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Concepts

The following is a tutorial essay for Frege’s Grundlagen with Fabian Pregel. In effect, it is a quick agenda for an hour-long discussion on a guiding question, which for this week was ‘Outline Frege’s distinction between concepts and objects. How can Frege maintain both that numbers are objects and that “the content of a statement of number is an assertion about a concept”?’. A free-standing and much more readable version of some of the material may eventually go up on my Substack.

I. Introduction

Frege’s distinction between concepts and objects is most clearly presented by invoking grammatical types (Bacon 2023: Ch. 1). Roughly, objects are the denotations of terms of type e (‘entities’), such as proper nouns like ‘Julius Caesar’; meanwhile, (monadic, first-order) concepts are the denotations of terms of type (e,t) (‘functions from entities to truth-conditions’), such as intransitive verbs like ‘laughs’. Frege himself used a simpler typing system with just one base type, but his distinction between concepts and objects survives in the type system with two base types e and t more commonly used now for natural language semantics (Heim & Kratzer 1998) and even logicist projects (Goodsell & Yli-Vakkuri m.s.). Some projects in higher-order logic—like Dorr 2016—do use a typing system like Frege’s; but this is somewhat more marginal.

If number statements are primarily about numbers, and numbers are objects rather than concepts, then it is hard to see how number statements could be primarily about concepts. But Frege holds that number statements are not primarily about numbers: “a property is assigned to the concept” (§46), and the number is “only an element in the predicate” (§57) expressing that property. So, there is no immediate tension between these two commitments.

§II presents Frege’s distinction between concepts and objects in a contemporary type system with two base types. §III explains why there is no tension between Frege’s commitment that numbers are objects and his commitment that number statements are about concepts, although it raises some worries about the latter view.

II. Concepts and objects

Here are the types: e is a type, corresponding to the grammatical category of noun phrases, which denote individuals (persons, places, things, …); t is a type, corresponding to the grammatical category of truth-evaluable sentences; and, if σ and τ are types, then ⟨σ,τ⟩ is a type, corresponding to the grammatical category of monadic predicates which take an argument of type σ to produce an output of type τ. To illustrate with some examples: ‘Ann’ is a proper name of type e; ‘the mother of’ is a function of type ⟨e, e⟩, so ‘the mother of Ann’ is a noun phrase of type e; ‘loves’ is a transitive verb of type ⟨e, ⟨e,t⟩⟩, so ‘loves Ann’ is an intransitive verb phrase of type ⟨e,t⟩. Combining these, ‘The mother of Ann loves Ann’ is a term of type t. Further, as neither two predicates of type ⟨e,t⟩ nor two subjects of type e can combine to form a sentence, the types are mutually exclusive.

Here is the informal English gloss on Frege’s distinction. Objects are in the domain of e: that is, they are what terms of type e may denote. Meanwhile, concepts are in the domain of ⟨e,t⟩: that is, they are what terms of type ⟨e,t⟩ may denote. But recall that such a term combines with a term for an individual to produce a truth-evaluable sentence. Thus, we may more intuitively gloss a concept as something that can be true or false of some individual. So, Ann herself is not a concept because she cannot be true or false of an individual. But being Ann is a concept because it can be true or false of an individual—it is true of Ann.

The English gloss, however, is not fully satisfactory. For instance, ‘… is a concept’ is an expression of type ⟨e,t⟩, as ‘Ann is a concept’ is a truth-evaluable (indeed, false) sentence. For ‘X is a concept’ to be a true sentence, it must firstly be a sentence of type t; but then this implies that ‘X’ is of type e. Conversely, ‘… is clever is a concept’ is not a grammatical sentence, and similarly for other English expressions of type ⟨e,t⟩ that we may want to substitute for ‘X’. Still, it is enough to gain a working understanding of Frege’s distinction.

Alternatively, to characterize Frege’s distinction more clearly, one could quantify into predicate position (or perhaps stipulate that philosophical talk of concepts and properties should be interpreted as doing so). To say that a is an object is to say what we might more formally put as ∃x(x=a). Similarly, to say that F is a concept is to say what we might more formally put as ∃X(X=F). (Contra Weiner’s (2004) reading of Frege, there is a notion of identity for non-objects; but the higher-order identity sign in the latter expression is of a different type to the first-order one in the former expression.) There is an interesting result in the vicinity: if one wants to interpret higher-order quantification over concepts as plural quantification, then one might in effect treat concepts as (possibly empty) pluralities. But recall that an attractive interpretation of Mill’s aggregates was as pluralities, and so upon inspection the Millian view that number statements are about aggregates and the Fregean view that number statements are about concepts turn out to converge.

