1.3: CHAPTER 2 KANTIAN ETHICS
Kantian Ethics
In spite of its horrifying title Kant’s Groundwork of the Metaphysic of Morals is one of the small books which are truly great; it has exercised on human thought an influence almost ludicrously disproportionate to its size. 1
1. An Introduction to Kantian Ethics
Immanuel Kant
was born in 1724 in Königsberg in East Prussia, where he died in 1804. Kant is famous for revolutionising how we think about just about every aspect of the world — including science, art, ethics, religion, the self and reality. He is one of the most important thinkers of all time, which is even more remarkable by the fact that Kant is a truly awful writer. His sentences are full of technical language, are very long, and are incredibly dense. You have been warned!
Kant is a rationalist writing during the Enlightenment (1685–1815). He thinks that we can gain knowledge from our senses and through our rational capacities. This means his general philosophical approach starts by asking what we can know
a priori
.
This is key to understanding his work but also makes his writing on ethics seem a bit odd. We think the study of ethics — unlike say maths — ought to direct our eye to what is going on around us in the world. Yet Kant starts by turning his eyes “inward” to thinking about ethical ideas.
Kant believes that in doing this people will come to recognise that certain actions are right and wrong irrespective of how we might feel and irrespective of any consequences. For Kant, actions are right if they respect what he calls the Categorical Imperative. For example, because lying fails to respect the Categorical Imperative it is wrong and is wrong irrespective of how we might feel about lying or what might happen if we did lie; it is actions that are right and wrong rather than consequences. This means that Kant’s theory is deontological rather than teleological. It focuses on our
duties
rather than our ends/goals/consequences.
There is, however, something intuitive about the idea that morality is based on reason rather than feelings or consequences. Consider my pet cat Spartan. He performs certain actions like scrabbling under bed covers, meowing at birds and chasing his tail. Now consider my daughter Beth, she performs certain actions like caring for her sister and helping the homeless.
Spartan’s actions are
not
moral whereas Beth’s actions are.Spartan’s thinking and actions are driven by his desires and inclination. He eats and plays and sleeps when he desires to do so, there is no reasoning on his part. Beth, in contrast, can reflect on the various reasons she has, reasons to care for her sister and the homeless.
We might think then that humans are moral beings not because we have certain desires but precisely because we are
rational
. We have an ability to “stand back” and consider what we are doing and why. Kant certainly thought so and he takes this insight as his starting point.
2. Some Key Ideas
Duty
Kant’s main works in ethics are his
Metaphysics of Morals
(1797) and the Groundwork of the Metaphysics of Morals (1785). Neither give practical advice about particular situations but rather through rational reflection, Kant seeks to establish the supreme principle of morality.
He starts from the notion of “duty” and although this is a rather oldfashioned term, the idea behind it should sound familiar. Imagine, your friend has told you that she is pregnant but asks you to promise to keep her secret. Through the coming weeks this juicy bit of gossip is on the tip of your tongue but you do not tell anyone because of your promise. There are things we recognise as being required of us irrespective of what we (really) desire to do. This is what Kant means by duty.
But this raises the question. If it is not desires that move us to do what is right (even really strong desires), what does? In our example, why is it that we keep our promise despite the strong desire to gossip? Kant’s answer is “the good will”.
Good Will
Kant gives the following characterization of the good will. It is something that is good irrespective of effects :
A good will is good not because of what it effects or accomplishes — because of its fitness for attaining some proposed end: it is good through its willing alone — that is, good in itself. 2
It is also good without qualification.
It is impossible to conceive anything at all in the world, or even out of it, which can be taken as good without qualification, except a good will. 3
What does Kant mean? Well, pick anything you like which you think might make an action good — for example, happiness, pleasure, courage, and then ask yourself if there are any situations you can think of where an action having those features makes those actions worse?
It seems there are. Imagine someone who is
happy
when kicking a cat; or someone taking pleasure in torture; or a serial killer whose courage allows her to abduct children in broad daylight. In such cases the happiness, pleasure and courage make the actions worse. Kant thinks we can repeat this line of thinking for anything and everything, except one thing —
the good will
.
