Skip to main content
SpringerLink
Log in

Computer verification for historians of philosophy

  • Original Research
  • Published:
Synthese Aims and scope Submit manuscript

A Correction to this article was published on 21 April 2023

This article has been updated

Abstract

Interactive theorem provers might seem particularly impractical in the history of philosophy. Journal articles in this discipline are generally not formalized. Interactive theorem provers involve a learning curve for which the payoffs might seem minimal. In this article I argue that interactive theorem provers have already demonstrated their potential as a useful tool for historians of philosophy; I do this by highlighting examples of work where this has already been done. Further, I argue that interactive theorem provers can continue to be useful tools for historians of philosophy in the future; this claim is defended through a more conceptual analysis of what historians of philosophy do that identifies argument reconstruction as a core activity of such practitioners. It is then shown that interactive theorem provers can assist in this core practice by a description of what interactive theorem provers are and can do. If this is right, then computer verification for historians of philosophy is in the offing.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

Notes

  1. In this paper I will use “interactive theorem provers” and “proof assistants” as rough synonyms, although there are non-interactive theorem provers; see Sect. 2.

  2. So far as I know, the phrase ‘philosophical history’ was coined by Kremer (2013, p. 294). Kremar (2013, pp. 298–299) offers this terminology in a way that builds on Bernard William’s distinction between two interrelated endeavors, “history of ideas” and “history of philosophy”: history of ideas produces something that is history first and philosophy second, the history of philosophy produces something that is philosophy first and history second (Williams, 1994, p. 19). Since I am convinced of Kremer’s claim that philosophical history is a valuable and distinctive intellectual enterprise, and since philosophical history is the enterprise that is my focus here, I will use ‘philosophical history’ and ‘philosophical historians’ interchangeably with ‘history of philosophy’ and ‘historians of philosophy’ respectively.

  3. The below paragraph essentially follows the clean presentation in Portoraro (2019, §1), though I replace Portoraro’s talk of problems and solutions with talk of premises and conclusions.

  4. By focusing on arguments here, it is not implied that historians of philosophy are either primarily or only concerned with arguments in this sense. See Sect. 3 for discussion.

  5. In contrast, non-deductive arguments are those wherein, probably, if the premises hold, then the conclusion holds. There is significant controversy over how to understand an argument’s premises making probable its conclusion, and in particular whether this should be couched in terms of an agent’s subjective probabilities (the Bayesian view) or instead, taking the states of affairs wherein the premises hold, in a measurement of how many such scenarios are such that the conclusion holds. See Hawthorne (2021, §2).

  6. Here I am leaving room for axiomatic arguments that do not quite fill in every step. Although it is often controversial whether an argument in a text really does contain gaps—see the criticisms of Reed (2005) towards Hilbert (1899) for instance—examples of such arguments arguably are found in Euclid’s Elements, Spinoza’s Ethics, Newton’s Principia Mathematica, or Whitehead and Russell’s Principia Mathematica. If the varying sorts of argument found in these texts are all to fall under the phrase ‘axiomatic arguments’, then it would be overly restrictive to confine the phrase to derivations or proofs in the logical sense.

  7. See Geuvers (2009, p. 3) and Harrison et al. (2014, p. 1).

  8. The interactive theorem prover Coq for example has the CoqHammer automated reasoning tool that searches for a proof of some goal using previously proven (user-provided) lemmas and rewriting available from databases. See https://coqhammer.github.io/ for further details.

  9. The kernel is the trusted core of an interactive theorem prover; it passes user commands inputted on the application program interface through the type checker. The kernel type-checks user declarations and rejects inappropriate ones. Much like the kernel of an operating system, the interactive theorem prover’s kernel controls the entire program application.

  10. Some interactive theorem provers take the further step of creating a proof object that can be verified by type-checking application files apart from the interactive theorem prover’s logical system. Those that have this feature satisfy what has become known as the de Bruijn criterion following (Barendregt & Barendsen, 2002, p. 323). Although de Bruijn did not coin that term for this property, de Bruijn initiated the Automath project, and one design principle for this project was that the software satisfy the de Bruijn criterion.

