Skip to main content

Identity and Aboutness


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 (PI1) 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 PI1 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.


  1. 1.

    Anderson, A.R., Belnap, N.D., & Dunn, J.M. (1976). Entailment: The Logic of Relevance and Necessity, 2nd edn. Vol. I. Princeton: Princeton University Press. ISBN 978-0-691-07192-3.

  2. 2.

    Angell, R.B. (1989). Deducibility, Entailment and Analytic Containment. In J. Norman R. Sylvan (Eds.) Directions in Relevant Logic, Reason and Argument. ISBN 978-94-009-1005-8 (pp. 119–143). Dordrecht: Springer Netherlands.

  3. 3.

    Bacon, A. (2019). Substitution structures. Journal of Philosophical Logic, 48, 1017–1075. ISSN 1573–0433.

    Article  Google Scholar 

  4. 4.

    Berto, F. (2019). Simple hyperintensional belief revision. Erkenntnis, 84, 559–575. ISSN 1572–8420.

    Article  Google Scholar 

  5. 5.

    Bou, F., & Rivieccio, U. (2011). The logic of distributive bilattices. Logic. Journal of the IGPL, 19, 183–216. ISSN 1367–0751.

    Article  Google Scholar 

  6. 6.

    Brast-McKie, B. (2020). Towards a Logic of Essence and Ground. Ph.D. thesis, The University of Oxford.

  7. 7.

    Caie, M., Goodman, J., & Lederman, H. (2019). Classical Opacity. Philosophy and Phenomenological Research n/a. ISSN 1933–1592.

  8. 8.

    Correia, F. (2004). Semantics for Analytic Containment. Studia Logica, 77, 87–104. ISSN 1572-8730.

    Article  Google Scholar 

  9. 9.

    Correia, F. (2016). On the logic of factual equivalence. The Review of Symbolic Logic, 9, 103–122. ISSN 1755-0203, 1755–0211.

    Article  Google Scholar 

  10. 10.

    Correia, F., & Skiles, A. (2019). Grounding, essence, and identity. Philosophy and Phenomenological research, 98, 642–670. ISSN 1933–1592.

    Article  Google Scholar 

  11. 11.

    Dorr, C. (2016). To be f is to be g. Philosophical Perspectives, 30, 39–134. ISSN 1520–8583.

    Article  Google Scholar 

  12. 12.

    Fine, K. (2015). Unified foundations for essence and ground. Journal of the American Philosophical Association, 1, 296–311. ISSN 2053–4477.

    Article  Google Scholar 

  13. 13.

    Fine, K. (2016). Angellic content. Journal of Philosophical Logic, 45, 199–226. ISSN 0022-3611, 1573–0433.

    Article  Google Scholar 

  14. 14.

    Fine, K. (2017a). A theory of truthmaker content i: conjunction, Disjunction and Negation. Journal of Philosophical Logic, 46, 625–674. ISSN 0022-3611 1573–0433.

    Article  Google Scholar 

  15. 15.

    Fine, K. (2017b). A theory of truthmaker content II: Subject-matter, common content, remainder and ground. Journal of Philosophical Logic, 46, 675–702. ISSN 0022-3611, 1573–0433.

    Article  Google Scholar 

  16. 16.

    Fine, K. (2017c). Truthmaker Semantics. In A Companion to the Philosophy of Language. ISBN 978-1-118-97209-0. 556–577). New York: Wiley.

  17. 17.

    Fine, K. (2020). Yablo on Subject-Matter. Philosophical studies, 177, 129–171. ISSN 1573–0883.

    Article  Google Scholar 

  18. 18.

    Fine, K., & Jago, M. (2019). Logic for exact entailment. The review of symbolic logic, 12, 536–556. ISSN 1755-0203, 1755–0211.

    Article  Google Scholar 

  19. 19.

    Fitting, M. (1989a). Bilattices and the Semantics of Logic Programming.

  20. 20.

    Fitting, M. (1989b). Bilattices and the theory of truth. Journal of Philosophical Logic, 18, 225–256. ISSN 0022-3611, 1573–0433.

    Article  Google Scholar 

  21. 21.

    Fitting, M. (1990). Bilattices in logic programming. In Proceedings of the Twentieth International Symposium on Multiple-Valued Logic., (Vol. 1990 pp. 238–246).

  22. 22.

    Fitting, M. (1991). Kleene’s Logic, Generalized. Journal of Logic and Computation, 1, 797–810. ISSN 0955-792X, 1465-363X.

    Article  Google Scholar 

  23. 23.

    Fitting, M. (1994). Kleene’s three valued logics and their children. Fundam. Inf., 20, 113–131. ISSN 0169–2968.

    Google Scholar 

  24. 24.

    Fitting, M. (2002). Bilattices are nice things. Self-reference, 53–77.

  25. 25.

    Ginsberg, M.L. (1988). Multivalued logics: a uniform approach to inference in artificial intelligence. Computational Intelegence, 4, 265–316.

    Article  Google Scholar 

  26. 26.

    Ginsberg, M.L. (1990). ISSN 0955-792X, 1465-363X. Journal of Logic and Computation, 1, 41–69.

    Article  Google Scholar 

  27. 27.

    Hawke, P. (2018). Theories of aboutness. Australasian. Journal of Philosophy, 96, 697–723. ISSN 0004–8402.

    Google Scholar 

  28. 28.

    Heim, I. (1990). E-Type Pronouns and donkey anaphora. Linguistics and philosophy, 13, 137–177. ISSN 1573–0549.

    Article  Google Scholar 

  29. 29.

    Kratzer, A. (1989). An investigation of the lumps of thought. Linguistics and philosophy, 12, 607–653. ISSN 1573–0549.

    Article  Google Scholar 

  30. 30.

    Kratzer, A. (1998). Scope or Pseudoscope? Are There Wide-Scope Indefinites?. In S. Rothstein (Ed.) Events and Grammar, Studies in Linguistics and Philosophy. ISBN 978-94-011-3969-4 (pp. 163–196). Dordrecht: Springer Netherlands.

  31. 31.

    Kratzer, A. (2002). Facts: Particulars or Information Units?. Linguistics and Philosophy, 25, 655–670. ISSN 1573-0549.

    Article  Google Scholar 

  32. 32.

    Lewis, D. (1988a). Relevant implication. Theoria, 54, 161–174. ISSN 1755–2567.

    Article  Google Scholar 

  33. 33.

    Lewis, D. (1988b). Statements partly about observation. Philosophical papers, 17, 1–31. ISSN 0556–8641.

    Article  Google Scholar 

  34. 34.

    Parry, W.T. (1989). Analytic Implication; Its History, Justification and Varietiess. In J. Norman R. Sylvan (Eds.) Directions in Relevant Logic, Reason and Argument. ISBN 978-94-009-1005-8. 101–118). Dordrecht: Springer Netherlands.

  35. 35.

    Perry, J. (1989). Possible Worlds and Subject Matter. In The Problem of the Essential Indexical and Other Essays (pp. 145–60). Palo Alto: CSLI Publications.

  36. 36.

    Rayo, A. (2013). The construction of logical space. Oxford: Oxford university press. ISBN 9780199662623, hbk.

    Book  Google Scholar 

  37. 37.

    von Fintel, K., & Partee, B.H. (2002). A Minimal Theory of Adverbial Quantification. In H. Kamp (Ed.) Context-Dependence in the Analysis of Linguistic Meaning, volume 11 of Current Research in the Semantics/Pragmatics Interface. ISBN 978-0-08-043694-4 (pp. 137–175). Amsterdam: Brill.

  38. 38.

    Yablo, S. (2014). Aboutness. Berlin: Princeton University Press. ISBN 978-1-4008-4598-9.

    Book  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Benjamin Brast-McKie.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.



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:

$$ \begin{array}{llrl} \textbf{Ref}\quad \quad A\equiv A.&\textbf{Trans}\quad (A\equiv B)\rightarrow[(B\equiv C)\rightarrow(A\equiv C)].\\ \textbf{Sym}\quad (A\equiv B)\rightarrow(B\equiv A).&\textbf{Imps}\quad (A\equiv B)\rightarrow(A\rightarrow B).\\ \textbf{LL}\quad (A\equiv B)\rightarrow(C\rightarrow C_{(B/A)}).&\textbf{Func}\quad (A\equiv B)\rightarrow[\mathcal{Q}(\vec{O})\equiv\mathcal{Q}(\vec{O}_{(B/A)})]. \end{array} $$

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.


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 \((A\equiv B)\rightarrow (C\equiv C_{[B/A]})\).


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 CC[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 ≤ in 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 AB, it follows for all 1 ≤ mn 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.


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. □

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Brast-McKie, B. Identity and Aboutness. J Philos Logic 50, 1471–1503 (2021).

Download citation


  • Identity
  • Subject-matter
  • Hyperintensionality
  • State semantics