The good will unlike anything else is good
unconditionally
and what makes a good will good is willing alone; not other attitudes, or consequences, or characteristics of the agent. Even Kant thinks this sounds like a rather strange idea. So how can he (and we) be confident that the good will even exists?
Consider
Mahatma Gandhi
’s (1869–1948) non-violent protest for Indian independence. He stood peacefully whilst the British police beat him. Here is a case where there must have been an overwhelming desire to fight back. But he did not. In this type of action Kant would claim that we “see” the good will — as he says — “shining like a jewel”.
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Seeing such resilience in the face of such awful violence we are humbled and can recognize, what Kant calls, its moral worth. Obviously not all actions are as significant as Gandhi’s! However, Kant thinks that any acts like this, which are
performed despite conflicting desires
, are due to the good will. Considering such actions (can you think of any?) means we can recognize that the good will exists.
3. Acting for the Sake of Duty and Acting in Accordance with Duty
From what we have said above about the nature of duty and good will we can see why Kant says that to act from good will is acting for the sake of duty. We act despite our desires to do otherwise. For Kant this means that acting for the sake of duty is the
only
way that an action can have moral worth. We will see below what we have to do for our actions to be carried out for the sake of duty. However, before we do this, we need to be really clear on this point about moral worth.
Imagine that you are walking with a friend. You pass someone begging on the street. Your friend starts to weep, fumbles in his wallet and gives the beggar some money and tells you that he
feels
such an empathy with the poor man that he just has to help him.
For Kant, your friend’s action has no moral worth because what is moving him to give money is empathy rather than duty! He is
acting in accordance with duty
. However, Kant does think your friend should be applauded as such an action is something that is of value although it wouldn’t be correct to call it a
moral
action.
To make this point clearer, Kant asks us to consider someone who has no sympathy for the suffering of others and no inclination to help them. But despite this:
…he nevertheless tears himself from his deadly insensibility and performs the action without any inclination at all, but solely from duty then for the first time his action has genuine moral worth.
5
In contrast to our friend, this person is acting
for the sake of
duty and hence their action is moral. We must be careful though.
Kant is not telling us to become emotionally barren robots!
He is not saying that before we can act morally we need to get rid of sympathy, empathy, desires, love, and inclinations. This would make Kant’s moral philosophy an absurd non-starter.
Let us see why Kant is not saying this. Consider an action such as giving to others. We should ask whether an action of giving to others
would have
been performed
even
if the agent lacked the desire to do so. If the answer is “yes” then the act has moral worth. This though is consistent with the agent
actually having
those desires.
The question for Kant is not whether an agent has desires but what moved the agent to act.
If they acted
because
of those desires they acted in accordance with duty and their action had no moral worth. If they acted for the sake of duty, and just happened to have those desires, then their action has moral worth.
4. Categorical and Hypothetical Imperatives
If we agree with Kant and want to act for the sake of duty what should we do? His answer is that we have to act out of
respect for the moral law
. He has two examples of how this works in practice: lying and suicide. We look at the former in Chapter 13, we will consider Kant’s example of suicide at the end of this chapter. However, before doing this we need to get a sense of what Kant has in mind when he talks about acting out of respect for the moral law.
The moral law is what he calls the “Categorical Imperative”. He thinks there are three formulations of this.
CI-1: …act only according to that maxim through which you can at the same time will that it become a universal law. 6
CI-2: So act that you use humanity, in your own person as well as in the person of any other, always at the same time as an end, never merely as a means. 7
CI-3: …every rational being must so act as if he were through his maxim always a lawmaking member in the universal kingdom of ends. 8
We will consider these in turn, showing how they are linked. Consider then, CI-1.
Kant’s idea is that we use this “test” to see what maxims are morally permissible. If we act in accordance with those then we are acting from duty and our actions have moral worth. Let us look at what this means.
Initially it is worth considering what “categorical” and “imperative” mean. An imperative is just a command. “Clean your room!” is an imperative I give my daughter every Saturday.
