Wittgensteinian truthmaker mereology

The passage in the blockquote is from Tractatus Logico-Philosophicus, 5.11–5.122, and it is in this sense (if any) that this post is “Wittgensteinian”.

Based on the ideas below, make up a formal theory of “truthmaker mereology” for propositions.

I will give the name truth-grounds of a proposition to those truth-possibilities of its truth-arguments that make it true.

If all the truth-grounds that are common to a number of propositions are at the same time truth-grounds of a certain proposition, then we say that the truth of that proposition follows from the truth of the others.

In particular, the truth of a proposition ‘p’ follows from the truth of another proposition ‘q’ if all the truth-grounds of the latter are truth-grounds of the former.

The truth-grounds of the one are contained in those of the other: p follows from q.

If p follows from q, the sense of ‘p’ is contained in the sense of ‘q’.

Here’s a compact, fully specified theory that matches your ideas and turns them into a clean piece of logic. I’ll call it Truthmaker Mereology for Propositions (TMM).

1. Ontology and primitives

• Situations (a.k.a. truth-possibilities): a nonempty set \( \Sigma \). Intuitively, these are partial states of affairs (they can be as fine-grained as you like).
• Mereology on situations: a binary relation \( \preceq \subseteq \Sigma\times\Sigma \) read “is part of”, with:

1. (Reflexive) \(s\preceq s\).
2. (Antisymmetric) \(s\preceq t\) and \(t\preceq s \Rightarrow s=t\).
3. (Transitive) \(s\preceq t\preceq u \Rightarrow s\preceq u\).

• Fusions (mereological sums): a partial binary operation \( s \sqcup t \in \Sigma \) whenever a least upper bound of \({s,t}\) exists; we assume:

4. (Existence of binary fusions) for all (s,t\in\Sigma), (s\sqcup t) exists and is an upper bound, and any common upper bound of \(s,t\) is above \(s\sqcup t\).
5. (Arbitrary fusions) for any nonempty \(X\subseteq\Sigma\), \(\bigsqcup X\) exists. (Together 1–5 make \((\Sigma,\preceq)\) a complete join-semilattice.)

• Propositions: a set \( \mathsf{Prop} \) with standard connectives \(\land,\lor,\to,\neg,\top,\bot\).
• Verification (truthmaking): a relation \( \Vdash \subseteq \Sigma\times \mathsf{Prop} \), read “\(s\) makes \(p\) true”.

2. Truth-grounds and up-closure

For any proposition \(p\), its truth-grounds (your “truth-grounds of a proposition”) are \[ G(p) \ := \ {, s\in\Sigma \mid s \Vdash p ,}. \] Axioms for \( \Vdash \) (the truthmaker postulates):

TM-U (Up-closure). If \(s\Vdash p\) and \(s\preceq t\), then \(t\Vdash p\). (Any extension of a maker is still a maker.)

TM-\(\top / \bot\). For all \(s\), \(s\Vdash\top\) and not \(s\Vdash\bot\).

TM-\(\land\). \(s\Vdash p\land q \iff \exists,u,v\preceq s\) with \(u\Vdash p\), \(v\Vdash q\), and \(s \succeq u\sqcup v\). (“Having both” is having a fusion of a maker for each.)

TM-\(\lor\). \(s\Vdash p\lor q \iff (\exists,t\preceq s,., t\Vdash p) \ \text{or}\ (\exists,t\preceq s,., t\Vdash q)\).

TM-\(\to\). \(s\Vdash p\to q \iff\) for all \(t\preceq s\), if \(t\Vdash p\) then \(t\Vdash q\).

TM-\(\neg\). \(s\Vdash \neg p \iff\) for all \(t\preceq s\), not \(t\Vdash p\). (So negative truths are supported by situations that preclude any part that makes \(p\).)

Extensionality. Identify propositions up to their grounds: \(p\equiv q \iff G(p)=G(q)\). (All semantic clauses below respect \(\equiv\).)

3. “Follows from” as containment of common grounds

For a finite (or arbitrary) set \(\Gamma\subseteq\mathsf{Prop}\), write \[ G(\Gamma)\ :=\ \bigcap_{p\in\Gamma} G(p) \] —the grounds common to all propositions in \(\Gamma\).

Definition (Consequence). \[ \Gamma \ \vdash_{\mathrm{TMM}} \ \varphi \quad\text{iff}\quad G(\Gamma)\ \subseteq\ G(\varphi). \] Particular case. \(q \vdash_{\mathrm{TMM}} p\) iff \(G(q)\subseteq G(p)\). This matches your: “If all the truth-grounds of \(q\) are truth-grounds of \(p\), the truth of \(p\) follows from the truth of \(q\).”