In any case, by now Frege’s distinction between concepts and objects should be more or less clear: it is something like the distinction between the meanings of adjectives and verbs on the one hand, and the meanings of nouns and pronouns on the other hand. Frege also generalizes concepts to other ‘higher order’ relational types (roughly, types that “end in t”); but the simplest concepts are those which are monadic properties of objects.

III. Number statements

With the object/concept distinction in hand, we can look a bit more carefully at what Frege claims about number statements. Let ‘n’ denote a number and ‘F’ denote a concept. Frege can hold that numbers are objects but that number statements assert something of concepts by holding that the logical form of a number statement is not n(F) (pace Goodsell & Yli-Vakkuri m.s.), but rather Ψ(n)(F), where Ψ is of type ⟨e,⟨⟨e,t⟩,t⟩⟩, so that the complex predicate Φ(n), rather than n itself, is of type ⟨⟨e,t⟩, t⟩. That is, Ψ is (λxλX. the number of instances of X is x), and so Φ(n) is (λX. the number of instances of X is n). In particular, Ψ is synonymous with something like ‘the number of instances of … is…’, where the latter dots are filled in first. This respects Frege’s claim that “a property is assigned to the concept” (§46), because the property Φ(n) is assigned to the concept F. Further, the number n is an object and is “only an element in the predicate” (§57), because it combines with the function Ψ to form the property assigned to F.

It does seem pre-theoretically right that ‘Jupiter has ninety-five moons’ is more about the property of being a moon of Jupiter than about the number ninety-five. But it’s not too clear what such aboutness amounts to. For every statement of the form Ψ(n)(F) we can construct an equivalent statement of the form X(F)(n). The grammatical subject of ‘n is the number of Fs’ and ‘The property of being F has n instances’ flips, but they mean the same thing. If all such statements are to count as number statements, it seems slightly odd to privilege one sentential structure as canonical; otherwise, number statements in Frege’s sense are only a small portion of our talk of numbers.

The latter worry is pressing in any case (Rumfitt 2002). Questions of measure rather than cardinality—how many gallons of water there are in a bucket—do not seem amenable to Frege’s treatment (if there are five gallons of water in the bucket, then there are many more than five instances of the concept ‘gallon of water in the bucket’). Abstract arithmetical statements—2 + 3 = 5—also seem marginalized by Frege’s approach; there is no natural sense in which such statements are about concepts, if the numbers themselves are objects. Either of these problems—too easy a notion of aboutness, and too narrow a scope—make Frege’s thesis weaker and so less interesting.

It also seems pre-theoretically right that numbers are objects; otherwise, in the typed setting, there are different numbers for different types, such that the number of cardinal numbers between one and ninety-five inclusive is not quite identical to the number of moons of Jupiter, which is an odd result (see for instance Goodsell & Yli-Vakkuri m.s.). The syntactic features of number-talk are also rather difficult to explain in a principled way if numbers are not objects, but higher-order properties. Overall, though, even if Frege’s two commitments are not obviously correct, it is certainly tenable to hold both of them together.

IV. Conclusion

I have outlined Frege’s distinction between concepts and objects in a modern type system, one which is used in formal semantics and in some contemporary attempts to lay logical foundations for mathematics and science, both strongly influenced by Frege’s work. In this system, like in Frege’s own, the difference between concepts and objects is that they are picked out by terms of different grammatical types. In particular, concepts are picked out by terms of type ⟨e,t⟩, while objects are picked out by terms of type e. A concept is, roughly, something which can be true or false of an object (namely, a monadic property). Higher-order concepts, of different relational types, are also explainable in this system. Other glosses of concepts, as we saw, included ‘properties’ and even ‘pluralities’, although talk of concepts is something like a proxy for quantifying into predicate position. There is no strong tension between Frege’s claim that numbers are objects and his claim that number statements make assertions about concepts, because Frege holds that number statements are about the instances of a concept, not that number statements are exclusively about numbers. (His most paradigmatic number statements are things like ‘There are zero moons of Venus’, rather than ‘2 + 3 = 5’.) The conjunction of these two views is thus tenable; there is no mystery as to how Frege can maintain both.

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