What about the “categorical” part? If a command is categorical then people ought to follow it irrespective of how they feel about following it, irrespective of what consequences might follow, or who may or may not have told them to follow it. For example, the command “do not peel the skin of babies” is categorical.
Contrast this with Hypothetical Imperatives. If I tell my daughter to clean her room, this is hypothetical. This is because whether she ought to clean her room is dependent on conditions about her and me. If she does not care about a clean room and about what her dad thinks, then it is not true that she ought to clean her room. Most commands are hypothetical. For example, “study!” You ought to study only
if
certain things are true about you; for example, that you care about doing well, that you want to succeed in the test etc.
Kant thinks that moral “oughts” — for example, “you ought not lie” — are
categorical
. They apply to people irrespective of how they feel about them.
The next thing we need is the idea of a “maxim”. This is relatively simple and is best seen through the following examples. Imagine I’m considering whether to make a false promise. Perhaps I think that by falsely promising you that l will pay you back I will be more likely to get a loan from you. In that case my maxim is something like “
whenever I can benefit from making a false promise I should do so
”.
Imagine I decide to exercise because I feel depressed, then I may be said to be acting on the maxim “
Whenever I feel depressed I will exercise
”. A maxim is a general principle or rule upon which we act. We do not decide on a set of maxims, perhaps writing them down, and then try to live by them but rather a
maxim is the principle or rule that can make sense of an action whether or not we have thought about it in these terms
.
5. The First Formulation of the Categorical Imperative
Let’s put these bits together in relation to CI-1
…act only according to that maxim through which you can at the same time will that it become a universal law. 9
The “test” that CI-1 prescribes is the following. Consider the maxim on which you are thinking about acting, and ask whether you can either (i) conceive that it become a universal law, or (ii) will that it become a universal law. If a maxim fails on either (i) or (ii) then there is no good reason for you to act on that maxim and it is morally impermissible to do so. If it passes the CI test, then it is morally permissible.
Kant is
not
saying that the CI-1 test is a way of working out what is and what is not moral. Presumably we can think of lots of maxims, which are non-moral, which pass the test, for example, “whenever I am bored I will watch TV”.
Equally he is not saying that if a maxim cannot be universalized then it is morally impermissible. Some maxims are just mathematically impossible. For example, “whenever I am going to exercise I will do it for an above the average amount of time”. This maxim cannot be universalized because we cannot conceive that everyone does something above “average”.
Finally, it is worth remembering that the maxim must be able to be
willed as
a universal law. This is important because maxims such as “if your name is Jill and you are 5ft 11, you can lie” will fail to be universalized because you cannot will that your name is Jill or that your height is 5ft11. It has to be possible to will as a universal law and for this to be true it must be at least possible for it actually to come about. This shows that the common concern that we can get any maxim to pass the CI-1 test by simply adding more and more specific details, such as names, heights or locations,
fails
. This is very abstract (what did we tell you about Kant’s work!). Let us consider an example.
6. Perfect and Imperfect Duties
Recall the example of making a false promise to secure a loan. The maxim is
“whenever I can benefit from doing so, I should make a false promise”
. The question is whether I could conceive or will that this become a universal law.
I could not. If everyone followed this maxim then we would all believe everyone else could make a false promise if it would benefit them to do so. Kant thinks such a situation is not conceivable because the very idea of making a promise relies on trust. But if “whenever it is of benefit to you, you can make false promises” was to become a universal law then there would be no trust and hence no promising. So by simply thinking about the idea of promising and lying we see the maxim will fail the test and, because we cannot universalize the maxim, then making a false promise becomes morally impermissible. This is true universally for all people in all circumstances for anyone can, in principle, go through the same line of reasoning.
A maxim failing at (i) is what Kant calls a
contradiction in conception
, and failing at (i) means we are dealing with what Kant calls a
perfect duty
. In our example we have shown we have a perfect duty not to make false promises.
Consider another example. Imagine that someone in need asks us for money but we decide not to help them. In this case our maxim is “whenever someone is in need and asks for money do not give them money”. Does this pass the CI-1 test?