  11. For comparison, a typical iPhone application has 50,000 lines of code; a Boeing 787 has 6.5 million; a Chevy Volt has 10 million; and an Android OS has 12–15 million (McCandless, 2015).

  12. As of April 2021, 3253 users have starred Coq’s Github repository and 194 people have contributed to it. And 376 authors have contributed to Isabelle’s Archive of Formal Proofs. The examples throughout this paper usually involve Coq or HOL descendants like HOL Light and Isabelle/HOL. These are some of the most popular interactive theorem provers and have been used in the significant applications to higher mathematics discussed below. It should be noted, however, that there are many different interactive theorem provers available and they are widely used in a variety of mathematical applications. A recent and comprehensive critical survey of 41 such applications is given by Saqib et al. (2019).

  13. Coq’s index of libraries has 579 entries. Isabelle/HOL’s Supplemental Library has 147 sections.

  14. The 49 Coq modules distributed with the usual opam package manager have 383,500 declarations (Müller et al., 2019, p. 172).

  15. This is not to imply that interactive theorem provers are new. They have been around since the 1960s. A history of such applications is given by Harrison et al. (2014). See also Geuvers (2009, §2) and Maric (2015, §2).

  16. See Maric (2015, §5) for other notable examples.

  17. Indeed, the late Vladimir Voevodsky’s work on Univalent Foundations brought fresh focus, energy, and publicity to the case for use interactive theorem provers (Voevodsky, 2015, p. 1278). A concise overview is given in Awodey et al. (2013, pp. 1165–1166).

  18. “...we have written a formal proof script that covers both the mathematical and computational parts of the proof. We have run this script through the Coq proof checking system, which mechanically verified its correctness in all respects. Hence, even though the correctness of our proof still depends on the correct operation of several computer hardware and software components (the processor, its operating system, the Coq proof checker, and the Ocaml compiler that compiled it)...All of them...can be (and are) tested extensively on other jobs, probably much more than the mind of an individual mathematician reviewing a proof manuscript could ever be” (Gonthier, 2005, p. 2).

  19. However, there are efforts to make using interactive theorem provers less laborious. For an example of some current work in this direction, see the Matryoshka Project website: https://matryoshka-project.github.io/.

  20. As Harrison (1996, §7) notes, there are two styles of input language for interactive theorem provers. The procedural style involves giving the application fully explicit instructions regarding what to do next. The declarative style involves giving the application a more general direction for what result to establish and perhaps an indication of a strategy for establishing it, but lets the application make explicit the needed steps. One might think of interactive theorem provers in the procedural style as sitting on an opposite pole from the automated theorem provers, whereas those in the declarative style are somewhere in between. Note that one can produce a proof in either the procedural or declarative style in some interactive theorem provers (Harrison, 1996, §4). Harrison et al. (2014, p. 39) rightly note that most users are better prepared to read the declarative style, particularly those unfamiliar with using interactive theorem provers.

  21. For discussion, see Russell (1914/1986, pp. 55–56), Klein (2020), and Elkind (2021).

  22. “The aim of historical contextualization consists in providing an interpretation of philosophical theories and the questions that they are supposed to answer that allows the reader to track the author’s philosophical intentions, taking into consideration the relevant aspects of their social, cultural and intellectual environment. In particular, it seeks to determine the role played by previous writings, events or situations in the production of the texts under consideration” (Lapointe & Pincock, 2017, p. 14).

  23. As Beaney (2013, p. 253) puts a similar point in terms of his notion of dialectical reconstruction, which phrase helpfully “suggests the interplay between rational and historical reconstruction that must continually go on in doing good history of philosophy.” Kremer (2013, p. 311) also puts a similar point in terms of his notion of philosophical history, whose goal of “the present philosophical understanding of its practitioners” is achieved “through understanding the philosophical past...”

  24. Since this example is for purposes of illustration, I will not venture into scholarly controversies over what the view actually is or who, if anyone, may have held it.

  25. “They evidently enjoy forbidding us to say that a man is good, and only letting us say that that which is good is good, or that the man is a man” (Plato, 1997, p. 251b; see also the following passages).

