On the transitivity of strict preference

The notion of comparing alternatives often comes up in philosophy, particularly when discussing practical reason. There are various names for this (we can talk about the reasons for choosing A over B, or how A is better than B, or how A is more desirable to B, or how A is preferred to B) but they all amount to the same thing.

The other day I was reading the SEP article on preference and was struck by this counterexample to transitivity of strict preference (I recall my friends mentioning it to me in the past, but I only thought about it critically this time around). In this quote, X≻Y represents that X is strictly preferred to Y, and X∼Y represents indifference between X and Y:

In an important type of counterexample to transitivity of strict preference, different properties of the alternatives dominate in different pairwise comparisons. Consider an agent choosing between three boxes of Christmas ornaments… Each box contains three balls, coloured red, blue and green, respectively; they are represented by the vectors ⟨R1,G1,B1⟩, ⟨R2,G2,B2⟩, and ⟨R3,G3,B3⟩. The agent strictly prefers box 1 to box 2, since they contain (to her) equally attractive blue and green balls, but the red ball of box 1 is more attractive than that of box 2. She prefers box 2 to box 3, since they are equal but for the green ball of box 2, which is more attractive than that of box 3. And finally, she prefers box 3 to box 1, since they are equal but for the blue ball of box 3, which is more attractive than that of box 1. Thus,

a. R1≻R2∼R3∼R1,
b. G1∼G2≻G3∼G1,
c. B1∼B2∼B3≻B1; and
d. ⟨R1,G1,B1⟩≻⟨R2,G2,B2⟩≻⟨R3,G3,B3⟩≻⟨R1,G1,B1⟩.

The described situation yields a preference cycle, which contradicts transitivity of strict preference.

(Note that I’ve added the labels to the listed conditions for the sake of this discussion.)

Now, I haven’t read much of the modern discussion on transitivity of preference (indeed, I didn’t even finish reading the article), so perhaps what I’m about to say is really obvious.

It seems clear to me that the above counterexample motivates the otherwise very natural distinction between (1) being better in some respect and (2) being better simply. Ultimately it has to do with why we prefer something over another. For instance, assume I prefer red balls over blue balls. Then I prefer this red ball over that blue ball simply, and I prefer this box of green and red balls over that box of green and blue balls in some respect.

I say this distinction is “very natural” because it seems necessary if we are to make sense of trade-offs, which are manifold in everyday experience. As a trivial example (which I find myself in often), imagine you need to pick one of two routes to your destination. Route A is longer but has prettier scenery and conversely route B is shorter but has uglier scenery. You have to pick one, but whatever choice you make will involve a trade-off. On account of what is this a trade-off? Well, surely it’s because shorter routes are preferable to longer ones and prettier routes are preferable to uglier ones. That is, A is better in some respect (prettiness) and B is better in some other respect (length).

This distinction resolves the above counterexample by showing us that (a)-(d) equivocate on “≻”. In (a)-(c) X≻Y means X is strictly preferred to Y simply, but in (d) it means X is strictly preferred to Y in some respect.

The SEP article immediately goes on to say the following:

These and similar examples can be used to show that actual human beings may have cyclic preferences. It does not necessarily follow, however, that the same applies to the idealizedrational agents of preference logic. Perhaps such patterns are due to irrationality or to factors, such as lack of knowledge or discrimination, that prevent actual humans from being rational.

Perhaps, but I’m inclined to think life’s more complicated than that. It seems pretty intuitive that there are various types of goods that are incommensurable. One way we might make this intuition precise is as follows: in general there seem to be two ways in which A is better than B:

  1. A and B are both means to C and A is a better means.
  2. B is a means to A.

(1) is where this whole business of comparing of alternatives comes in. Given our above discussion we realise that A can be a better means either in some respect or simply. Aristotle mentions something like (2) at the beginning of the Nicomachean Ethics. The guiding intuition here is that ends are preferred to means because “it is for the sake of the former that the latter are pursued” (I.1 1094a15-16).

Now, combining this with our previous discussion on basic human goods, the fact that there are multiple basic goods suggests that at least sometimes two goods will be incommensurable.

Virtual existence

It might not seem like it, but a proper understanding of virtual existence can be significantly helpful when trying to understand the structure of human communities. To this end, I’d like to spend some time thinking about this puzzling notion here.