No it fails the CI-1 test. Although it is true that the maxim passes (i) not giving to the needy does not threaten
the very idea
of giving money away. Kant thinks that anyone thinking about this will see that that maxim will fail at (ii) and hence it is morally impermissible. Here is why.
You cannot know if you will be in need in the future and presumably you would want to be helped if you were in need. In which case you are being inconsistent if you willed that “people should not help those in need” should become a universal law. For you might want people to help those in need in the future, namely,
you
.
So we cannot will the maxim “whenever someone is in need do not help them” to become a universal moral law. Again this is a thought process that
anyone
can go through and it means that this moral claim is true universally for all people in all circumstances. Failing at (ii) is what Kant calls a
contradiction in will
, and failing at (ii) means we are dealing with what Kant calls an
imperfect duty
.
It is absolutely key to recognize that CI-1 is not simply asking “what if everyone did that?” CI-1 is
not a form of Utilitarianism
(see Chapter 1). Kant is not saying that it is wrong to make false promises because if people did then the world would be a horrible place. Rather Kant is asking about whether we can
conceive or will
the maxim to become a universal law.
7. Second Formulation of the Categorical Imperative
The second formulation (CI-2) is the following:
So act that you use humanity, in your own person as well as in the person of any other, always at the same time as an end, never merely as a means. 10
Kant thinks that CI-1 and CI-2 are two sides of the same coin, though precisely how they are related is a matter of scholarly debate. Put very simply CI-2 says you should not use people, because if you do, you are failing to treat them as a rational agent and this is morally wrong.
For example, if I use your essay without your knowledge then I have not treated you as a rational agent. I would have done had I asked you for your essay and you had freely chosen to let me have it. But given that I did not ask you, I was in a sense
making choices on your behalf
and thus did not treat you as a rational agent. So according to Kant I should always treat you as an end not a means. I should always treat you as a free rational agent.
Kant’s theory then has a way of respecting the
dignity
of people. We should treat people with respect and with dignity purely on the basis that they are rational agents, and not because of their race, gender, education, upbringing etc. From this you can also see that Kant’s theory allows us to speak about “rights”. If someone has a right then they have this right irrespective of gender, education, upbringing etc. For example, Jill has a right to free speech because she is a person, consequently that right will not disappear if she changes her location, personal circumstances, relationship status, political viewpoint etc.
Importantly, CI-2 does not say that you
either
treat someone as a means
or
an end. I could treat someone as an end
by
treating them as a means. Suppose that you have freely decided to become a taxi driver. If I use you as a means by asking you to take me to the airport I am also treating you as an end. But Kant does not believe this to be morally wrong because I am respecting you as a rational agent; after all, you
chose
to be a taxi driver. Of course, if I get into your car and point a gun at your head and ask to be taken to the airport then I am not treating you as an end but rather solely as a means, which is wrong.
8. The Third Formulation of the Categorical Imperative and Summary
The final formulation of the Categorical Imperative is a combination of CI-1 and CI-2. It asks us to imagine a kingdom which consists of only those people who act on CI-1. They never act on a maxim which cannot become a universal law. In such a kingdom people would treat people as ends, because CI-2 passes CI-1. This is why CI-3 is often called the “Kingdom of Ends” formulation:
…every rational being must so act as if he were through his maxim always a lawmaking member in the universal kingdom of ends. 11
In summary, we have seen that Kant thinks that acts have moral worth only if they are carried out for the sake of duty. Agents act for the sake of duty if they act out of respect for the moral law, which they do by following the Categorical Imperative in one of its formulations.
Consequently, Kant thinks that acts are wrong and right universally, irrespective of consequences and desires. If lying is wrong then it is wrong in all instances. From all this, it follows that we cannot be taught a set of moral rules for each and every situation and Kant believes that it is up to us to work it out for ourselves by thinking rationally.
There have been, and continue to be, many books and journal articles written about Kant’s ethics. He has a profound and deep insight into the nature of morality and he raises some fundamental questions about what it is to be human. Kant’s moral theory is radically
Egalitarian
as his theory is blind to individual personal circumstances, race, gender and ethnicity. Everyone is equal before the moral law!
