Fregean Concepts

2024-2 Frege and Russell Midterm Exam
#1) What is special about those functions that Frege calls concepts. Explain the role of truth-values, the True and the False in Frege notion of a concept. Explain how it is that concepts can play the role of what mathematicians nowadays call “characteristic functions.” Could Frege identify the True with, say 0 and identify the False with 1? Would this be metaphysically circular, given his account of what 0 and 1 are as objects?
 
(A) Arguably, the notion of a function has a pivotal role in Frege’s thought, including his philosophy of language, philosophy of mathematics, philosophy of logic, and ontology. Investigating logic through mathematics (functions) is a leitmotif running through his entire philosophy. But what, for Frege, is the nature of a function? Despite its fundamental significance, this question is often neglected due to misleading parallels between Frege’s notation and modern notation. One common misguided interpretation is to consider Frege’s notion of a function as a relation like:

(∀x)(∀y)(∀z)(xRy ⊃ xRz .⊃. y=z)

However, Frege never adopted this view, as he regarded functions as primitive, indefinable entities (Landini 2012: 2). As Landini argues, “Frege’s work does not identify functions with special relations…For Frege, the ontology of functions, not relations, is the foundation of logic (and arithmetic and analysis)” (Landini 2012: 3). In this ontological aspect, what is essential regarding functions is that they are unsaturated as opposed to the saturated entities that he called objects. In “Function and Concepts,” Frege wrote:

I am concerned to show that the argument does not belong with a function, but goes together with the function to make up a complete whole; for a function by itself must be called incomplete, in need of supplementation, or unsaturated [ungesättigt]. And in this respect functions differ fundamentally from numbers (Frege 1891: 133).

It is well known that he applied this analysis to our ordinary language as well. “Statements in general…can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or unsaturated” (Frege 1891: 139). Take, for instance, ‘Caesar conquered Gaul.’ According to Frege, it is divided into an unsaturated function, ‘(―) conquered Gaul,’ and a saturated object (the argument), ‘Caesar’ (Frege 1891: 139). If we visualize this function, what does it look like? One tempting answer would be:

For it is natural to assume that the value of such a function is a complete whole, ‘Caser conquered Gaul.’ However, it should be noted that he holds a rigorous distinction between sense [Sinn] and reference [Bedeutung] in his mature symbolism. The basic idea here is that the reference of a sign is what the sign designates while the sense of the sign contains the mode of presentation (Frege 1892a: 152). Thus, “the Bedeutung of ‘Evening Star’ would be the same as that of ‘Morning Star,’ but not the sense” (Frege 1892a: 152). Since “no senses are functions” (Landini 2012: 132), the function ‘(―) conquered Gaul’ must be analyzed in terms of the Bedeutung of an expression, as follows:

According to Frege, the values of the function ‘(―) conquered Gaul’ are the two truth-values considered as logical objects. “A statement contains no empty place, and therefore we must take its Bedeutung as an object. But this Bedeutung is a truth-value. Thus the two truth-values are object” (Frege 1891: 140). In “On Sense and Reference,” Frege further articulates this idea as follows:

We are therefore driven into accepting the truth-value of a sentence as constituting its Bedeutung. By the truth-value of a sentence I understand the circumstance that it is true or false. There are not further truth-values. For brevity I call one the True, the other the False. Every assertoric [declarative] sentence concerned with the Bedeutung of its words is therefore to be regarded as a proper name, and its Bedeutung, if it has one, is either the True or the False (Frege 1892a: 157-158).

One might argue that the Fregean idea—that every declarative sentence concerned with the Bedeutung of its words should be regarded as a proper name—is seriously flawed, as it breaches the fundamental distinction between a term and a well-formed formula (wff) in modern logic (See, e.g., Dummett 1973: 183-184). Although I will explore this issue in more detail below (see the second question and the answer), it is worth mentioning here that this objection assumes that, for Frege, a function Fa is a wff, not a term. However, this assumption is mistaken, as Frege’s notion of a function is derived from what mathematicians today refer to as ‘characteristic functions.’ For instance,

Where both ‘n is even’ and ‘n is odd’ are formulas, while fn is a term. Likewise,  

Thus, Frege’s function sign is a term, not a formula. A concept is a function whose values are always exclusively either the object ‘the True’ or the object ‘the False,’ as illustrated above (Landini 2012: 30). Let me briefly summarize Frege’s notion of a function as described thus far:

(1) Ungesättigtheit Concept is an unsaturated entity.

(2) Truth-values Concept is a function whose values are always exclusively either the object ‘the True’ or the object ‘the False.’