Substances and aggregates again

You’ll recall that, in our discussions about substantial activities, we spent a fair amount of time introducing the notion of substance. There we said that a substance is something which has intrinsic directedness towards an end, or equivalently something which has intrinsic causal powers.

The guiding intuition here is that when a substance does something it is the substance itself doing it, as opposed to its parts or something external. So, the doing is intrinsic to the substance as opposed to its parts or something outside. Now, we use the term characteristic behaviours to pick out what something does always, or for the most part, given the kind of thing that it is; that is, how it behaves so long as it’s not being “blocked” in some way. Plants and animals grow to become healthy adults unless prevented by genetic defect or environmental factors, hydrogen combusts under certain circumstances unless prevented, the phosphorous in a match head will ignite when struck unless prevented, and so on.[1] We mentioned before that the only way such characteristic behaviours can be made intelligible is in terms of that thing’s being directed toward that behaviour by virtue of the kind of thing that it is. And this directedness, you’ll recall, needn’t be the result of conscious deliberation.

Putting this all together we get that if the doing of something always or for the most part is intrinsic to a thing, then so is the directedness towards this behaviour. Similarly, since a thing can’t do something without the power to do it, if the doing of something is intrinsic to a thing then so are the causal powers needed to do it. It is roughly along these lines that we (following Aristotle and the Scholastics) come to understand substances in the terms mentioned above. For a longer discussion of this, as well as responses to objections, I suggest you read Edward Feser’s Scholastic Metaphysics.

In contrast to substances, aggregates are those things which have only extrinsic directedness or causal powers. That is, an aggregate’s causal powers are reducible to the sum of the causal powers of its parts and what is imposed on it from outside.

A pile of rocks would be an obvious example of an aggregate. Its power to hold something 2 meters above the ground is merely the sum of the individual rocks that make it up. Above Aristotle used an example of a bed, which is merely an aggregate of the materials (wood and metal) that make it up.

Some aggregates, because of their complexity, are less obviously aggregates. Examples of these are things like watches and computers. A watch’s power for time-telling is imposed on it by us, and its power for the circular motion of its hands is merely the sum of the powers of its parts such as the conduction of electricity and so on. Similarly for a computer or a calculator.

The pile of rocks would be a table to the extent that it is intended as such by the individual or community that has access to it. Here it would be the sum of the parts together with an outside intention that make the pile of rocks a table.

With the watch there is nothing intrinsic to it or its parts that enables it to tell the time. This is something derived from us on the outside, as interpreters of the mechanical symbols we used in the watch’s construction. If you took us out of the equation, all that would remain are cogs, electrically stimulated, moving other cogs and pieces of metal at a fairly constant rate.

I suspect this is less clear to many of us in the case of computers. But this is more a function of our ignorance about how computers work than anything substantial about computers themselves. In this case various electrical components alternate their charges by interacting with one another, typically terminating in patterns of colours on a screen or sounds from a speaker. Sure, we’ve managed to do this faster and with smaller components, but there is nothing of significant difference (at least not for our purposes) between this and purely mechanical computers. We impose meaning on these patterns of colour and sound, and thereby impose on the computer the ability to compute things that are not intrinsic to the metal or electrical currents themselves. This is not unlike we impose material symbols and utterances with meaning in written and spoken communication.

Now, between substances and aggregates the substances are more ontologically fundamental. Or, as it has been put, substances are the most fundamentally real things. Of course, both aggregates and substances depend on their parts, but (1) aggregates are always made up of substances, and (2) with substances there is also a sense in which the parts depend on the whole. A full examination of (1) will require a deeper understanding of per se causal chains than we have space here to discuss so, as before, we’ll put this off until another time. We will spend the rest of our space here attempting to make inroads to understanding (2). Throughout these attempts we will being using the insight that what a thing is (its nature) is closely tied up with its directedness, characteristic behaviours, and causal powers. Indeed, we said last time and have noted elsewhere, that a thing’s nature just is what it is directed towards.

The actual existence of parts in aggregates

Now, an aggregate’s causal powers and directedness are by definition not intrinsic, but rather extrinsically derived from its parts and from outside. Thus we find that aggregates don’t really have a nature or existence over and above the substances that make them up and the ends imposed on them from outside. That is, their nature and existence are wholly reducible to extrinsic sources, and it is therefore by reference to these extrinsic sources that these aggregates are intelligible.