Related to this, his theory respects the rights of individuals and, relatedly, their dignity. Any theory that is to have a hope of capturing our notion of rights needs to be able to respect the thought that a right is not something that disappears if circumstances change. Jill has a right to life, period; we do not say Jill has a right to life “if…” and then have to fill in the blanks. This is precisely something that Kant’s theory can give us. CI-1 generates maxims which do not have exceptions and CI-2 tells us that we should
always
treat everyone as an end in themselves and never solely as a means to an end. It tells us, for example, that we ought not to kill Jill, and this holds true in all circumstances.
There are, though, a number of tough questions that Kant’s work raises. We consider some of these below. However, as with all the philosophical ideas we discuss in this book, Kant’s work is still very much alive and has defenders across the world. Before we turn to these worries, we work through an example that Kant gives regarding suicide.
9. Kant on Suicide
Kant is notoriously stingy with examples. One he does mention is suicide (another is lying see Chapter 13). This is an emotive topic and linked to questions about mental health and religion. An attraction of Kant’s view is the ability to apply his Categorical Imperatives in a dispassionate way. His framework should allow us to “plug in” the issue and “get out” an answer. Let’s see how this might work.
Kant thinks that suicide is always wrong and has very harsh words for someone who attempts suicide
He who so behaves, who has no respect for human nature and makes a thing of himself, becomes for everyone an Object of freewill. We are free to treat him as a beast, as a thing, and to use him for our sport as we do a horse or a dog, for he is no longer a human being; he has made a thing of himself, and, having himself discarded his humanity, he cannot expect that others should respect humanity in him. 12
But why does he think this? How does this fit with Kant’s Categorical Imperatives? We will look at the first two formulations.
Fundamental to remember is that for Kant the motive that drives all suicide is “avoid evil”. By which he means avoiding suffering, pain, and other negative outcomes in one’s life. All suicide attempts are due to the fact that we love ourselves and thus want to “avoid evils” that may befall us.
 Given what we have just said about my motives this means I will be acting on this maxim: “From self-love I make as my principle to shorten my life when its continued duration threatens more evil than it promises satisfaction”.
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Following CI-1 the question then is whether it is possible to universalise this maxim? Kant thinks not. For him it is unclear how we could will it that all rational agents as the result of
self-love
can
destroy themselves
when their continued existence threatens more evil than it promises satisfaction. For Kant self-love leading to the destruction of the self is a
contradiction
. Thus he thinks that we have a
perfect
(rather than an imperfect) duty to ourselves not to commit suicide. To do so is morally wrong. This is how Kant puts it:
One sees at once a contradiction in a system of nature whose law would destroy life [suicide] by means of the very same feeling that acts so as to stimulate the furtherance of life [self-love], and hence there could be no existence as a system of nature. Therefore, such a maxim cannot possibly hold as a universal law of nature and is, consequently, wholly opposed to the supreme principle of all duty.
Notice a few odd things here in relation to CI-1. The point about universalisation seems irrelevant. Kant could have just said it is a contradiction to will from self-love the destruction of oneself. It seems that there is nothing added by asking us to consider this point universalised. It does not add weight to the claim that it is a contradiction.
Second, it is not really a “contradiction” at all! It is different to the lying promise example. In this it seems that the very concept of a promise relies on trust, which lying would destroy. In contrast in the suicide case the “contradiction” seems more like a by-product of Kant’s assumption regarding the motivation of suicidal people. So we can avoid the “contradiction” if we allow for the possibility that suicide need not be driven by self-love. If this were true then there would be no “contradiction”. Hence, it seems wrong to call the duty not to kill oneself — if such a duty exists — a “perfect” duty. So the first formulation does not give Kant the conclusion that suicide is morally wrong.
