predicate:
Inst(x, k) = “individual x instantiates kind k”. This is going to be important later.
So Inst(b, kH) reads: “Bucephalus is a horse.” This lets us talk about Aristotle’s secondary substances rigorously, without treating kinds as just more individuals in D (which would be Platonism in disguise).
3.5 Instantiation (Kind-Membership)
Inst(x, k): “individual x instantiates kind k”. As I said, note this definition as it's essential in what follows.
Example: Inst(b, kH) means Bucephalus is a horse.
3.6 Form-Tokens and Matter-Tokens
We introduce “internal principles” as values of functions on individuals.
FormOf(x): the form-token of x - (returns the structural/organising principle in the individual.)
MatOf(x): the matter-token of x - (returns the underlying stuff/potentiality in the individual.)
Example: FormOf(b) is Bucephalus’s horse-form-token; MatOf(b) is Bucephalus’s particular flesh-and-bone matter-token.
Extending the model to type the function FormOf properly
A technical but important correction: in standard model-theoretic semantics, every function symbol must have a well-defined domain and codomain that are explicitly present in the model structure.
In the earlier definition, FormOf(x) was introduced informally as “the form-token of x”. To make that precise (and to avoid collapsing forms either into individuals or into kinds), we extend the model with a third domain.
So, instead of:
M = ⟨ W, w0, D, K, I ⟩
we use:
M = ⟨ W, w0, D, K, F, I ⟩
where:
D is the domain of individuals (primary substances).
K is the domain of kinds (species/genera; secondary substances).
F is the domain of form-tokens (immanent structural/organising principles instantiated in individuals).
We can now type the functions and relations cleanly:
FormOf : D → F (maps an individual to its form-token).
MatOf : D → M (optionally: a separate domain M of matter-tokens; or treat matter-tokens as another subset/domain for full typing symmetry).
FType : F × K → {true, false} (classifies a form-token as being of the form-type corresponding to a kind).
Example: b ∈ D (Bucephalus), and FormOf(b) ∈ F is Bucephalus’s individual horse-form-token. The key Aristotelian bridge can now be written in a fully well-typed way:
Inst(x, k) ↔ FType(FormOf(x), k)
Interpretation: kind-membership is grounded in an immanent form-token (FormOf(x)), while kinds (K) merely index the classification of forms; we have not reintroduced a Platonic realm of self-standing Forms.
3.7 Form-Types as Kind-Indexed Predicates on Form-Tokens
HorseForm(f): “f is a horse-form-token”. More generally, we will use a relation:
FType(f, k): “form-token f is of form-type corresponding to kind k”.
Example: FType(FormOf(b), kH) says Bucephalus’s form-token is a horse-form-token.
3.8 Accident Predicates (For Illustration)
White(x): “x is white”. This will stand for an accidental feature.
Example: White(b) might be true (or false) depending on which horse you chose.
3.9 Necessity
□φ means: φ is true in all admissible worlds. (Standard Kripke semantics assumed.)
4. Aristotle’s Constraints (The “Design Specs”)
Aristotle needs four constraints to hold together:
(C1) Kinds must be objective. There are genuine kinds k in K such that Inst(x, k) is not mere convention.
(C2) No separate realm of Forms is required. Whatever explains Inst(x, k) must be grounded in x.
(C3) Essence is necessity. To say “P is essential to the kind k” is to say: necessarily, anything of kind k has P. So P is a necessary property for a thing to be of that kind.
(C4) Accidents vary. Some predicates (like White) are not necessary for kind-membership.
Now we encode these constraints as axioms/definitions, and see what follows.
5. The Core Aristotelian Bridge: Kind-Membership Is Fixed by Form
This is Aristotle’s anti-Plato move stated cleanly:
Axiom (Immanence - Form grounds kind-membership):
For all individuals x and kinds k, and all worlds w:
∀x ∀k ∀w ( E(x, w) → ( Inst(x, k) ↔ FType(FormOf(x), k) ) )
Example instance:
∀w ( E(b, w) → ( Inst(b, kH) ↔ FType(FormOf(b), kH) ) )
Interpretation: x is a horse (instantiates Horse) because its internal organising principle is of the horse-form type. No transcendent exemplar is doing the explanatory work.
The “universal” is not a separate thing; it’s the form-type realised in each horse.
