Meaning and Truth
Course Notes
(2) Concepts, Objects and Numbers
(A) Concept and Object
(i) Stages in the formalization of ‘All A’s are B’s’
(A) All A’s are B’s.
(A') Every A is a B.
(A") Whatever is an A is a B.
(A†) If anything is an A, then it is a B.
(A‡) For all x, if x is an A, then x is a B.
(A*) ("x) (Ax ® Bx).
In the transition from (A") to (A†), note the use of the propositional connective ‘if ... then’; in the transition from (A†) to (A‡), note the use of function-argument notation; and in the transition from (A‡) to (A*), note the interpretation of the ‘if ... then’ connective as the truth-functional material conditional.
(ii) Analysis of ‘All A’s are B’s’
Propositions of the form ‘All A’s are B’s’ are understood as involving the subordination of concepts (A being subordinate to B) rather than the subsumption of an object or objects (all A’s) under a concept (B); e.g. (Wa) is understood as (Wb):
(Wa) All whales are mammals.
(Wb) The concept whale is subordinate to the concept mammal.
Advantage: (Wa) can both be understood and be true even if there are no whales.
(iii) First-level and second-level concepts
Objects fall under first-level concepts; first-level concepts fall within second-level concepts. Quantifiers are then to be construed as second-level concepts; e.g. existential statements are understood in terms of the second-level concept is instantiated. Thus (Ga) is regarded as better expressed by (Gb):
(Ga) God exists.
(Gb) The concept God is instantiated. [ ($x) Gx. ]
Existence is not, on this conception, construed as a predicate; and hence the ontological argument for the existence of God is seen as flawed. Such a conception also solves the problem of how to account for negative existential statements, such as ‘Unicorns do not exist’ or ‘Round squares do not exist’.
As far as proper names are concerned, however, their use in stating propositions with a truth-value presupposes the existence of the objects referred to by the names.
(iv) Absoluteness of the distinction between concept and object
Problem of the unity of the proposition: If both objects and concepts (functions with one argument) are ‘saturated’, then some way of binding them together is required for a thought to be expressed by a proposition. Taking a relation (a function with two arguments) to do this only threatens an infinite regress. So something must be ‘unsaturated’; and the problem may as well be solved by treating all functions as ‘unsaturated’, and only objects as ‘saturated’. (Cf. FR, p. 193.)
(v) The paradox of the concept horse
According to Frege, whilst ‘X’ may be a concept-word, an expression of the form ‘The X’ is a proper name, i.e. it refers to an object whenever it is used to make a statement. But now consider:
(CHC) The concept horse is a concept.
According to Frege, ‘The concept horse’ indicates an object, but then, given the absoluteness of the distinction between concept and object, (CHC) is false, which is counterintuitive. Frege blames the paradox on the inadequacies of ordinary language, in which we cannot properly refer to concepts; but a possible solution is to regard (CHC) as expressing what is better expressed by:
(CHC') Everything is either a horse or not a horse. [ ("x) (Hx Ú Ø Hx). ]
This works on the assumption that concepts are sharp, i.e. that they divide all objects (of the domain) into those that do and those that do not fall under them (cf. e.g. FR, pp. 80, 259). Vague concepts remain a problem for Frege.
(B) The Analysis of Number Statements
(i) The Foundationalist Project in Mathematics
The 19th century witnessed a growing concern with the foundations of mathematics (prompted, in particular, by attempts to make mathematical analysis more rigorous). Very crudely, the project might be summarized thus. Non-Euclidean geometry could be modelled within Euclidean geometry; Euclidean geometry, through analytic geometry, could be grounded in number theory; transfinite numbers, hypernumbers, complex numbers and irrational numbers could be derived from the rational numbers; and rational numbers, both positive and negative, were definable in terms of the natural numbers (cf. FMS, §3.1). With Frege, the final step in the reductive process was undertaken: the definition of the natural numbers in purely logical terms. In his Begriffsschrift, he had been successful in providing a logical analysis of mathematical induction (cf. FR, pp. 75-8). In his second book, Die Grundlagen der Arithmetik (The Foundations of Arithmetic), he turned to the concept of number itself.
(ii) Analytic/synthetic and a priori/a posteriori distinctions
These distinctions, according to Frege, concern not the content of a judgement, but the justification for making it; and he provides the following characterizations (cf. GL, §3/FR, pp. 92-3):
These characterizations immediately rule out there being any analytic a posteriori truths (GL, §12), since general logical laws and definitions are assumed to ‘neither need nor admit of proof’. Analyticity implies apriority, in other words, but not vice versa.
(iii) Endorsement of Leibniz
Frege agrees with Leibniz’s view that numerical formulae are provable, via axioms and definitions, though he criticizes Leibniz’s own proofs for missing out the associative law (GL, §6). He also endorses the view that arithmetical truths are analytic and a priori (GL, §§ 11, 15), though he recognizes the difficulty in Leibniz’s conception of the provability of contingent truths (GL, §15).