Moving to the second formulation. This helps us understand Kant’s harsh assessment of people attempting suicide. Remember he calls such people “objects” or “beasts” or “things”. So, what is the difference between beasts or objects or things, and humans? The answer is that we are
rational
. Recall, that for Kant our rationality is of fundamental value. If anyone’s actions do not recognize someone else’s rationality then they have done something morally wrong. This amounts to treating them as merely means to our own end. Given all this you can see what Kant is getting at. For him committing suicide is treating yourself as a mere means to some end — namely the end of avoiding pain and suffering etc. — and not an end in itself. You are treating yourself as a “beast” a “thing” an “object”, not as a human being with the gift of reason. This is morally wrong.
Moreover, if you do this, then others treating you with respect as a
rational person
can conclude that you also want others to treat you in this way. Because if you are rational then you must think that it is OK to universalise the maxim that we can treat others as objects, beast and thing. They can thus treat you as a beast, object, and thing and still be treating you with respect as a rationale agent. With regard to attempting suicide your action is wrong because you have ignored your own rationality. You have treated yourself as a mere means to an end.
But, like the first formulation this is very weak. It is unclear why in attempting suicide you are treating yourself as a mere means to an end. You might think you are respecting your rationality by considering suicide. Recall, Kant says that it is sometimes fine to treat people as a means to an end, e.g. a taxi driver. It is fine where people have given consent for you to treat them that way. In that case, suicide might be like the taxi driver case. We have freely decided to treat ourselves as a means to an end. We are, then, treating ourselves as a rational agent and not doing something morally wrong by committing suicide.
There are some other things that Kant says about the wrongness of suicide that do not link to the Categorical Imperatives. For example, he talks about humans being the property of God and hence our lives not being something we can choose to extinguish. However, we need not discuss this here.
There is a consensus between Kant scholars that, as it stands, Kant’s argument against suicide fails. There are some though who use Kant’s ideas as a starting point for a more convincing argument against suicide. For example, see
J. David Velleman
(1999) and
Michael Cholbi
(2000).
10. Problems and Responses: Conflicting Duties
If moral duties apply in
all
circumstances, then what happens when we have duties which conflict?  Imagine then that an SS officer knocks at your door and asks if you are hiding Jews? What might Kant’s theory tell us to do? Our duty is to refrain from lying so does this mean we are morally required to tell the SS officer our secret? If this is the conclusion then it makes Kant’s theory morally repugnant.
However,
there is no requirement in Kant’s theory to tell the truth
, there is just a requirement not to lie. Lying is about intentional deceit, so maybe in this example there is a way not to lie. For example, if we simply stayed silent (see Chapter 13).
Even if we respond in this sort of way in this example, presumably we can engineer an example that would not allow for this. For example, perhaps we are in a law court and the SS officer asks us under oath. In that example, silence would not be an option. This certainly would seem to count against Kant’s theory for it does seem morally wrong to reveal the location of the Jewish people.
The main point though is that Kant thinks we need to take the features of each individual situation into account. He does not just want us to mindlessly apply generic rules whilst paying no attention to what is before us. So
Peter Rickman
writes regarding these types of cases:
…it should be plain that more than one imperative/moral principle is relevant to the situation. Certainly we should tell the truth; but do we not also have a duty to protect an innocent man from harm? Further, do we not have an obligation to fight evil? We are confronted with a conflict of values here. Unfortunately, as far as I know, there is no explicit discussion of this issue in Kant. One could assume, however, that his general approach of distinguishing the lesser from the greater evil should be applied. I think Kant might say that although lying is never right, it might be the lesser evil in some cases. 14
So the point is not that these sorts of examples are “knock down” criticisms of Kant’s theory but rather that Kant’s theory is underspecified and fails to give guidance with these specific sorts of cases. In fact, we might think that this is an advantage of his theory that has given us the supreme principle of morality and the general way of proceeding but has left it up to us to work out what to do in each situation. We will leave the reader to see if this can be done and in particular, whether it can be done in a way consistent with the other aspects of his moral theory.
11. Problems and Responses: The Role of Intuitions
One of the most common criticisms levelled at Kant’s theory is that it is simply counter intuitive. For example, lying, for him, is morally impermissible in all instances irrespective of the consequences. Yet we seem to be able to generate thought experiments that show that this is a morally repugnant position.