6. Define Essence as Modal Invariance of a Kind (important!)
We now define “P is essential to kind k” in the most literal modern way: P - some predicate on individuals - holds of all existing instances of k in every admissible world.
Definition (Kind-essence - 'Essential-to'):
EssTo(k, P) ↔ □ ∀x ∀w ( (E(x, w) ∧ Inst(x, k)) → P(x) )
Here P(x) is any predicate on individuals (e.g., White(x), HasFourLegs(x), etc.).
Example 1 (accident not essence): ¬EssTo(kH, White)
This says: it is not necessary that all existing horses are white. (Correct.)
Example 2 (form-linked essence): Let PForm,k(x) abbreviate FType(FormOf(x), k). Then:
EssTo(k, PForm,k)
This says: necessarily, anything of kind k has a form-token of the k-type.
That’s not “horses are horses”; it’s the claim that the essence of membership in a kind is tracked by form rather than matter or accident.
Logic gloss: essence vs accident (in the same formal category)
In the formal reconstruction, both essence and accident are treated as predicates of individuals — i.e. they are in the same logical category (properties that can be true of substances). The difference is their modal status.
An essential predicate is modally invariant for a kind: P is essential to kind k iff necessarily, every existing instance of k has P:
EssTo(k, P) ↔ □ ∀x ((E(x) ∧ Inst(x, k)) → P(x)).
By contrast, an accidental predicate is modally variable: it is possible to have two instances of the same kind that differ with respect to P:
◇ ∃x ∃y (Inst(x, k) ∧ Inst(y, k) ∧ P(x) ∧ ¬P(y)).
Same predicate type, different necessity profile — which is exactly why Aristotle can oppose essence to accident without inventing a second realm.
7. Matter Explains Variability, Not Definition
To represent the idea that accidents vary among members of the same kind, we add a modal possibility statement:
◇ ∃x ∃y ( E(x, w) ∧ E(y, w) ∧ Inst(x, kH) ∧ Inst(y, kH) ∧ (White(x) ∧ ¬White(y)) )
Plain English: it is possible that there exist two horses, one white and one not. Therefore White is not essential to horsehood.
This is exactly the Aristotelian distinction between what belongs to a thing as such (essence) and what belongs to it contingently (accident). Matter (and circumstance) supplies the room for difference; form supplies the stable identity.
8. The Argument as a Sequence
Step 1: Plato needs universals for explanation, but makes them transcendent, which threatens explanatory contact with particulars.
Step 2: Aristotle imposes immanence: whatever makes Inst(x, k) true must be in x itself.
Step 3: Introduce form-tokens - FormOf(x) - as internal grounds, and form-types - FType(–, k) - as the kind-indexed classification of those internal grounds.
Step 4: Axiomatically connect kind-membership to form: Inst(x, k) ↔ FType(FormOf(x), k).
Step 5: Define essence as modal invariance: EssTo(k, P) ↔ □∀x∀w((E(x,w)∧Inst(x,k))→P(x)).
Step 6: Show accidents fail the test: there are admissible worlds with horses differing in White, so ¬EssTo(Horse, White).
Step 7: Show what survives the test is form-linked: EssTo(k, x) ↦ FType(FormOf(x), k)).
Conclusion: Aristotle’s substance–essence picture can be rendered as: primary substances are form–matter composites; kind-membership is grounded in immanent form; essence is the modal profile of a kind grounded in that form; matter and circumstance explain accidental variation.
9. What This Captures (And What It Doesn’t)
This formalisation captures the heart of the Aristotelian move: the universal is immanent and explanatory, and essence is necessity rather than a heavenly object.
What it does not yet capture—because it would require additional apparatus—is Aristotle’s full-blooded teleology: forms do not merely classify; they explain capacities and ends (the four causes).
If we wanted to go further, we would add disposition predicates (e.g., CanDigestGrass(x), HasTypicalEquineLocomotion(x)) and connect those systematically to FType(FormOf(x), k). That is where “essence” becomes not just modal invariance but also explanatory depth.
10. Closing Thought
Plato made universals too far away to do their job without metaphysical baroque. Aristotle’s fix is to keep the job (explanation) but move the machinery (form) inside the thing. Once you do that, the modern modal definition of essence stops being a foreign import and starts looking like a tidy restatement of what Aristotle was aiming at all along: not a second realm, but a disciplined way of saying what cannot vary if this thing is to remain the kind of thing it is.
Here are some thoughts on the above essay from Gemini 3.
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