Plus, since every declarative sentence should be regarded as a proper name of the True or the False, a concept sign must be a predicate in such a sentence. Therefore,

(3) The Correspondence between Concept/Object and Predicate/Subject the concept is the Bedeutung of a predicate while an object is the Bedeutung of a subject (Frege 1892b: 187).

In this way, Frege makes a sharp contrast between concepts and objects. According to Frege:

(4) The Predicative Nature “The behavior of the concept is essentially predicative even where something is being asserted about it”^[각주:1] (Frege 1892b: 189).

Finally, he maintains that the distinction between concepts and objects even holds when it comes to second-level functions:

Second level concepts, which concepts fall under, are essentially different from first-level concepts, which objects fall under. The relation of an object to a first-level concept that it falls under is different from the (admittedly similar) relation of a first-level to a second-level concept. To do justice at once to the distinction and to the similarity, we might perhaps say: An object falls under a first-level concept; a concept falls within a second-level concept. The distinction of concept and object thus still holds with all its sharpness (Frege 1892b: 189).

But the fact that Frege’s concepts are characteristic functions raises a serious concern about ontological circularity. According to Landini,

Within traditional mathematics characteristic functions have the values are 0, 1. If the True = 0 and the False = 1, it would simply make Frege’s Grundgesetze in line with the rather customary mathematical practice involved with its adopting characteristic functions (Landini 2022: 373).

Thus, one might ask whether Frege identified the True with, say, 0 and the False with 1. If the answer is affirmative, then we could conclude that a concept is a characteristic function whose values are always exclusively either the object ‘0’ or the object ‘1.’ But for Frege, “the logical object that is 0 is that objects correlated uniquely with the second-level numerical concept 0uɸu” (Landini 2022: 374). Consequently, 0 as a logical object is only knowable through concept-correlation with numeric second-level concepts, while the definition of a concept again requires a logical object, 0 (Landini 2012: 96). This clearly involves an ontological circularity.

Frege’s entertains in his Grundgesetze that concepts might yield as their values (the True/the False) entities that are themselves values of his value-range function z’ɸz within which a concept fξ falls. To allow this is to allow ontological circularity. We thus have direct textual evidence that Frege’s leaves it open whether, e.g., the True is the number 0. (Landini 2022: 374).

However, it is a separate question whether such circularity is truly vicious. Given that Frege deliberately avoids the problem of over-determination^[각주:2] by allowing ontological circularity, it might not be as problematic (Landini 2022: 374).

 

References

Dummett, Michael (1973). Frege: Philosophy of Language. London: Duckworth.
Enderton, Herbert (1977). Elements of Set Theory. New York: Academic Press.
Frege, Gottlob (FA). The Foundations of Arithmetic. A Logico-Mathematical Enquiry into the Concept of Number. translated by J. L. Austin. New York: Harper Torchbooks, 1960.
Frege, Gottlob (BLA). The Basic Laws of Arithmetic. Exposition of the System. translated and edited by Montgomery Furth, Berkeley and Los Angeles: University of California Press, 1967.
Frege, Gottlob (1891). “Function and Concepts.” in: The Frege Reader, edited by M. Beaney, Oxford: Blackwell, 1997.
Frege, Gottlob (1892a). “On Sinn and Bedeutung.” in: The Frege Reader, edited by M. Beaney, Oxford: Blackwell, 1997.
Frege, Gottlob (1892b). “On Concept and Object.” in: The Frege Reader, edited by M. Beaney, Oxford: Blackwell, 1997.
Hunter, Geoffrey (1971). Metalogic: an introduction to the metatheory of standard first order logic. Berkeley, CA: University of California Press.
Klement, Kevin C. (2001). Frege and the Logic of Sense and Reference. New York: Routledge.
Landini, Gregory (2011). Russell. New York: Routledge.
Landini, Gregory (2012). Frege’s Notations: What They Are and How They Mean. London and Basingstoke: Palgrave-Macmillan.
Landini, Gregory (2022) “Stipulations Missing Axioms in Frege’s Grundgesetze der Arithmetik,” History and Philosophy of Logic, 43:4, 347-382.
Zach, Richard (2019). Sets, Logic, Computation: An Open Introduction to Metalogic. Open Logic Project.

1. This is the main point of Frege’s reply to Benno Kerry’s ‘the concept horse’ objection (see Frege 1892b). [본문으로]
2. If there are two competing logical options equally epistemically viable, then it is not epistemically justified to set out an axiom favoring one option over the other (Landini 2022: 353). [본문으로]