However, in Kant’s defence we might ask why we should use our intuitions as any form of test for a moral theory. Intuitions are notoriously fickle and unreliable. Even if you pick the oddest view you can think of, you would probably find some people at some point in time that would find this view “intuitive”. So how worried should we be if Kant’s theory leads to counter intuitive consequences? This then raises a more general methodological question to keep at the forefront of your mind when reading this book. What role, if any, should intuitions have in the formation and the testing of moral theory?
12. Problem and Responses: Categorical Imperatives and Etiquette
Kant argues that what we are morally required to do is a matter of reason. If people reason in the right way then they will recognise, for example, that lying is wrong. However, some philosophers, for example
Philippa Foot
(1920–2010), have worried about this link to reason. The strength of Foot’s challenge is that she agrees that morality is a system of Categorical Imperatives but says that this
need not be due to reason
.
Foot uses the example of
etiquette
to motivate her argument.15 Rules of etiquette seem to be Categorical Imperatives but are
not
grounded in reason. Consider an example. I had a friend at university who was a sportsman. He was in many teams, his degree was in sports and exercise and if there were ever a spare minute he would be running, on his bike or in the pool. Unsurprisingly he wore a tracksuit and trainers all the time!
During our second year at university a mutual friend died. There was a big formal funeral arranged. My friend decided to go to this funeral in his tracksuit and trainers. I asked him about this and his response was that it was what he liked wearing. However, to my mind at least, this reason, which was based on his desire, did not change the fact that he really ought not have worn a tracksuit. Foot would agree and thinks that rules of etiquette are categorical because they are not dependent on any particular desires someone would have.
However, even if they are categorical, Foot thinks that rules of etiquette are not rules of reason. We do not think that if we reasoned correctly we would recognise that we ought not to wear tracksuits to funerals, or (to think of some other rules of etiquette) we ought not to reply to a letter written in the third person in the first person, or we ought not to put our feet on the dinner table during a meal etc. It is not simply a matter of thinking in the right way but rather to recognise these “oughts” as part of a shared cultural practice.
So although this does not show that Kant is wrong, it does throw down a challenge to him. That is, we need independent reasons to think that the
categorical
nature of moral “oughts” are based on
reason
and not just part of a shared cultural practice. To respond to this challenge, the Kantian would have to put forward the argument that in the particular case of moral “oughts”, we have a good argument to ground the categorical nature in reason rather than institutional practices.
13. Problems and Responses: The Domain of Morality
Kant thinks that the domain of morality is merely the domain of reasons and as far as we are agents who can reason then we have duties and rights and people ought to treat us with dignity. The flip side of this is that non-rational agents, such as non-human animals, do not have rights and we can, according to Kant, treat them as we like!
The challenge to Kant’s theory is that the scope of morality seems bigger than the scope of reasons. People do think that we have moral obligations toward non-rational agents. Consider someone kicking a cat. We might think that
morally
they ought not to do this. However, Kant’s theory does not back this up because, as far as we know, cats are not rational agents. Despite it not being wrong to treat animals in this way, Kant
still thinks that we should not
, because if we did, then we would be more likely to treat humans in this way.
SUMMARY
|
Kant’s moral theory is extremely complicated and badly expressed. However, it is hugely influential and profound. As a system builder Kant’s work starts with rational reflection from which he attempts to develop a complete moral system.
He starts from the notion of duty. He shows that what allows us to act for the sake of duty is the good will, and that the good will is unconditionally good. If we want to act for the sake of duty we need to act out of respect for the moral law and this amounts to following the Categorical Imperative. Kant argues that in following the Categorical Imperative, agents will converge on what is morally permissible. Hence Kant can talk about absolute and objective moral truths. |
COMMON STUDENT MISTAKES
• Confusing acting in
accordance with
duty and acting
for the sake of
duty.
• Thinking that Kant’s theory has no room for emotions.
• Thinking that Kant’s Categorical Imperative can be summed up in the question: “how would you like it if everyone did that”?
• Thinking that the Categorical Imperative is a form of Utilitarianism.
• Thinking Kant believes you can never treat someone as a means to an end.
ISSUES TO CONSIDER
1. Think about your life. