Boudewijn de Bruin - The Logic of Valuing





Published as "The Logic of Valuing." Economics, Rational Choice and Normative Philosophy. Ed. Thomas Boylan and Ruvin Gekker. London: Routledge, 2008.

1. Introduction

The aim of this paper is to analyse the logical form of valuing. I will argue that valuing a concept or property, such as gold, is a universal statement qua logical form, that valuing an object, such as a soccer ball, is an existential statement qua logical form, and, furthermore, that a correct analysis of the logical form of valuing contains doxastic operators. I will show that these ingredients give rise to an interesting interplay between uniform and ununiform quantification, on the one hand, and de dicto and de re beliefs, on the other.

Boudewijn de Bruin

Boudewijn de Bruin is assistant professor in the Faculty of Philosophy of the University of Groningen. De Bruin did his undergraduate work in musical composition at Enschede, and in mathematics and philosophy at Amsterdam, Berkeley, and Harvard. He obtained his Ph.D. in philosophy from The Institute for Logic, Language, and Computation in Amsterdam. His doctoral dissertation, under supervision of Johan van Benthem and Martin Stokhof, was on the logic of game theoretic explanations. read more

This project contrasts with two other projects traditionally referred to by the terms 'logic of value' and 'logic of valuation'. I am not concerned with logical principles governing propositions involving value modalities, logics of value. And I am not interested in the validity, or lack of validity, of arguments on the border of the fact-value dichotomy, logics of valuation.[1]

[1] Early references are Hartmann (1951) and Chisholm&Sosa (1966) for logics of value, and Perry (1957) for a logic of valuation.

Adopting a doxastic outlook on valuing, the proffered analysis remains nonetheless neutral with respect to many issues from meta-ethics. As Michael Smith has persuasively argued, to say that valuing is a form of believing is not to exclude non-cognitivism, non-relativism or irrealism.[2]

Nor is the analysis particularly narrow. Although for the sake of argument I take the bearers of value to be concepts and objects, exactly the same structural observations can be made if you attach value to events, propositions, or even to possible worlds.

[2] Smith (2002).

I will stress one area of application, the value of political freedom. The received view is that the value of freedom lies in the value of the specific things one is free to do. But Ian Carter has recently shown that freedom has irreducible, 'non-specific' value, too. I will show that underlying the debate between the proponents of the received view and their critics is a disagreement about logical form: ununiform de dicto beliefs about freedom as a concept, for the received view, and uniform half-de dicto-half-de re beliefs about freedom as an object, for its critics.[3]

Section 2 defends the analysis in terms of universal and existential quantifiers and doxastic operators. Section 3 brings it all together in a two-dimensional typology of uniform-ununiform and de dicto-de re. Section 4 is on freedom. Section 5 concludes.

2. Quantification and Beliefs

2.1 Valuing Concepts

Smith values gold. When he visited the Egyptian Museum in Cairo he was struck by the beauty of Tutankhamen's death mask, the way gold is combined, there, with lapis lazuli and turquoise to give the Pharaoh his proud yet mild and gentle look. So, Smith says, he values gold because he values Tutankhamen's death mask. But Smith does not show any particular interest in other objects of gold. He ignores, at the exhibition, the other golden masks, the bracelets, the necklaces, the emblems. He wears a silver wedding ring because he does not like the colour of gold. He has never bought any gold plates or cutlery or sculptures for himself.

Smith, I would say, is mistaken about what he really values. He does not value gold. Indeed, he values some particular object (partly) made of gold. To value gold he would have to value more golden things. The logical form of valuing gold G is

&forall x (G(x) -> Value x),

'Smith values all golden objects', rather than

&exist x (G(x) & Value x),

'There exists a golden object Smith values'.

This may seem to put excessive demands on Smith. Of course, to value gold, Smith has to think highly of more than one golden object; but can't he value gold without having to treasure all objects made of gold? As it stands, the universal reading sounds too strong. But we will see that it all depends on how to analyse 'Value x' in this context. It is appropriate to suspend criticism for a moment.

2.2 Valuing Objects

Smithies values the official soccer ball signed by 1999 US World Cup team star Brandi Chastain. Although she bought it at an auction herself, the soccer player Chastain does not matter much to her. Actually, she has not even watched the match with China in 1999. In addition, she does not like Chastain's extravagant style of playing soccer. And she finds the $195 she spent on the ball quite excessive. But she does not regret having bought it because it adds to her collection of celebrity autographs the signature of a sportswoman.