(iv) Critique of Kant
Complex numerical formulae (e.g. 135664 + 37863 = 173527) are not self-evident, but this, says Frege, shows that they must be provable rather than (as Kant thought) synthetic. Since it is ‘awkward’ to draw a distinction between small and large numbers, there is no reason to suppose that simple formulae are not provable (GL, §5). Kant thought that intuition was required to do arithmetic, but even if it is pure intuition that is involved, it is hard to see how we could have intuitions of very large numbers (cf. GL, §§ 5, 12, 89). In fact, Frege argues, arithmetic governs not just the actual and intuitable, but everything thinkable (GL, §14).
(v) Critique of Mill
Frege regards Mill’s views as highly confusing. It seems absurd to treat definitions of large numbers as stating physical facts, but if these definitions are then seen as generated inductively from simpler cases, there remains the problem of formulating the general law. In the simple cases, such as the definition of the number 3, the truth of ‘3 = 2 + 1’ appears to be dependent on whether collections of objects can be rearranged in the manner required, and even if some can, there is then the question as to how such truths apply in other cases. There is also the problem as to what the physical facts might be that underlie the numbers 0 and 1. We may require experience to become aware of the truths of arithmetic, but this does not make those truths empirical as that term is used in opposition to a priori. (GL, §§7-8.)
Mill, Frege argues, confuses the applications of an arithmetical proposition with that pure proposition itself; ‘+’, for example, does not mean ‘heaping up’ (GL, §9). And induction, writes Frege, presupposes arithmetic, since it depends on the theory of probability (GL, §10).
Number, thinks Mill, is a property of external things, but if so, it is quite a different kind of property to colour, which, says, Frege ‘belongs to a surface independently of any choice of ours’, whereas what number we apply to a conglomeration of objects depends on our way of viewing it – e.g. conceiving a pile of cards as 1 pack or 52 cards or 40 points in bridge (GL, §§ 22-5).
(vi) Critique of Psychologism
In his ‘Introduction’ to the Foundations, Frege stresses his opposition to the incursion of psychology into logic and mathematics, and states explicitly as his first fundamental principle: ‘There must be a sharp separation of the psychological from the logical, the subjective from the objective’ (FR, p. 90). His main objection to psychologism or subjectivism is that construing numbers as private ideas would rule out communication and make argument worthless (GL, §27; cf. GG, I, pp. xvi-xviii/ FR, pp. 202-5). Frege distinguishes between ‘objectivity’ and ‘actuality’, understanding by the former ‘what is independent of sensation, intuition and imagination ... but not of reason’ (GL, §26; cf. FR, pp. 204-5).
(vii) Number statements as containing assertions about concepts
In the course of his critique of rival conceptions of arithmetic in the first half of the Foundations (§§ 5-44), Frege establishes the following (mainly negative) preliminary points:
(a) Certain traditional arguments for construing arithmetical propositions as synthetic truths (whether a priori or a posteriori) are flawed (§§ 5, 7-10, 12). The only other possible option, that they are analytic a priori truths, remains viable (§§ 3, 6, 11, 15).
(b) Numbers are not properties of external things, since ascriptions of number depend on the concepts under which the things are classified (§§ 21-5). What is numerable is everything thinkable, not just the sensible or intuitable (§§ 14, 24, 40).
(c) Numbers are not subjective ideas; they are objective, though non-actual (§§ 26-7).
(d) Numbers cannot be construed either as sets of objects or as sets of ‘units’. The problem as to whether to treat units as the same or different shows that ‘unit’ (‘Einheit’) and ‘one’ (‘eins’) must be distinguished. (§§ 28, 34-9.)
The difficulties revealed in the various views considered are resolved, according to Frege, by treating a number statement as containing an assertion about a concept (GL, §46/FR, pp. 98-9). Taking the points just summarized in order, the following clarifications can then be offered:
(a) It is more plausible that assertions about concepts should be analytic. Whilst ‘All whales are mammals’, for example, might at first sight appear to be about animals (and hence be merely synthetic), in fact, Frege suggests in §47 (FR, pp. 99-100), it involves the subordination of concepts rather than the subsumption of objects under a concept.
(b) We can explain how it might be thought that numbers are properties of external things, our ideas of both numbers and properties being abstracted from things themselves, just because we do abstract concepts this way (though this is not the only way of acquiring concepts), the concepts then being what number statements are assertions about. We can also make sense of the universal applicability of number, since its domain is as wide as that of conceptual thought. (§§ 48-9/FR, pp. 100-1.)
(c) To make assertions about concepts is not to say something about one’s subjective ideas, since concepts are objective. It is an objective fact, for example, that the concept whale is subordinate to the concept mammal. (Cf. §47/FR, pp. 99-100.)
(d) Something can only be called a ‘unit’ relative to a concept that precisely delimits what falls under it – what is now termed a sortal concept. The concept red, for example, is not such a concept, since what falls under it (e.g. a surface) can be arbitrarily divided into parts all of which themselves fall under the concept. No finite number could therefore attach to this concept. The concept moon of Jupiter, on the other hand, is a sortal concept, each of the four moons of Jupiter being a unit relative to this concept. ‘Units’ are thus the same in so far as it is the same concept that is being applied, and different in so far as the objects numbered (falling under that concept) are different. (§54/FR, pp. 104-5.)
© Mike Beaney
February 1999
Meaning and Truth 
Mike Beaney