Armstrongian states of affairs are thought to be composed unmereologically. Develop a formal philosophical theory of states of affairs that composes them mereologically by postulating abstract cases. For instance, the real Romeo has “cased counterparts” nominative-Romeo, accusative-Romeo, genitive-Romeo, etc. Then the state of affairs “Romeo loves Juliet” is the mereological sum of nominative-Romeo with the relation of love and accusative-Juliet, while “Juliet loves Romeo” is a state of affairs composed of the relation of love, accusative-Romeo, and nominative-Juliet. This solves the asymmetry problem. The relation between an object and its cased counterparts naturally cannot be identity or parthood, and we may call it the relation of reference. Develop formal features of this theory as well as how it might be motivated and challenged and how it might answer challenges.

Here’s a compact but fully-worked theory that does exactly what you asked: it treats states of affairs (SoAs) as mereological fusions whose parts include a relation and “cased counterparts” of the relata. I’ll call it Case-Mereological Truthmaking (CMT).

1) Ontological palette

We posit four basic categories:

Objects: \(O\).
Relations/Properties (Universals): \(U=\bigcup_{n\ge 1}U_n\) where \(U_n\) are \(n\)-adic universals.
Case types: a fixed finite set \(K\) (e.g., \(\mathrm{Nom},\mathrm{Acc},\mathrm{Gen},\ldots)\). These are abstract cases (argument-roles), not linguistic artifacts.
Cased counterparts: \(C = O\times K\). Intuitively, \(\langle o,k\rangle\) is “\(k\)-Romeo”, “\(k\)-Juliet”, etc.

There is a single extensional mereology \(\langle E,\leq\rangle\) (parthood \(\leq\); fusion/sum “\(\bigsqcup\)”) whose domain \(E\supseteq O\cup U\cup C\cup S\) contains all entities we need, including the set \(S\) of states of affairs (defined below).

2) Case machinery and reference

Case formation: for each \(k\in K\), a total function \(c_k:O\to C\) with \(c_k(o)=(o,k)\).
Decasing: \(\mathrm{dec}:C\to O\) with \(\mathrm{dec}(o,k)=o\).
Case-type: \(\mathrm{case}:C\to K\) with \(\mathrm{case}(o,k)=k\).
Reference \(primitive binary relation\): \(\mathrm{Ref}(c,o)\). Axioms:
1. \(\forall c,\exists!o,\mathrm{Ref}(c,o)\).
2. \(\forall o,k,\mathrm{Ref}(c_k(o),o)\).
3. Non-identity: \(\forall o,k;c_k(o)\neq o\).
4. No cross-category parthood: \(\neg(c\leq o)\) and \(\neg(o\leq c)\) whenever \(\mathrm{Ref}(c,o)\).

(We could identify \(\mathrm{Ref}(c,o)\) with \(\mathrm{dec}(c)=o\). Keeping it relational makes room for variants where counterparts depend on times or contexts.)

3) Relational signatures (case profiles)

Each \(n\)-adic relation \(R\in U_n\) carries a case profile \(\sigma(R)=\langle k_1,\ldots,k_n\rangle\in K^n\). Think of \(\sigma\) as the abstract argument-role typing for that universal.

Example: \(\sigma(\mathrm{Love})=\langle \mathrm{Nom},\mathrm{Acc}\rangle\).
For symmetric relations like Adjacent, we still pick two distinct role-types \(k_1,k_2\) (even if the relation is symmetric in extension). The symmetry shows up in truth conditions, not in identity of parts.

4) States of affairs as mereological fusions

An atomic SoA is any fusion of the form
\[
\mathbf{s} = \bigsqcup {,R,; c_{k_1}(o_1),\ldots,c_{k_n}(o_n),}
\]
where \(R\in U_n\) and \(\sigma(R)=\langle k_1,\ldots,k_n\rangle\).

Identity for SoAs: Extensionality—SoAs are identical iff they have exactly the same parts. So order is not encoded; role information is carried by case-types on the parts.
Category disjointness: No cased counterpart is a universal, and vice versa; neither is identical to a state.

