## Abstract

This paper develops a theory of propositional identity which distinguishes necessarily equivalent propositions that differ in subject-matter. Rather than forming a Boolean lattice as in extensional and intensional semantic theories, the space of propositions forms a non-interlaced bilattice. After motivating a departure from tradition by way of a number of plausible principles for subject-matter, I will provide a Finean state semantics for a novel theory of propositions, presenting arguments against the convexity and nonvacuity constraints which Fine (*Journal of Philosophical Logic*, *4545*, 199–226 13, 14, 15) introduces. I will then move to compare the resulting logic of propositional identity (PI^{1}) with Correia’s (*The Review of Symbolic Logic, 9*, 103–122 9) logic of generalised identity (GI), as well as the first degree fragment of Angell’s (2) logic of analytic containment (AC). The paper concludes by extending PI^{1} to include axioms and rules for a subject-matter operator, providing a much broader theory of subject-matter than the principles with which I will begin.

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I am greatly indebted to Mathias Böhm, Kit Fine, James Kirkpatrick, Ofra Magidor, Michail Peramatzis, James Studd, and Tim Williamson for helpful comments and discussion.

## Appendix

### Appendix

I will begin by considering propositional languages of the form \({\mathscr{L}}=\angle {\mathbb {L},\vec {\mathcal {Q}}}\rangle \) where each \(\mathcal {Q}_{i}^{n}\in \vec {\mathcal {Q}}\) is an *n*-ary sentential operator for some \(n\in \mathbb {N}\), and \(\vec {\mathcal {Q}}\) includes the extensional connectives ‘¬’, ‘∧’, and ‘∨’ along with the propositional identity operator ‘≡’. The *well-formed sentences* in \(\texttt {wfs}(\mathbb {L})\) consist of the sentence letters in \(\mathbb {L}\) along with any \(\mathcal {Q}_{i}^{n}(\vec {O})\), where \(\mathcal {Q}_{i}^{n}\in \vec {\mathcal {Q}}\) and \(\vec {O}\) is a sequence of *n* well-formed sentences. Assuming a background classical logic in which \(\ulcorner {A\rightarrow B}\urcorner \) abbreviates \(\ulcorner {\neg A\vee B}\urcorner \), we may consider the following:

As above, \({\mathscr{L}}\) is *transparent* just in case all instances of **Func** hold, where \(\ulcorner {\mathcal {Q}(\vec {O}_{(B/A)})}\urcorner \) is the result of replacing one or more instances of \(\ulcorner {A}\urcorner \) which occur as members of the sequence \(\ulcorner {\vec {O}}\urcorner \) with \(\ulcorner {B}\urcorner \), where similarly \(\ulcorner {\mathcal {Q}(\vec {O}_{[B/A]})}\urcorner \) is the result of replacing all instances of \(\ulcorner {A}\urcorner \) which occur as members of the sequence \(\ulcorner {\vec {O}}\urcorner \) with \(\ulcorner {B}\urcorner \). Additionally, I will take \(\ulcorner {C_{(B/A)}}\urcorner \) to be the result of replacing one or more instances of \(\ulcorner {A}\urcorner \) as it occurs anywhere in \(\ulcorner {C}\urcorner \) with \(\ulcorner {B}\urcorner \), as well as taking \(\ulcorner {C_{[B/A]}}\urcorner \) to be the result of replacing all instances of \(\ulcorner {A}\urcorner \) as it occurs anywhere in \(\ulcorner {C}\urcorner \) with \(\ulcorner {B}\urcorner \). We may now prove the following propositions.

**P1** If \({\mathscr{L}}\) is transparent, then **Ref** and **Imps** entail both **Sym** and **Trans**.

###
*Proof*

Assuming that \({\mathscr{L}}\) is transparent, \((A\equiv B)\rightarrow [(A\equiv A)\equiv (B\equiv A)]\) follows, where \([(A\equiv A)\equiv (B\equiv A)]\rightarrow [(A\equiv A)\rightarrow (B\equiv A)]\) holds by **Imps**, and so \((A\equiv B)\rightarrow (B\equiv A)\) follows from **Ref** by propositional logic. Again by transparency, \((A\equiv B)\rightarrow ([(A\equiv C)\rightarrow (A\equiv C)]\equiv [(B\equiv C)\rightarrow (A\equiv C)])\), and so \((A\equiv B)\rightarrow ([(A\equiv C)\rightarrow (A\equiv C)]\rightarrow [(B\equiv C)\rightarrow (A\equiv C)])\) by **Imps**. Since \((A\equiv C)\rightarrow (A\equiv C)\) holds by propositional logic, we may conclude as desired that \((A\equiv B)\rightarrow [(B\equiv C)\rightarrow (A\equiv C)]\). □

**P1** If \({\mathscr{L}}\) is transparent, then **Ref** and **Imps** entail \((A\equiv B)\rightarrow (C\equiv C_{[B/A]})\).

###
*Proof*

Assuming that \({\mathscr{L}}\) is transparent, the proof proceeds by induction on the complexity of \(C\in \texttt {wfs}(\mathbb {L})\). Of course, if \(C\in \mathbb {L}\), then \((A\equiv B)\rightarrow (C\equiv C_{[B/A]})\) holds by propositional logic if *A* occurs in *C*, and *C* ≡ *C*_{[B/A]} holds by **Ref** otherwise, where \((A\equiv B)\rightarrow (C\equiv C_{[B/A]})\) follows by propositional logic.

Assume for induction that \((A\equiv B)\rightarrow (C\equiv C_{[B/A]})\) holds whenever comp(*C*) ≤ *n*, and let comp(*C*) = *n* + 1. Assuming that \({C}={\mathcal {Q}^{n}(\vec {D})}\), we may observe that for all 1 ≤ *i* ≤ *n* that \((A\equiv B)\rightarrow (D_{i}\equiv D_{i[B/A]})\) follows by hypothesis, where \((D_{i}\equiv D_{i[B/A]})\rightarrow [\mathcal {Q}^{n}(\vec {E})\equiv \mathcal {Q}^{n}(\vec {E}_{[D_{i[B/A]}/D_{i}]})]\) by the transparency of \({\mathscr{L}}\) for any \(\vec {E}\). Assuming *A* ≡ *B*, it follows for all 1 ≤ *m* ≤ *n* that \(\mathcal {Q}^{n}(\vec {D}_{[D_{1[B/A]}/D_{1}]\ldots [D_{m[B/A]}/D_{m}]})\equiv \mathcal {Q}^{n}(\vec {D}_{[D_{1[B/A]}/D_{1}]\ldots [D_{m+1[B/A]}/D_{m+1}]})\). Given that **Ref** and **Imps**, it follows by **P1** that **Trans** holds, and so by *n* − 1 applications of **Trans**, \(\mathcal {Q}^{n}(\vec {D})\equiv \mathcal {Q}^{n}(\vec {D}_{[D_{1[B/A]}/D_{1}]\ldots [D_{n[B/A]}/D_{n}]})\). We may then observe that \({\mathcal {Q}^{n}(\vec {D}_{[D_{1[B/A]}/D_{1}]\ldots [D_{n[B/A]}/D_{n}]}}={\mathcal {Q}^{n}(\vec {D})_{[B/A]}}\), and so it follows by discharging our assumption that \((A\equiv B)\rightarrow (\mathcal {Q}^{n}(\vec {D})\equiv \mathcal {Q}^{n}(\vec {D})_{[B/A]})\). Since \({C}={\mathcal {Q}^{n}(\vec {D})}\), we may conclude that \((A\equiv B)\rightarrow (C\equiv C_{[B/A]})\). □

**P1** Assuming **Ref** and **Imps**, then \({\mathscr{L}}\) is transparent just in case **LL** holds.

###
*Proof*

Assume **Ref** and **Imps**. Letting **LL** hold in \({\mathscr{L}}\) where \(\mathcal {Q}\) is an operator in \({\mathscr{L}}\), it follows that \((A\equiv B)\rightarrow ([\mathcal {Q}(\vec {O})\equiv \mathcal {Q}(\vec {O})]\rightarrow [\mathcal {Q}(\vec {O})\equiv \mathcal {Q}(\vec {O})_{(B/A)}])\). However, given **Ref**, \(\mathcal {Q}(\vec {O})\equiv \mathcal {Q}(\vec {O})\), and so \((A\equiv B)\rightarrow [\mathcal {Q}(\vec {O})\equiv \mathcal {Q}(\vec {O})_{(B/A)}]\). In particular, \((A\equiv B)\rightarrow [\mathcal {Q}(\vec {O})\equiv \mathcal {Q}(\vec {O}_{(B/A)})]\) as in **Func**. Generalising on \(A,B,\vec {O},\) and \(\mathcal {Q}\), we may conclude that \({\mathscr{L}}\) is transparent.

Assume instead that \({\mathscr{L}}\) is transparent. Letting \(p\in \mathbb {L}\) where *p* does not occur in *B* or *C*, it follows by **L1** that \((p\equiv B)\rightarrow (C_{(p/A)}\equiv C_{(p/A)[B/p]})\). However, *C*_{(p/A)[B/p]} = *C*_{(B/A)}, and so \((p\equiv B)\rightarrow (C_{(p/A)}\equiv C_{(B/A)})\). Thus it follows that, \((A\equiv B)\rightarrow (C\equiv C_{(B/A)})\) and so **LL** follows by **Imps**. □

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### Cite this article

Brast-McKie, B. Identity and Aboutness.
*J Philos Logic* **50, **1471–1503 (2021). https://doi.org/10.1007/s10992-021-09612-w

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### Keywords

- Identity
- Subject-matter
- Hyperintensionality
- State semantics