Furthermore, the substances that make up (or impose on) an aggregate retain their intrinsic directedness, characteristic behaviours, and causal powers. We use the term “actual existence” to refer to the way in which these substances exist, and say that they are “actually present” in (or around) the aggregate.

This is consistent with saying that, by virtue of being part of an aggregate these substances have their characteristic behaviours and causal powers influenced by one another. In this case they don’t take on new behaviours or powers, but rather have their behaviours and powers redirected through interaction with one another. Think, for instance, of how cogs influence each other in mechanical clocks or how pipes redirect the flow of water.

The virtual existence of parts in substances

What can we say, then, about parts of substances? A substance does have a nature and existence over and above its parts and outside imposition. It’s nature and existence are not wholly reducible to extrinsic sources, and it is therefore to some extent intelligible apart from these extrinsic sources. At this point we must tread carefully, for it is easy to misunderstand what is being said. In the interest of clarity, then, we make a distinction: a part of a substance can be considered in two ways, either (1) in itself or (2) as a part. In the former case we consider the part in isolation from the substance it belongs to, and in the latter case we consider the part in the context of the substance it belongs to.

We can illustrate this distinction with some examples. A previously mentioned example of a substance was a water molecule. We pointed out that “water boils at 100°C while hydrogen, considered in itself, boils at -252.9°C and oxygen, considered in itself, boils at -183°C.” On the other hand, hydrogen and oxygen, considered as a parts of water, boil at 100°C. In some ways this is obvious, for the continued existence of the water depends on the continued existence of the hydrogen and oxygen that make it up, and since water doesn’t combust when it comes into contact with fire it follows that neither do its parts. On the other hand, I’m sure that for many of us hearing something like “the boiling point of hydrogen is 100°C” causes somewhat of a knee-jerk reaction. In a way this highlights the point of the distinction. Presumably we have such knee-jerk reactions because of what we learnt in chemistry class. But chemistry, like many sciences, seeks primarily to understand the essential features of the objects it studies, and therefore typically studies these objects in isolation from outside influence. They therefore have little to say about these objects when considered as part of another. So such knee-jerk reactions are neither surprising nor hindering to our discussion.

Another example of a substance previously mentioned was an animal. For each of the organs an animal has we can consider it in itself or as a part. In this case we typically have the reverse intuitions as above: with the molecules we are accustomed to thinking about them in themselves, but with organs we are accustomed to thinking about them as parts. Consider, for instance, the claims “hydrogen boils at -252.9°C” and “the heart pumps blood”. The former tells us how hydrogen behaves in isolation from outside influence, and the latter tells us what the heart does in the context of the rest of the body. Now, considered in itself, “an organ is merely a clump of flesh which decomposes if left to its own devices.” Indeed, don’t we see this all the time with severed limbs and corpses? On the other hand, considered as parts organs “are each capable of their individual functions in the body (walking, grasping, thinking, sensing, pumping blood, and so on) and they are all capable of participating in the life of the animal”, where by “life” we mean the ability of a thing to “produce, conserve and repair its proper functioning as the kind of thing it is”. For example, compare what happens when you cut a severed hand (or some other non-living thing) and when you cut a living thing. The latter will repair itself to some extent whereas the former will do nothing.

You’ll notice from the examples given that the characteristic behaviours, directedness, and causal powers of things can differ quite significantly depending on whether we’re considering them in isolation and or as parts. Therefore, so do their natures: a clump of flesh is a significantly different kind of thing to a heart pumping blood, a free hydrogen atom is a different kind of thing to a hydrogen atom in a water molecule, and so on. An implication of this is that the things considered in themselves are not actually present in the substances they belong to, at least not in the same sense that the things considered as parts are. In these cases, we say that the things in themselves are “virtually” present in the wholes they belong to.

What of the parts considered as parts? They are directed by the nature of the substance and derive their causal powers from the substance. I don’t mean that the substance is something separable from its parts, for of course it is constituted by them. Rather, it is on account of the parts being configured together so as to produce a whole which is capable of more than the mere sum of its parts, each considered in itself, that the parts cease to behave like they would in isolation and take on a new nature grounded in the overall configuration itself. It is because of this configuration of the substance that the parts have different causal powers, behaviours, and directedness. It is in this sense that the parts all together take on the nature of the substance and share in it’s existence. And it is this sense that parts depend on the substance they belong to.

A hylomorphic account of virtual existence

Let’s summarise what we’ve said so far. In aggregates the parts considered in themselves actually exist, since they continue doing what they do in isolation. And although they continue behaving as they would in isolation, they can nonetheless have this behaviour redirected by the other parts of the aggregate. Finally, in aggregates there isn’t really a distinction between the parts considered in themselves and the parts considered as parts.

In substances the parts considered in themselves virtually exist, since they do not continue doing what they do in isolation. By virtue of how they are configured in the substance, they behave in a new ways which share in the existence of the substance. Finally, in substances there is a distinction between the parts considered in themselves and the parts considered as parts.

Now, we might be tempted to see the word “virtual” and think that what’s being claimed is that the parts of substances, considered in themselves, do not make any causal contribution to the substance they belong to. On the contrary, Scholastics call this “merely logical” existence and distinguish it from virtual existence. Something has merely logical existence when its existence is wholly dependent upon intellectual activity. Examples would be fictional stories, imaginary friends, hallucinations, and dreams. On the other side of the spectrum are substances, which, as we’ve been saying, have actual existence. In this context this entails that they “fully” exist independently of intellectual activity. Virtual existence sort of stands in between these two opposites: to some extent they have mind-independent existence, and to some extent they are dependent upon intellectual activity. This may sound slightly strange, but this conclusion is implicit in what we’ve been saying. Their partial dependence on intellectual activity derives from the fact that while they are part of a substance they do not behave as they do in isolation, and so it requires intellectual activity to “fill in” what’s currently absent. Their partial independence of intellectual activity derives from the fact that the parts could not be configured so as to constitute the substance unless they were as they are in themselves. For instance, it is precisely because of how hydrogen and oxygen molecules are in themselves, that they can come together to form a water molecule.

We can shed some light on this somewhat strange property of virtual existence by means of the Aristotelian theory of hylomorphism.[2] According to hylomorphism every material thing is composed of “form” and “matter”, where by matter we mean some otherwise indeterminate substratum and by form we mean the configuration of the matter that determines it to this rather than that. So stated, hylomorphism is completely general, and we illustrate it with three very different cases.

First, there’s the sense in which we’re talking about material substances like trees, dogs, humans, water molecules, wood, and so on. Our matter is the “stuff” we’re all made out of, and our forms are the configurations of this matter into the various kinds of material things there are. We humans have our matter configured in a way quite differently from how the matter in the tree outside or in my pet cat is configured. It is on account of these different forms that we have our distinctive behaviours, directedness, and causal powers, and on account of which we are called humans, trees, and cats.

Second, there are things like written or spoken sentences or pieces of music. With the sentence the matter would be the words or letters and the form would be the syntax together with some kind of “semantic coherence” (since syntax alone isn’t enough). With the music we have something similar, but I suspect there syntax is enough.

Third, there are actions. Here, I think, is where we begin to see the generality of the form-matter distinction. Consider the motion of my hand into your face. This movement itself is indeterminate between at least two possibilities: either I am punching you or I am doing something else and have hit you by mistake. As such, the movement is the matter of my action. What is the form? Surely it’s my intention. If I intend to hurt you then the action is me punching you, otherwise it is a mistake. So, while the form-matter distinction primarily applies to material substances, at the end of the day it goes far beyond this to almost everything.[3]

Now, using the form-matter distinction we can say the following. Something merely logically exists if it has neither form nor matter in reality, but is only understood or imagined in such terms. Something actually exists if it is constituted by the composition of form and matter which are intrinsic to it; that is, the thing is made up of its own form and matter. And something virtually exists if it is constituted by the composition of intrinsic matter and extrinsic form; that is, the thing contributes its matter to the form of something else, or equivalently the thing’s matter is informed by something else.

This captures, in the technical jargon of hylomorphism, what we were talking about earlier with the parts making contribution to the configuration of the substance and thereby participating in its existence. It also enables us to make sense of the partial dependence of virtually existing things on intellectual activity: such activity is necessary to “fill in” the missing intrinsic form, but not the matter.


  1. We’ve spoken about these kinds of generalisations before, at which point we called them “Aristotelian categoricals”. John Haldane, in his talk Aquinas and Realism, calls them “generics”. David Oderberg, in his paper Essence and Properties, calls them “properties”, along with Scholastics more generally.
  2. We won’t be able to do complete justice to the theory now, so if you’re interested in more than what I have to say here I recommend all the resources I’ve listed on my resources page under the sections “Hylomorphic dualism” and “Hylomorphism in general”.
  3. One might be tempted to equate form and structure. While in some cases these are the same (music pieces, for example), this is not true in general. See David Oderberg’s paper Is Form Structure? for a more detailed discussion of this.

Goods, basic goods, and faculties

We’ve mentioned before that the goodness of some thing is relative to that thing’s nature. It is good for a human to have two legs because our biology is structured in such a way that having two legs is conducive to our flourishing. By the same token, it is not good for a cat to have two legs.

Now, these various goods can be grouped together and structured hierarchically: colour sensitivity, amoung other things, is a good which is subsumed under the good of seeing. Good seeing is itself subsumed under the goods of sensing, which in turn is subsumed under the goods of animal life.

A breakdown of the goods that are subsumed under the good of animal life.
A breakdown of the goods that are subsumed under the good of animal life.

At this point three things can be said. First, the goodness of the lower goods is dependent on the higher goods. Put another way, the lower goods are for the sake of the higher goods. Colour sensitivity, for instance, is for the sake of seeing and is good to the extent that it enables us to see well.

In the opening passage of the Nicomachean Ethics Aristotle, while discussing human acts, makes the same point:

Now, as there are many actions, arts, and sciences, their ends also are many; the end of the medical art is health, that of shipbuilding a vessel, that of strategy victory, that of economics wealth. But where such arts fall under a single capacity — as bridle-making and other arts concerned with the equipment of horses fall under the art of riding, and this and every military action under strategy, in the same way other arts fall under yet others — in all of these the ends of the master arts are to be preferred to all the subordinate ends; for it is for the sake of the former that the latter are pursued. (I.1 1094a7-16)

We might visualise his scenario as follows:

A breakdown of the acts that are subsumed under the act of strategy.
A breakdown of the acts that are subsumed under the act of strategy.

The second thing to note is that this hierarchy has a limit. That is, the tree does not go up indefinitely. Aristotle says that “we do not choose everything for the sake of something else (for at that rate the process would go on to infinity, so that our desire would be empty and vain)” (I.2 1094a21-23). If you are familiar with the distinction between per se and per accidens causes, what Aristotle is getting at here is that the relationship between the lower goods and the higher goods forms a per se final causal chain, and as such has an endpoint. If you are unfamiliar with this distinction, unfortunately space doesn’t not allow me to argue for this here, so you’ll just have to trust me.

Aristotle called these highest goods chief goods, and Aristotelians these days typically call them basic goods. Basic goods are desired for their own sake and not for the sake of another. Now, in simple things there might be only one basic good, but for many things there is more than one, and they can often be quite broad. David Oderberg, for instance, thinks the basic human goods are life, knowledge, friendship, work and play, appreciation of beauty, and religious belief and practice (The Structure and Content of the Good).

In some sense the basic goods
In some sense the basic goods “make up” the nature of a thing.

So how do we figure out what the basic goods of a thing are? This relates to the third thing to be said: the basic goods correspond to the various distinctive faculties something has according to its nature. After all, broadly speaking, being a human is an activity and the basic goods represent the broadest aspects of this activity by which we measure it good or bad. Given that the way a thing acts correspond to the faculties it has, it seems that the basic goods and faculties of a thing would correspond to each other.

Now, at this broad level it’s not always clear how we are to carve up reality, but we can make some comments that will help us on our way. First, it isn’t particularly informative to say that “given that humans are rational animals, the basic human goods must be rationality and animality”. What we’re looking for are the various aspects of what being a good rational animal involves. Second, the basic goods might overlap, but their faculties should not be wholly reducible to one another. This would be a clear sign that we’re not thinking at a broad enough level. Third, Aristotle’s Nichomachean Ethics can be seen as his attempt at studying the basic goods by means of studying the various human virtues.

One clear example of a basic good, given our recent discussion about substantial activities, would be what Aristotle and Oderberg call “friendship”, which corresponds to our faculty for working together toward a common end. More on this another time.