  26. The logic of development here is assumed to be a classical logic extended by the modal operator ‘impermissible’ and with the quantifiers ranging over (at least) everything in the spatial–temporal universe (and their properties). It is worth bearing in mind that formalization is always formalization in some logical system, but as we will see, the Late-Learners would seemingly be opposed to formalization within any system (unless one could be devised that is free of predication period).

  27. Indeed, the Late-Learners arguably would reject any formalization, or even natural language formulation, of an argument for their view. The Visitor argues in fact that the Late-Learner’s view cannot be stated because any predication is impossible on their view, at least according to Castagnoli (2010, Chap. 13).

  28. This name is slightly unfortunate because ‘computational hermeneutics’ had some years earlier been used to describe hermeneutic practices involving very different computational tools (Harnad, 1990). Since this alternative usage seems to be well-established among digital humanists (Mohr et al. , 2015; Rockwell and Sinclair , 2016; Piotrowski and Neuwirth , 2020), and it predates the newer usage by over 20 years, I will only use ‘computational hermeneutics’ in describing Fuenmayor and Benzmüller’s methodology.

  29. As Beaney (2013, p. 247) has noted, Rorty (1984, pp. 52–53, Footnote 1) talks of a hermeneutic circle with respect to the relationship between rational and historical reconstruction. This additional layer of nuance seems consistent with the computational hermeneutics framework.

  30. Note that by “compositional structure,” they mean this primarily in the Tarskian sense that a recursive specification of the truth evaluation of formulas should be available. They do not mean “compositional structure” in the Fregean sense that the meaning of a formula is built up from the meanings of its parts, that is, they do not endorse what Szabó (2020, §1.6.5) calls “the building principle.”

  31. This is not to say the programs are exactly analogous; Word and TeXdo not solve logic problems or engage in heuristic search as interactive theorem provers do. But both sorts of software applications are alike in this respect: they are both technologies that greatly assist with specific tasks (though not without certain costs), and history of philosophy was done (and still could be done in principle) without them.

  32. One might be worried about the paradox reported in Kirchner et al. (2020, §5). The paradox arose there within the context of a shallow embedding of abstract object theory, a hyperintensional second-order modal logic founded upon the logic of relations, within an extensional higher order logic founded upon the logic of functions. The paradox that arises, however, is due to the formation of complex terms using \(\lambda \)-expressions and definite descriptions; Principia’s simply-typed grammar does not permit the comprehension of universal properties, but only allows the formation of terms that are properties of all members of some universal class through its comprehension schema .

  33. One reason given for this in Principia is to abbreviate tedium (1910, pp. 93, 96). Another plausible reason, not mentioned explicitly in the work, is mitigating the cost of printing, which was substantial to the authors—each author paid £50 in 1910, which is about £6022.92 in 2020 terms (7090.77, or US$8086.50, or C$10,506.17), according to the Bank of England’s inflation calculator: https://www.bankofengland.co.uk/monetary-policy/inflation/inflation-calculator.

  34. The repository is available through https://www.principiarewrite.com/.

  35. Substitutions were the most common omissions because substitutions into axioms or theorems used as lemmas occur in practically every proof.

  36. There are many full proofs that are faithful in this sense, and sometimes the shortest proof faithful to the recipe will not be unique. Further, some proof recipes contain steps that are unnecessary.

  37. Curiously, this seems to be an empirical claim. Is there a general-use interactive theorem prover that can detect the inconsistency in Frege’s Grundgesetze without direct instructions from a human user? Not many attempts to formalize Frege’s systems in interactive theorem provers have been made, but this would be a worthwhile test case for an interactive theorem prover’s inferential engine.

  38. Isabelle is a generic proof assistant. Isabelle comes in different instances, that is, the base logic for the theorem-proving environment can differ; the usual instance of Isabelle used is Isabelle/HOL, which is a higher-order logic theorem-proving environment. Here I will use ‘Isabelle’ generically without specifying the instance. More information about Isabelle is available at https://isabelle.in.tum.de/overview.html.

  39. For more information, see the Isabelle reference manual (Wenzel, 2021, §9.3–§9.4).

  40. Although this might not be an example of doing history of philosophy, Lokhorst (2010, 2011) has leveraged interactive theorem provers (specifically, Prover9) in doing meta-ethics; Lokhorst has applied Prover9 to repairing apparent defects in Mally’s formulation of deontic logic and to modeling a robot capable of ethical reasoning. Additional examples are cited in Fuenmayor and Benzmüller (2019b).

  41. As glossed by Beyleveld (1991, p. 1), “purposive agents and prospective purposive agents (PPAs) are rationally committed, by virtue of conceiving of themselves as PPAs, to assenting to a moral principle, “the principle of generic consistency” (PGC), which states that all PPAs have claim (or “strong”) rights to their freedom and well-being.”

  42. This is also true of much modern mathematical writing at the research level. Without going so far as Gonthier (2005, p. 2) and Hales et al. (2015, p. 3) do and claiming that superior logical safety belongs to Coq-checked proofs over the human-reviewed ones, we can nonetheless say that the formalization requirements for mathematical research today fall far short of the standards laid out in the QED Project (Boyer, 1994, p. 238). Indeed, the late Vladimir Voevodsky’s work on Univalent Foundations brought new focus and energy into the QED Project (Voevodsky, 2015, p. 1278). So long as mathematicians today are publishing proof recipes (if they are), then there is room for such formalization work as Gonthier, Hales, and others have carried out on modern texts—work no different in kind than what we carried out on Principia.

  43. “It is suggested that the time is ripe for a new branch of applied logic which may be called “inferential” analysis, treating proofs as numerical analysis does calculations. This discipline, it is believed, will in the not too remote future lead to proofs of difficult theorems by machine” (Wang, 1963, p. 224).

References

  • Alama, J., Oppenheimer, P. E., & Zalta, E. N. (2015). Automating Leibniz’s theory of concepts. In International conference on automated deduction (pp. 73–97). Springer.

  • Awodey, S., Pelayo, A., & Warren, M. A. (2013). Voevedsky’s univalence axiom in homotopy type theory. Notices of the American Mathematical Society, 60(9), 1164.

    Article  Google Scholar 

  • Barendregt, H., & Barendsen, E. (2002). Autarkic computations in formal proofs. Journal of Automated Reasoning, 28(3), 321–336.

    Article  Google Scholar 

  • Beaney, M. (2013). Chap. 9: Analytic philosophy and history of philosophy: The development of the idea of rational reconstruction. In E. H. Reck (Ed), The historical turn in analytic philosophy (pp. 231–260). https://doi.org/10.1007/978-1-137-30487-2

  • Benzmüller, C., Claus, M., & Sultana, N. (2015). Systematic verification of the modal logic cube in Isabelle/HOL. arXiv preprint. arXiv:1507.08717

  • Benzmüller, C., & Paleo, B. W. (2015). Interacting with modal logics in the Coq proof assistant. In L. D. Beklemishev & D. V. Musatov (Eds), Computer science—Theory and applications. Lecture notes in computer science, CSR 2015 (Vol. 9139, pp. 398–411). Springer. https://doi.org/10.1007/978-3-319-20297-6

  • Beyleveld, D. (1991). The dialectical necessity of morality. University of Chicago Press.

    Google Scholar 

  • Beyleveld, D. (2012). The principle of generic consistency as the supreme principle of human rights. Human Rights Review, 13, 1–18. https://doi.org/10.1007/s12142-011-0210-2

    Article  Google Scholar 

  • Blumson, B. (2021). Mereology. Archive of Formal Proofs. Formal proof development. https://isa-afp.org/entries/Mereology.html

  • Boyer, R. (1994). The QED manifesto. In A. Bundy (Ed), Automated deduction—Cade-12: 12th International conference on automated deduction, proceedings. Lecture notes in artificial intelligence, in the panel discussion “A Mechanically Proof-Checked Encyclopedia of Mathematics: Should We Build One? Can We?”, Berlin (Vol. 12, pp. 237–251).

  • Castagnoli, L. (2010). Ancient self-refutation. Cambridge University Press.

    Google Scholar 

  • Davidson, D. (1973). Radical interpretation interpreted 8(3/4), 313–328. https://www.jstor.org/stable/42968535

  • Davidson, D. (1994). Radical interpretation interpreted 8, 121–128. https://www.jstor.org/stable/2214166

  • Elkind, L. D. C. (2021). Chap. 15: Russell on the ethical value of logic. In T. J. Madigan & P. Stone (Eds), Bertrand Russell: Public intellectual (2nd ed., pp 251–267). Tiger Bark Press.

  • Fitelson, B., & Zalta, E. N. (2007). Steps toward a computational metaphysics. Journal of Philosophical Logic, 36(2), 227–247.

    Article  Google Scholar 

  • Fleuriot, J. (2001). A combination of geometry theorem proving and nonstandard analysis with application to Newton’s Principia. In Distinguished Dissertations. Springer. https://doi.org/10.1007/978-0-85729-329-9

  • Fuenmayor, D., & Benzmüller, C. (2017). Automating emendations of the ontological argument in intentional higher-order modal logic. In Joint German/Austrian conference on artificial intelligence (Künstliche Intelligenz) (pp. 114–127). Springer.

  • Fuenmayor, D., & Benzmüller, C. (2018a). Computational hermeneutics: An integrated approach for the logical analysis of natural-language arguments. In International conference on logic and argumentation (pp. 187–207). Springer.

  • Fuenmayor, D., & Benzmüller, C. (2018b). Formalisation and evaluation of Alan Gewirth’s proof for the principle of generic consistency in Isabelle/HOL. https://www.isa-afp.org/entries/GewirthPGCProof.html

  • Fuenmayor, D., & Benzmüller, C. (2019a). A computational-hermeneutic approach for conceptual explicitation. In Á. Nepomuceno-Fernández, L. Magnani, F. J. Salguero-Lamillar, C. Barés-Gómez, & M. Fontaine (Eds.), Model-based reasoning in science and technology (pp. 441–469). Springer.

    Chapter  Google Scholar 

  • Fuenmayor, D., & Benzmüller, C.(2019b). Harnessing higher-order (meta-)logic to represent and reason with complex ethical theories. In A. Nayak & A. Sharma (Eds), PRICAI 2019: Trends in artificial intelligence. Lecture notes in computer science (Vol. 11670, pp. 418–432). Springer. https://doi.org/10.1007/978-3-030-29908-8_34

  • Fuenmayor, D., & Benzmüller, C. (2020). A case study on computational hermeneutics: E. J. Lowe’s modal ontological argument (pp 195–228). Springer. https://doi.org/10.1007/978-3-030-43535-6_12

  • Geuvers, H. (2009). Proof assistants: History, ideas and future. Sadhana, 34, 3–25. https://doi.org/10.1007/s12046-009-0001-5

    Article  Google Scholar 

  • Gewirth, A. (1981). Reason and morality. University of Chicago Press.

    Google Scholar 

  • Gonthier, G. (2005). A computer-checked proof of the Four Colour Theorem. https://www.cl.cam.ac.uk/~lp15/Pages/4colproof.pdf

  • Gonthier, G. (2008). A computer-checked proof of the four colour theorem. Notices of the American Mathematical Society, 55(11), 1382–1393. http://www.ams.org/notices/200811/tx081101382p.pdf

  • Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Roux, S. L., Mahboubi, A., O’Connor, R., Ould, S. B., Pasca, I., Rideau, L., Solovyev, A., Tassi, E., & Théry, L. (2013). A machine-checked proof of the odd order theorem. In S. Blazy, C. Paulin-Mohring & D. Pichardie (Eds), Interactive theorem proving. Lecture notes in computer science (pp 163–179). Springer. https://hal.inria.fr/hal-00816699

  • Hales, T., Adams, M., Bauer, G., Dang, D. T., Harrison, J., Hoang, T. L., Kaliszyk, C., Magron, V., McLaughlin, S., Nguyen, T. T., Nguyen, T. Q., Nipkow, T., Obua, S., Pleso, J., Rute, J., Solovyev, A., Ta, A. H. T., Tran, T. N., Trieu, D. T., Urban, J., Vu, K. K., & Zumkeller, R. (2015). A formal proof of the Kepler Conjecture. arXiv e-prints. https://arxiv.org/abs/1501.02155, eprint arXiv:1501.02155

  • Harnad, S. (1990). Against computational hermeneutics 4, 167–172. http://cogprints.org/1576/

  • Harrison, J. (1996). Proof style. In International workshop on types for proofs and programs (pp. 154–172). Springer.

  • Harrison, J., Urban, J., & Wiedijk, F. (2014). History of interactive theorem proving. Computational Logic, 9, 135–214.

    Article  Google Scholar 

  • Hawthorne, J. (2021). Inductive logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, spring (2021 edition). Metaphysics Research Lab: Stanford University.

    Google Scholar 

  • Hilbert, D. (1899). Grundlagen der Geometrie. Teubner: Verlag von B. G.

    Google Scholar 

  • Kirchner, D., Benzmüller, C., & Zalta, E. N. (2019). Computer science and metaphysics: A cross-fertilization. Open Philosophy, 2(1), 230–251.

    Article  Google Scholar 

  • Kirchner, D., Benzmüller, C., & Zalta, E. N. (2020). Mechanizing Principia Logico-Metaphysica in functional type-theory. The Review of Symbolic Logic, 13(1), 206–218. https://doi.org/10.1017/S1755020319000297

    Article  Google Scholar 

  • Klein, A. (2020). Philosophy at war: Nationalism and logical analysis. https://aeon.co/essays/philosophy-at-war-nationalism-and-logical-analysis

  • Koutsoukou-Argyraki, A. (2019). Aristotle’s assertoric syllogistic. Archive of Formal Proofs, Formal proof development. https://isa-afp.org/entries/Aristotles_Assertoric_Syllogistic.html

  • Kremer, M. (2013). Chap. 11: What is the good of philosophical history? In E. H. Reck (Ed), The historical turn in analytic philosophy (pp. 294–325). https://doi.org/10.1007/978-1-137-30487-2

  • Lapointe, S., & Pincock, C. (2017). Introduction. In S. Lapointe & C. Pincock (Eds.), Innovations in the history of analytical philosophy (pp. 1–23). Palgrave Macmillan.

    Chapter  Google Scholar 

  • Lokhorst, G. J. C. (2010). Where did Mally go wrong? In Deontic logic in computer science (Vol. 6181). Springer. https://doi.org/10.1007/978-3-642-14183-6_18

  • Lokhorst, G. J. C. (2011). Computational meta-ethics. Minds and Machines, 21(2), 261–274.

    Article  Google Scholar 

  • Maric, F. (2015). A survey of interactive theorem proving. Zbornik radova, 18(26), 173–223.

    Google Scholar 

  • McCandless, D. (2015). Codebases, research by Pearl Doughty-White and Miriam Quick. https://www.informationisbeautiful.net/visualizations/million-lines-of-code/

  • Mohr, J. W., Wagner-Pacifici, R., & Breiger, R. L. (2015). Toward a computational hermeneutics. Big Data and Society. https://doi.org/10.1177/2053951715613809

    Article  Google Scholar 

  • Müller, D., Rabe, F., & Sacerdoti Coen, C. (2019). The Coq library as a theory graph. In C. Kaliszyk, E. Brady, A. Kohlhase & C. Sacerdoti Coen (Eds), Intelligent computer mathematics (pp. 171–186). Springer.

  • Novak, N. (2015). Practical extraction of evidence terms from common-knowledge reasoning. Electronic Notes in Theoretical Computer Science, 312(24), 143–160. https://doi.org/10.1016/j.entcs.2015.04.009

    Article  Google Scholar 

  • O’Leary, D. J. (1988). The propositional logic of Principia Mathematica and some of its forerunners. Russell: The Journal of Bertrand Russell Studies, 8(1–2), 92–115.

  • Parsons, T. (2021). The traditional square of opposition. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, Fall (2021 edition). Metaphysics Research Lab: Stanford University.

    Google Scholar 

  • Piotrowski, M., & Neuwirth, M. (2020). Prospects for computational hermeneutics. In Atti del IX Convegno Annuale AIUCD. https://doi.org/10.6092/UNIBO/AMSACTA/6316

    Article  Google Scholar 

  • Plato. (1997). Sophist. In J. M. Cooper (Ed.), Plato: Complete works (pp. 235–293). Hackett Publishing Company.

  • Portoraro, F. (2019). Automated reasoning. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, spring (2019 edition). Metaphysics Research Lab: Stanford University.

    Google Scholar 

  • Reed, D. (2005). Figures of thought: Mathematics and mathematical texts. Routledge.

  • Rockwell, G., & Sinclair, S. (2016). Hermeneutica: Computer-assisted interpretation in the humanities. MIT Press.

  • Rorty, R. (1984). Chap. 3: The historiography of philosophy: Four genres. In R. Rorty, J. B. Schneewind & Q. Skinner (Eds), Philosophy in history (pp. 49–76). Cambridge University Press.

  • Russell, B. (1914/1986). Chap. 5: On scientific method in philosophy. In J. G. Slater (Ed), The philosophy of logical atomism and other essays, 1914–19, the collected papers of Bertrand Russell (Vol. 8, pp. 55–73). George Allen & Unwin LTD.

  • Saqib Nawaz, M., Malik, M., Li, Y., Sun, M., & Ikram Ullah Lali, M. (2019). A survey on theorem provers in formal methods. arXiv e-prints. arXiv:1912.03028, 1912.03028

  • Shapiro, S., & Kissel, T. K. (2018). Classical logic. In Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/spr2018/entries/logic-classical/

  • Smith, R. (2020). Aristotle’s logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, Fall (2020 edition). Metaphysics Research Lab: Stanford University.

    Google Scholar 

  • Szabó, Z. G. (2020). Compositionality. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, Fall (2020 edition). Metaphysics Research Lab: Stanford University.

    Google Scholar 

  • Voevodsky, V. (2015). An experimental library of formal mathematics based on the univalent foundations. Mathematical Structures in Computer Science, 25(5), 1278–1294. https://doi.org/10.1017/S0960129514000577

    Article  Google Scholar 

  • Wang, H. (1963). Chap. IX: Toward mechanical mathematics. In A survey of mathematical logic. reprinted from 1960 (Vol. 1, pp. 224–268). Science Press and North-Holland Publishing Company.

  • Wenzel, M. (2021). The Isabelle/Isar reference manual. https://isabelle.in.tum.de/documentation.html

  • Whitehead, A. N., & Russell, B. (1910). Principia Mathematica (1st ed., Vol. I). Cambridge University Press.

  • Williams, B. (1994). Descartes and the historiography of philosophy. In J. Cottingham (Ed.), Reason, will, and sensation: Studies in Descartes’s metaphysics. Clarendon Press.

    Google Scholar 

Download references

Acknowledgements

I thank participants of the 2021 Meeting of the Society for the Study of the History of Analytical Philosophy for their helpful comments. I also thank Katalin Bimbó, Bernard Linsky, and four anonymous reviewers with Synthese for their excellent feedback and criticisms. I am grateful to Samuel C. Fletcher for organizing this special issue of Synthese and for his patience with my revisions. Some of the research on this paper was done while I was an Izaak Walton Killam Postdoctoral Fellow in Philosophy at the University of Alberta.

Author information

Authors and Affiliations

  1. Department of Political Science, Western Kentucky University, Cherry Hall 306, 1906 College Heights Blvd, Bowling Green, KY, 42101-1086, USA

    Landon D. C. Elkind

Authors
  1. Landon D. C. Elkind
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Landon D. C. Elkind.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

The original online version of this article was revised: formula.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elkind, L.D.C. Computer verification for historians of philosophy. Synthese 200, 198 (2022). https://doi.org/10.1007/s11229-022-03678-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11229-022-03678-y

Keywords

Search

Navigation