Examples

“Romeo loves Juliet”: \(\mathbf{s}{RJ}=\bigsqcup\{\,\mathrm{Love},; c{\mathrm{Nom}}(\mathrm{Romeo}),; c_{\mathrm{Acc}}(\mathrm{Juliet})\,\}\).
“Juliet loves Romeo”: \(\mathbf{s}{JR}=\bigsqcup\{\,\mathrm{Love},; c{\mathrm{Nom}}(\mathrm{Juliet}),; c_{\mathrm{Acc}}(\mathrm{Romeo})\,\}\).

By extensionality and the different case-typed parts, \(\mathbf{s}{RJ}\neq \mathbf{s}{JR}\). That is the asymmetry solution.

5) Truth conditions and truthmaking

Let \(W\) be a set of worlds. Each \(R\in U_n\) has a world-relative extension \(||R||_w\subseteq O^n\).

Obtention: An atomic SoA \(\mathbf{s}\) obtains at \(w\) iff
\[
\exists \langle o_1,\ldots,o_n\rangle\bigg(\bigwedge_{i=1}^n \mathrm{Ref}(c_{k_i}(o_i),o_i);\land; \langle o_1,\ldots,o_n\rangle \in ||R||_w \bigg).
\]
Truthmaking: \(\mathbf{s}\) is a truthmaker for the proposition \(R(o_1,\ldots,o_n)\) (with roles given by \(\sigma(R))\). Truthmaking is hyperintensional: if \(R\) and \(R'\) are necessarily coextensive but distinct universals, then the SoAs that use \(R\) vs \(R'\) are distinct because the relation part differs.

6) Mereological background (ECM)

Adopt classical extensional mereology:

Reflexive, antisymmetric, transitive parthood.
Fusion: For any nonempty \(X\subseteq E\), there exists the unique \(y\) such that \(\forall z,(z) overlaps (y \leftrightarrow \exists x\in X, z) overlaps (x)\). Write \(y=\bigsqcup X\).

SoAs are special fusions constrained by §4.

7) Formal properties and theorems

T1 (Asymmetry)
If \(\sigma(R)=\langle k_1,\ldots,k_n\rangle\) and \(i\neq j\), then, for fixed objects \(o_1,\ldots,o_n\), swapping \(o_i,o_j\) while keeping case-types fixed yields a different SoA unless \(o_i=o_j\) and \(k_i=k_j\).
Reason: the cased parts differ (e.g., \(c_{k_i}(o_i)\) vs \(c_{k_i}(o_j)\)), hence the fusion differs.

T2 (Role-invariance of the sum)
The order of mereological summation is irrelevant. Case-types encode the argument mapping; no appeal to sequences is needed for identity.

T3 (No Bradley regress)
We do not need a further “nexus of exemplification” as a constituent: the role-typing \(\sigma(R)\) and the case-marks on counterparts determine the predication structure. No additional tie gives rise to a regress.

T4 (Hyperintensionality)
If \(R\neq R'\), then for any tuple \(\bar{o}\),
\(\bigsqcup{R, c_{k_1}(o_1),\dots}\neq \bigsqcup{R', c_{k'_1}(o_1),\dots}\), even if \(||R||_w=||R'||_w\) at all worlds.

8) Quantification and “gappy” frames

To handle quantification and predicates with open argument places, introduce frame-SoAs (templates):

For \(R\in U_n\) and a subset \(I\subseteq{1,\ldots,n}\), a frame is a fusion of \(R\) with cased counterparts \(c_{k_j}(o_j)\) for \(j\notin I\), leaving “gaps” at positions in \(I\).
A λ-case abstraction builds a property from a frame:
\(\lambda x_{k_i},., [R;\ldots,,\ldots]\) is the property had by an object \(o\) iff the completed SoA (fusing \(c{k_i}(o)\) into the gap) obtains.

Then:

“Someone loves Juliet” is true at \(w\) iff \(\exists o\,\) the frame \(\bigsqcup{ \mathrm{Love}, c_{\mathrm{Acc}}(\mathrm{Juliet})}\) can be completed with \(c_{\mathrm{Nom}}(o)\) to an obtaining SoA at \(w\).

9) Monadic and higher adicity

For monadic \(F\in U_1\) set \(\sigma(F)=\langle \mathrm{Nom}\rangle\).
“Romeo is mortal”: \(\bigsqcup{F, c_{\mathrm{Nom}}(\mathrm{Romeo})}\).
For adverbials, events, etc., either:
- treat them as relational modifiers (new universals with enriched signatures), or
- add distinct adjunct cases (e.g., \(\mathrm{Manner},\mathrm{Instrument}\)).

10) Symmetric and plural relations

If \(R\) is symmetric, \(\sigma(R)=\langle k_1,k_2\rangle\) with \(k_1\neq k_2\). Then distinct SoAs can both obtain for \(\langle a,b\rangle\) and \(\langle b,a\rangle\). This is compatible with truthmaker pluralism.
For plurals and mass terms, let \(O\) include pluralities (closed under fusion). Then \(c_k\) applies to pluralities; distributive/collective readings can be encoded by distinct universals (e.g., Lift_collective vs Lift_distributive) with their own case profiles.

11) Motivations

Solves order/asymmetry: Unlike Armstrong’s unmereological “exemplification” tie or bundles lacking order, CMT encodes argument-structure via case-typed parts. “Romeo loves Juliet” vs “Juliet loves Romeo” differ mereologically.
Truthmaker-friendly: SoAs are concrete truthmakers (in the ontological sense) with clear identity conditions; no extra nexus entity.
Hyperintensional granularity: Coextensive universals still yield distinct SoAs when the relation part differs—useful for attitude reports and fine-grained propositions.
Simplicity of composition: All structure is delivered by standard extensional mereology plus a modest role apparatus (cases and signatures).
Category discipline: The reference relation prevents collapsing objects and their role-bearers (no identity or parthood between \(o\) and \(c_k(o)\)).

12) Anticipated challenges and replies

C1. “Aren’t cases just linguistic?”
Challenge: If cases are grammatical, the ontology looks language-dependent.
Reply: Cases here are abstract roles (akin to thematic or argument positions). Linguistic case is evidence, not identity. Formally, (\sigma) assigns role-types that need not mirror any particular language.

C2. “Explosion of entities (too many counterparts)!”
Challenge: For each \(o\) and \(k\) there is \(c_k(o)\).
Reply: (i) These are lightweight abstracta—ordered pairs \((o,k)\) in the simplest model. (ii) They are indispensable bookkeeping devices replacing a primitive nexus. (iii) They are dependent on \(o\): add an existence-dependence axiom if desired: \(\exists c_k(o)\Rightarrow \exists o\).

C3. “Why isn’t reference identity or parthood?”
Challenge: If \(c_k(o)\) is not identical with or part of \(o\), what ties them together?
Reply: Exactly reference does: a sui generis, functional relation implementing decasing. It prevents category mistakes (no object is a role-marked proxy) and blocks trivialization in mereology (no \(o\) contains all its roles as parts).

C4. “Bradley regress lurking?”
Challenge: Doesn’t the relation need a further tie to its relata?
Reply: The tie is encoded by \(a\) the presence of the relation universal itself as a part, plus \(b\) the case profile \(\sigma(R)\) and the cased counterparts as distinct parts. No additional item is required to ‘unify’ them; fusion does that.

C5. “Permutation invariance vs identity of indiscernibles.”
Challenge: In mereology, unordered sums collapse structure.
Reply: We re-introduce structure via typed parts. Two fusions that differ only by swapping (\mathrm{Nom})- vs (\mathrm{Acc})-parts are numerically distinct (T1).

C6. “Passivization and alternations.”
Challenge: “Juliet is loved by Romeo” uses different surface roles.
Reply: Keep \(\sigma(\mathrm{Love})=\langle \mathrm{Nom},\mathrm{Acc}\rangle\) as the deep case profile; passive morphology concerns sentence form, not the SoA. The same obtaining SoA makes both active and passive sentences true.