Basic meta-properties. From set-theoretic facts:

• Reflexivity: \(p\vdash p\).
• Monotony: If \(\Gamma\subseteq\Delta\) and \(\Gamma\vdash\varphi\), then \(\Delta\vdash\varphi\).
• Cut/Transitivity: If \(\Gamma\vdash\psi\) and \(\Gamma\cup{\psi}\vdash\varphi\), then \(\Gamma\vdash\varphi\).

4. Mereology of propositions (ground-mereology)

Define a parthood order on propositions by \[ p \ \sqsubseteq \ q \quad :\iff\quad G(q)\ \subseteq\ G(p). \] Read “\(p\) is at most as informative as \(q\)”—the stronger a proposition, the smaller its ground set; so the order reverses set inclusion (as your notes suggest).

With this order:

• Meet is disjunction: \(p\ \wedge_{\sqsubseteq}\ q = p\lor q\) since \(G(p\lor q)=G(p)\cup G(q)\).
• Join is conjunction: \(p\ \vee_{\sqsubseteq}\ q = p\land q\) since \(G(p\land q)=\mathrm{Up}\{u\sqcup v : u\in G(p),,v\in G(q)\}\subseteq G(p)\cap G(q)\), and TM-U yields equality at the level of common grounds. Hence \((\mathsf{Prop}/{\equiv},\sqsubseteq)\) is a complete lattice anti-isomorphic to the lattice of ground-sets \( \langle \{G(p)\},\subseteq\rangle \).

Entailment = parthood. \(q \vdash_{\mathrm{TMM}} p \iff p \sqsubseteq q\).

5. “Sense containment”

Define the sense (core content) of \(p\) as the mereological greatest lower bound of its grounds: \[ \mathsf{Sense}(p)\ :=\ \bigwedge G(p)\ =\ \bigcap\nolimits_{\preceq} \{ s\in\Sigma : s\Vdash p \}, \] i.e., the largest situation \(c\) such that \(c\preceq s\) for every \(s\in G(p)\). (This exists because \((\Sigma,\preceq)\) has arbitrary fusions, hence dually arbitrary meets of up-sets.)

Then: \[ q \vdash_{\mathrm{TMM}} p \quad\Rightarrow\quad \mathsf{Sense}(p) \ \preceq\ \mathsf{Sense}(q). \] Reason: \(G(q)\subseteq G(p)\) implies \(\bigcap_{\preceq}G(p) \preceq \bigcap_{\preceq}G(q)\). This is your final clause: “If \(p\) follows from \(q\), the sense of \(p\) is contained in the sense of \(q\).”

6. Worked laws (all provable from the clauses above)

• Conjunction: \(G(p\land q)= \uparrow\{u\sqcup v : u\in G(p), v\in G(q)\}\); in particular \(p\land q \sqsubseteq p\) and \(p\land q \sqsubseteq q\).
• Disjunction: \(p \sqsubseteq p\lor q\) and \(q \sqsubseteq p\lor q\).
• Modus ponens as ground-containment: If \(G(q)\subseteq G(p)\), then \(q\vdash p\). If \(G(p)\subseteq G(r)\) and \(G(p\to q)=\Sigma\) (i.e., \(p\to q\) is valid), then \(G(r)\subseteq G(q)\), so \(r\vdash q\).
• Negation: \(G(\neg p)={s : \text{no } t\preceq s \text{ is in } G(p)}\) (up-closed), giving \(\neg\neg p \sqsubseteq p\) (and equality if \(\Sigma\) is Boolean in the obvious way).

7. Optional representation (to ground the picture)

A concrete model is:

• Let \(A\) be a set of atomic facts.
• Let \(\Sigma\) be the set of consistent, deductively closed sets of literals over \(A\) (finite or infinite), ordered by \(\subseteq\).
• Fusion is set-theoretic union (when consistent; if you want totality, take the closure of the union).
• A valuation \(s\) verifies an atom \(a\) iff \(a\in s\); extend to all formulas by TM-clauses above.
  This satisfies TM-U, the connectives, and the mereology (intersections and unions are the meets/joins).

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What you get

• A truthmaker-first semantics: propositions are determined by their up-closed sets of verifying situations.
• A mereology of truth: consequence is literally containment of common grounds, and the core sense of a proposition is the meet of its grounds—hence “if \(p\) follows from \(q\) then the sense of \(p\) is contained in the sense of \(q\)”.
• A tidy algebra: the lattice \((\mathsf{Prop}/{\equiv},\sqsubseteq)\) is anti-isomorphic to the lattice of ground-sets under \(\subseteq\).

If you’d like, I can also add a sequent calculus that’s sound/complete for \( \vdash_{\mathrm{TMM}} \) or work through a couple of concrete examples with specific atoms to illustrate the order and the “sense” operator.