Analytic Philosophy

In October 2025, Russell’s ‘On Denoting’ celebrates the 120th anniversary of its publication (Russell, 1905). It is most famous for the theory of definite descriptions. Many will agree with Ramsey that it is a paradigm of philosophy. The present contribution compares and contrasts it with the first axiomatic theory of definite descriptions: Frege’s. To do so, both theories will be clad in a modern proof-theoretic dress. There is a sense in which ‘On Denoting’ marks the inception of analytic philosophy. The theory of definite descriptions was the breakthrough that allowed Russell to part with the Meinongianism he espoused around the time of the Principles (Russell, 1903): instead of taking propositions at face value, Russell subjects them to logical analysis. The theory of definite descriptions continues to be debated to the present day. This motivates the combination of historic and systematic aspects of our contribution. A definite description is an expression of the form ‘the F’. If there is a unique F, it is proper, if not, improper. It appears to be a singular term that purports to refer to the unique F. According to Russell this is only apparent. Definite descriptions disappear upon analysis of complete sentences in which they occur. ‘The F is G’ means ‘There is a unique F and it is G’. No definite description occurs in the paraphrase, hence no referent of ‘the F’ is required to specify the meaning of ‘The F is G’. Definite descriptions are governed by a contextual definition (Whitehead and Russell, 1910,∗14.01): [ιxFx]G(ιxFx) =df. ∃x(∀y(Fy ↔x = y) ∧Gx) Russell’s notation is clumsy. Arthur Prior proposed a more elegant method (Prior, 1963): Formalise complete sentences ‘The F is G’ by a binary quantifier I as Ix(F ,G). In this form, Russell’s theory has been defended forcefully by Neale (Neale, 1990). We’ll present a formalisation of I in sequent calculus. In ‘Uber Sinn und Bedeutung’ (Frege, 1892), Frege proposed a chosen object theory, later advocated by Carnap (Carnap, 1947): improper definite descriptions refer to a more or less arbitrarily chosen object, say 0. The theory Grundgesetze der Arithmetik (Frege, 1893) is a version of this theory where the object isn’t chosen arbitrarily. It is rather idiosyncratic. The definite description operator \ does not apply to predicates. \ξis the unique object that falls under a concept, if ξ is a name of the extension of a concept under which a unique object falls, and if it is not, \ξis its argument. \is governed by the axiom a = \´ϵ(a = ϵ): a is identical to the unique thing that falls under the concept ‘being identical to a’. The axiom only specifies how to use \when it is applied to a singleton given by an identity. Dummett observes that Frege only uses \when it has been shown that a unique object is in ξ(Dummett, 1991). In practice, this is how mathematicians often proceed. But it remains odd: the axiom does not cover all cases. Frege’s axiomatisation is incomplete. We will present Frege’s theory in a less idiosyncratic fashion and show how to complete it with rules of sequent calculus that cover all cases. The sequent calculi for Frege’s and Russell’s theories of definite descriptions satisfy desirable proof theoretic criteria, in particular cut elimination. There are also corresponding systems of natural deduction, more convenient in practice. We’ll compare the two formalisations by noting differences in theorems and the validity of inferences. References Carnap, R. (1947). Meaning and Necessity. A Study in Semantics and Modal Logic. Chicago, Illinois: University of Chicago Press. Dummett, M. (1991). Frege. Philosophy of Mathematics. London: Duckworth. Frege, G. (1892). Über Sinn und Bedeutung. Zeitschrift f ¨ ur Philosophie und philosophische Kritik 100, 25–50 Frege, G. (1893). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleited, Volume I. Jena: Hermann Pohle. Kürbis, N. (2024). Definite descriptions. In H. Nesi and P. Milin (Eds.), The Encyclopedia of Language and Linguistics (3 ed.)., Chapter published online first https://doi.org/10.1016/B978-0-323-95504-1.00381-1. Elsevier. Neale, S. (1990). Descriptions. Cambridge, Mass.: MIT Press. Pelletier, F. J. and B. Linsky (2005). What is Frege’s theory of definite descriptions? In B. Linsky and G. Imaguire (Eds.), On Denoting: 1905-2005. Munich: Philosophia Verlag. Prior, A. (1963). Is the concept of referential opacity really necessary? Acta Philosophica Fennica 16, 189–199. Russell, B. (1903). The Principles of Mathematics. Cambridge University Press. Russell, B. (1905). On denoting. Mind 14(56), 479–493. Russell, B. (1919). Introduction to Mathematical Philosophy. London: George Allen and Unwin. Whitehead, A. and B. Russell (1910). Principia Mathematica, Volume 1. Cambridge: Cambridge University Press.