C7. “Empty names and negative existentials.”
Options:
(i) Actualist: counterparts exist only for existing objects; negative existentials are made true by lack of obtaining SoAs.
(ii) Meinong-friendly: allow non-existents in \(O\) with a separate existence predicate \(E!\). The truth-conditions then quantify over \(O\) with \(E!\) filters. The core CMT is neutral; pick your favorite.

C8. “Symmetric relations double-count truthmakers.”
Reply: Accept truthmaker pluralism: two distinct obtaining SoAs can both make “\(a\) is adjacent to \(b\)” true. If you dislike that, stipulate \(\sigma(R)\) uses a single multiset case so that \(\{c_k(a),c_k(b)\}\) fuses into a unique multiset-typed part; CMT is flexible.

C9. “Are SoAs too fine-grained?”
Reply: Fine granularity is a feature: we often want \(\mathrm{Loves}(r,j)\) and \(\mathrm{Loves}(j,r)\) distinct in attitude and causal explanations. Coarser individuation can be recovered by quotienting over case-permutations if desired.

13) Variants and extensions

Temporalized counterparts: let \(C=O\times K\times T\) for times \(t\); \(\mathrm{Ref}((o,k,t),o)\). Then SoAs are fusions including time-slices.
Event semantics: add an event universal \(e\) as a part, with \(\sigma(\mathrm{Agent})=\langle \mathrm{Ag}\rangle\), \(\sigma(\mathrm{Theme})=\langle \mathrm{Th}\rangle\), etc.; adverbs/locations become further case-typed parts.
Category-theoretic gloss: view SoAs as labeled hyperedges in a typed hypergraph; mereological sum corresponds to edge union; case labels ensure role sensitivity.

14) Worked examples

Romeo loves Juliet
Parts: \({\mathrm{Love}, c_{\mathrm{Nom}}(\mathrm{Romeo}), c_{\mathrm{Acc}}(\mathrm{Juliet})}\).
Obtains at \(w\) iff \((\mathrm{Romeo},\mathrm{Juliet})\in ||\mathrm{Love}||_w\).
Juliet loves Romeo
Parts: \({\mathrm{Love}, c_{\mathrm{Nom}}(\mathrm{Juliet}), c_{\mathrm{Acc}}(\mathrm{Romeo})}\).
Distinct from the first; obtains iff \((\mathrm{Juliet},\mathrm{Romeo})\in ||\mathrm{Love}||_w\).
Juliet is admired by Romeo (passive surface form)
Same SoA as “Romeo admires Juliet” given \(\sigma(\mathrm{Admire})=\langle \mathrm{Nom},\mathrm{Acc}\rangle\).

15) Compact axiom list (for reference)

Mereology: Classical extensional mereology over \(E\).
Sorts: \(O,U,C,S\subseteq E\); \(C=O\times K\).
Reference: (unique-reference, coverage for all \(c_k(o)\), non-identity, no cross-category parthood).
Signatures: \(\sigma:U_n\to K^n\).
SoA formation: If \(R\in U_n\) and \(o_i\in O\), the fusion \(\bigsqcup{R,c_{k_1}(o_1),\ldots,c_{k_n}(o_n)}\in S\).
SoA identity: extensionality of parts.
Obtention: as in §5.
Hyperintensionality: Distinct universals yield distinct SoAs (no collapsing by necessary equivalence).

Bottom line. CMT gives you a rigorously mereological construction of states of affairs that cleanly handles argument order—asymmetry is captured by case-typed counterparts—while preserving truthmaking, avoiding a regress, and remaining flexible about linguistic surface forms. It’s ontologically modest (adding only lightweight case-proxies) yet hyperintensional enough to support fine-grained theorizing. If you’d like, I can also sketch a natural deduction system or a proof theory for λ-case abstraction over these frames.

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Friday, November 21, 2025

Mereological SoAs via ontologized cases