Does Smithies value the soccer ball? Shouldn't she place a value on other aspects of the ball as well? Shouldn't she appreciate everything, or almost everything, about the ball? Of course not. To value the ball it is sufficient for her that some aspect of the ball strikes her as valuable. She may not have a high opinion of Chastain, she may not like playing soccer at all, she may think the colours of the ball are awful and Chastain's handwriting childish, as long as there is one thing about the ball she delights in, it would be wrong to deny that she values the ball. It would be wrong to analyse valuing the soccer ball b as

&forall C (C(b) -> Value C),

'Every property C of the soccer ball b is valuable'. It suffices that

&exist C (C(b) & Value C),

'At least one property C of the ball b is valuable'.

2.3 Beliefs

Smithson values Vermeer's View of Delft. He really got infatuated with the canvas when he first visited the Mauritshuis in The Netherlands, and he has been interested in it ever since. He has bought numerous books on Vermeer and on Dutch seventeenth century art and culture. He has learned all kinds of oil painting techniques so as to be able to study the painting the old-fashioned way by copying it. But, when prompted, he doesn't know just how to explain what it is that makes View of Delft dear to him. Yet he knows that there is something that appeals to him, something hard or impossible, for him, to describe, but still something that somehow explains why he values it. He values the painting not for a specific characteristic, but just so. That is, the logical form of his valuing is

Believe &exist C (C(d) & Value C),

'Smithson believes that there exists a valuable concept C which is true of View of Delft'.

Now suppose Smithson learns a lot about aesthetics and theories of perception, and, thinking harder, comes to realize that what strikes him most in the oil painting is quite simply the warm red and brownish tones of the roofs of some of the houses on the left of the canvas. He values the painting for some very specific characteristics. The logical form of his valuing correspondingly changes to

&exist C Believe (C(d) & Value C),

'Of some particular C, Smithson believes that it applies to View of Delft, and that it is valuable'.

I believe that this is quite a general phenomenon. To develop a taste for French cheese, to learn to like Charles Ives's Concord Sonata, to become aware of the benefits of political freedom after the fall of the Berlin Wall, there will rather often be a moment where your state of mind or attitude is best described involving de dicto beliefs about value which are not de re. You do not know what you like about the cheese and you do not have beliefs about what you specifically value in the music or the freedom, but all the same you do value something.

[3] The received view is found in the writings of Ronald Dworkin and Will Kymlicka. See, for instance, Dworkin (1979) and Kymlicka (1988). Carter (1999), Ch. 2, is the fullest expression of the criticism. Others siding with Carter include Matthew H. Kramer (2003), Serena Olsaretti (1999), and Martin van Hees (2000). The 'half-and-half' variety will be explained below.

The same is true of the logical form of valuing concepts. Smiths values gold. She values all instances of gold without being familiar with all these instances of course. It would be presumptuous to analyse her valuing as

&forall x Believe (G(x) -> Value x),

ascribing to her precisely this kind of familiarity de re. Rather, Smiths's state of mind should be phrased as

Believe &forall x (G(x) -> Value x).[4]

[4] One may ask whether the beliefs have to be true. Can I value water for its soporific qualities? If you answer affirmatively, I may object that it does not have these qualities. If you answer negatively, I may object that I may believe that water has these qualities. The present analysis can accommodate plain and possibly false beliefs as well as true beliefs and even knowledge by adjusting the intended interpretation of the Believe-operator.

3. Typology

The logical form of valuing a concept is a universal statement about valuing objects. The logical form of valuing an object is an existential statement about valuing concepts. Combined this means that the logical form of valuing a concept is a universal statement in which somewhere an existential statement about valuing an object comes in, and that the logical form of valuing an object is a existential statement in which somewhere a universal statement about valuing a concept comes in. Where is this 'somewhere'?[5]

Ignoring for a moment the second dimension, the doxastic operator, there are two ways to combine: a uniform and an ununiform one. Plugging in existential quantification in universal quantification yields, for valuing gold G,

&forall x (G(x) -> &exist C (C(x) & Value C)),

'Every golden object has a property I value'. The verbal rendering is intentionally opaque. Does it mean that different golden objects may have different valuable properties for me? Or is there one single property that applies to all golden things? To sort things out, let us rewrite the above sentence as

&forall x &exist C ((G(x) -> C(x)) & Value C),

'For every object x there is a valuable concept C such that if x is gold it is also C'. Clearly, this says that different objects may have different valuable properties. Very different is what we get when we swap the quantifiers:

&exist C &forall x ((G(x) -> C(x)) & Value C),

'There is some valuable concept C such that all golden objects x fall under it'. It is crucial to be aware of the differences between the two kinds of valuing. Let us use the term 'uniform' for the latter variety, and the slightly artificial 'ununiform' for the former. You value the concept gold ununiformly if you find something valuable in every single instance: you take the golden ring as a traditional symbol of marriage only, you use a golden plate to show off, you dote upon a golden sculpture because of its florid kitsch, and so on, but there is nothing valuable that all golden objects share. You would value gold uniformly if, for instance, you were to value gold's resistance to corrosion or its malleability and ductility, or if you were to love it that even the tiniest pieces of gold can be exchanged on the gold market for money, or if you liked its olive-brown or yellow colour, its lustrous sheen.

Let us turn to valuing objects, and let us start with the uniform way of valuing object b taking the form

&exist C &forall x (C(b) & (C(x) -> Value x)).

You value a soccer ball signed by Brandi Chastain because you are fond of women's sports memorabilia. And you value it uniformly because you value all other women's sports memorabilia, too.

[5] Valuing a concept because of its objects, valuing the objects because of certain concepts, and so on ad inf . . . or not? Whether this regress stops, and where, and whether this happens because of a summum bonum or because of a vicious circle, is up to the individual whose valuing is analysed and not for us to decide. There is no infinite regress in my proposal itself.

Ununiformly valuing an object b is of the form &forall x &exist C (C(b) & (C(x) -> Value x)), You value the soccer ball, but you are unable to specify one single valuable concept. You would say you are fond of women's sports memorabilia. But at the same time there are women's sports memorabilia you do not value such as the hockey stick of Nadine Ernsting, Germany's 2004 olympic top scorer. You would say you like soccer not hockey. But at the same time you do not value the shirt worn by Pele in the 1958 World Cup final against Sweden. You would say Pele is a man not a woman. You value the soccer ball nonetheless, and what makes it valuable is that it has two properties which are only valued when combined, not in isolation, women's sports, and soccer.[6]

[6] The question whether ununiformity implies uniformity (for concepts as well as objects) will depend on the extent to which combining concepts by (possibly infinite) disjunction results in concepts. This may seem a bit of formal cheese-paring, because in ordinary second-order logic the biconditional &forall x&exist C &Phi(x, C) <-> &exist C&forall x &Phi(x, C) is a tautology (the union of the ununiformly obtained sets from the left hand side unproblematically serves the uniform right hand side). For concepts this is far from obvious.

Having set apart uniform and ununiform valuing, we have lost track of the doxastic dimension. So where to place the doxastic operator? Given two quantifiers, there are, in principle, three possible positions for the operator: a purely de dicto prefix and a purely de re suffix, and a half-and-half infix. Table 1 displays the resulting typology. It works for concepts and objects, for it shows the quantifiers and the doxastic modality only.[7]

de dictoBelieve &forall x &exist CBelieve &exist C &forall x
half-and-half&forall x Believe &exist C&exist C Believe &forall x
de re&forall x &exist C Believe&exist C &forall x Believe

4. Application to Political Freedom

If speaking of value in relation to Tutankhamen's death mask sounds hyperbolic, and attaching value to Brandi Chastain's autograph vulgar and tasteless, if differences between de dicto and de re valuing Vermeer's View of Delft seem snobbish or outre, the value of political freedom provides a more serious and significant illustration of the different kinds of valuing.

[7] The framework can also be applied to valuing other bearers of value, such as events, propositions, and even possible worlds. First, observe that as soon as the uniform-ununiform dimension is ignored, valuing a concept C is of the form &exist C (C is contained in D & Value D). To analyse valuing events, for instance, the only question to be answered is what is the analogue of the relation 'is contained in'. A possible answer is 'is caused by', which yields &exist C (C causes D & Value D) as the logic of valuing events. Prefix or suffix the existential quantifier with the Believe-operator, and you are done.

As I said in the beginning of this paper, the received view of Ronald Dworkin and Will Kymlicka reduces the value of freedom to the value of the specific things one is free to do. This view, however, has undergone vehement criticism. Its critics point to its failure to account for the fact that often a decrease of our overall freedom is experienced as showing no 'respect for [our] autonomy and discretion' or for our capacity to make responsible decisions.[8] Even if my actual desires are not frustrated at all by removing one of my liberties, say the liberty to read a novel I definitely do not want to read, I will attach less value to my overall freedom after the censor has banned the book. The value of my overall freedom, then, cannot be reduced to the value of all specific liberties from which it is made up. Or in the words of Ian Carter:

'The difference between my view and that of Dworkin and Kymlicka . . . is that their view entails denying that our freedoms have value independently of the value we attach to the specific things they leave us free to do. . . . To say that freedom is non-specifically valuable is to say that it is valuable "as such" '.[9]

I will not take a stand in this debate. But I will demonstrate that underlying the debate is a difference in logical form of valuing: proponents of the received view value freedom as an object, and they value it in an ununiform de dicto fashion; the critics value freedom as a concept, and they value it in a uniform half-and-half way.

The received view, first. The concept freedom F is valuable whenever all its instances, which I call 'liberties', are valuable:

&forall x (F(x) -> Value x).

But this is only a rough indication of how I interpret the proponents of the received view. First, all uniform ways of valuing freedom (of the form &exist C &forall x, with a Believe-operator affixed at one of three positions) can be ruled out. For it is definitely not the point of the received view to ascribe one and the same valuable property to all liberties. Freedom of speech is valuable because, say, it gives you the opportunity to write the kind of novels you want to write, whereas the value of freedom of worship has to do with what is demanded by your religion. On the other hand, purely de re ununiform valuing (that is, &forall x &exist C Believe) would ask too much of the received view. For every single liberty its champions would have to have in mind a specific property making it valuable. And, unless this case reduces to uniform valuing, more than one valuable property would have to be provided. Half-and-half (&forall x Believe &exist C) is possible, but still requires some specific de re familiarity with all liberties. Purely de dicto (Believe &forall x &exist C), then, is the most plausible phrasing op the position. The received view, then, comes down to the belief that any liberty has some valuable property.

Second, the critics of the received view. As I said, my claim is that the critics value freedom f as an object. Valuing freedom as a concept would imply valuing freedom's specific instantiations, the liberties, and this is precisely what the critics of the received view reject. So their valuing is roughly of the form

&exist C (C(f) & Value C).

Uniform, ununiform, de dicto, de re? A rendering of the critics' position in terms of the ununiform variety (&forall x &exist C) would force them to admit not to be able to identify a unique valuable property of freedom. This is a coherent position, but except in the form of the purely de re version it is a relatively weak one. Moreover, to some extent the critics do mention concrete valuable properties. Ian Carter describes a connection between freedom's value and the value of human agency. Matthew H. Kramer asserts that freedom is an indicator of respect for a person's autonomy or rationality. And Martin van Hees emphasizes human responsibility. The critics of the received view, then, value freedom uniformly, and they value it purely de re.

5. Conclusion

The purpose of this paper was to analyse the logical form of valuing concepts or properties and objects. I argued that valuing a concept is a belief about a universal statement, and that valuing an object is a belief about an existential statement. Exploiting the differences between de dicto and de re beliefs, and the differences between uniform and 'ununiform' quantification, a two-dimensional typology of six types of valuing was obtained. An application to valuing political freedom sketched an example of the use of the analysis.


Carter, Ian 1999: A Measure of Freedom. Oxford: Oxford University Press.

Chisholm, Roderick and Ernest Sosa 1966: 'On the Logic of "Intrinsically Better"'. American Philosophical Quarterly, 3, pp. 244-49.

Cunningham, R. L. 1979: Liberty and the Rule of Law, College Station: Texas A&M University Press.

Dworkin, Ronald 1979: 'We Do Not Have a Right to Liberty' in Cunningham (1979), pp.161-81.

Hartman, Robert S. 1961: 'The Logic of Value'. Review of Metaphysics, 14, pp. 389-432.

Kramer, Matthew H. 2003: The Quality of Freedom. Oxford: Oxford University Press.

Kymlicka, Will 1988: 'Liberalism and Communitarianism'. Canadian Journal of Philosophy, 18, pp. 181-204.

Olsaretti, Serena 1999: 'The Value of Freedom and Freedom of Choice'. Notizie di Politeia, 56, pp. 114-21.

Perry, Orval L. 1957: 'The Logic of Moral Valuation'. Mind, 66, 42-62.

Smith, Michael 2002: 'Exploring the Implications of the Dispositional Theory of Value'. Philosophical Issues: Realism and Relativism, 12, 329-47.

van Hees, Martin 2000: Legal Reductionism and Freedom. Dordrecht: Kluwer Academic Publishers.

[8] Kramer (2003), p. 241. For responsibility, see van Hees (2000), p. 154.

[9] Carter (1999), pp. 33-34. Given the overall claim of the book, Carter should have spoken about the value of 'our freedom' rather than of 'our freedoms'. The next sentence in the quote corrects the mistake.