A New Look on the Priority Dispute between Wittgenstein and Carnap Wittgenstein’s allegations against Carnap in plagiarism from the summer of 1932 are well-known. Wittgenstein claimed that in his 1932 paper “Physicalistic Language as the Universal Language of Science” Carnap borrowed material both from the Tractatus, as well as from the protocols of his talks with Schlick and Weismann. Furthermore, Wittgenstein held that Carnap is plagiarizing him on such a large scale that “soon [he will] be in a situation where [his] own work shall be considered as a reheated version or plagiarism of Carnap’s”. Carnap decisively denied the allegations. To the clarification of the Prioritätsstreit between Wittgenstein and Carnap was dedicated extensive literature (Coffa 1991, Hintikka 1996, McGuinness 2002, Pears 1988, Stern 2013, Uebel 1995). In this paper I suggest a new interpretation, based on a fresh reading of Wittgenstein’s Tractatus which will be published in a book-form in 2026. According to it, first, Wittgenstein’s book discussed one world: the world of language—not the “external world” and the language and how they relate to one another. What Wittgenstein meant under “the world”, i.e. his ontology, is not something external to language but what language can possibly refer to. Secondly, language has two faces: the already mentioned ontology, and its flip side—the Tractarian concept script. The two ways of speaking effectively amount to two different yet equivalent languages. My claim is that Wittgenstein saw exactly the latter position of the Tractatus as plagiarized in Carnap’s introduction of two forms of language: formal and material (physicalistic). In his letter to Schlick from August 8, 1932, Wittgenstein clearly says that he dealt with physicalism in the Tractatus—but under other name. Moreover, he further claims that Carnap “makes no step beyond him” on this point. It is well-known that in 1925–26 and 1926–27 the core Vienna Circle, Schlick, Waismann, Neurath, Hahn and Carnap, intensively discussed Wittgenstein’s Tractatus line-by-line. However, in 1931 came a new wave of influence of Wittgenstein’s book on them—when Waismann’s “Theses” luminously presented its ideas in a form easy to understand (McGuinness 1984). The “Theses” too were industrially studied by its members and discussed at meetings of the Circle. The importance of this short work results not only from the fact that they combined in a succinct form the ideas was the Tractatus with some new ideas of Wittgenstein. They also presented his book in most clear form. Only now could two members of the core Vienna Circle, Neurath and Carnap, understand two main ideas of the Tractatus. Firstly, following the reading of Wittgenstein’s newly exposed ideas, Neurath, who openly criticised the Tractatus at the time, realized that there is only one thing—language; not language and the external world. In his 1932 paper Carnap openly followed Neurath on this. Secondly, Carnap understood and adopted the language dualism of ontology and conceptual script, defended in the Tractatus, transforming it into dualism between formal and material (physicalistic) language. Exactly it led to the priority dispute.

A few decades ago, Arthur Fine’s natural ontological attitude (NOA) preoccupied philosophers of science as a legitimate, naturalist third alternative to both scientific realism and scientific anti-realism. With NOA, the concept of truth is “a concept already in use and [we] abide by the standard rules of usage”, rules “governed by the very same standards of evidence and inference that are employed by science itself” (Fine, 1986). Alan Musgrave (1989) and Stathis Psillos (1999) subsequently pointed out that the relevant science in this context needs to be interpreted realistically, contrary to the neutralism of NOA, and so argue that Fine’s NOA collapses into a form of scientific realism. My task in this paper is to defend Fine’s NOA from this Collapse argument. NOA, contends Fine, “sanctions ordinary referential semantics, and commits us, via truth, to the existence of the individuals, properties, relations, processes, and so forth referred to by the scientific statements that we accept as true” (Fine 1986). To proponents of the Collapse argument, this statement appears as an endorsement of scientific realism. Here one might salvage NOA from this critique, as Kyle Stanford (2006) does, by reading Fine as an instrumentalist. My alternative, non-instrumentalist answer is to interpret NOA as a semantical thesis, following the lead of Gottlob Frege. Fine says that “NOA passes a minimal standard for philosophy of science” (Fine 1986). We can say that NOA expresses a minimal realism: what counts as an independently existing entity in the world is not entirely vacant. Comparatively, anti-realism maintains that the set of worldly entities is vacant: everything is a product, ultimately, of our minds. Frege maintains that this independent, real world contains what he terms ‘thoughts’, understood objectively and non-psychologically. Thoughts are propositions, and often for Frege they express mathematical claims, though they can express abstract claims of other sorts. Fregean semantics, I argue, pastes very easily onto NOA. Fine is explicit that the standards of usage employed by scientists “involve a Davidsonian-Tarskian, referential semantics”, one that supports “a thoroughly classical logic of inference” along with “the customary ‘grammar’ of ‘truth’ (and its cognates)” (1986). Frege’s Begriffschrift is the ultimate origin of this semantics, and in “Thought” Frege argues at length that truth does not involve the correspondence of a picture with reality. Similarly, Fine rejects a “special correspondence theory of realism”, the “correspondence with the external world” so longed for by “the realist” (1986). Rather, for both philosophers, truth is sui generis. Frege says this expressly in “Thought”. Comparatively, Fine's NOA “[rejects] all interpretations, theories, construals, pictures, etc. of truth”: “the concept of truth is the fundamental semantical concept” (1986). As sui generis, Frege infamously defends the view that truth (and falsity) is an object: the Bedeutung (or meaning) of a sentence is either the True or the False. Though few follow Frege here, I argue that Fregean, referential semantics effectively rebuts Musgrave’s and Psillos’ Collapse argument, distinguishing Fine’s NOA as a semantic, as opposed to a scientific, realism.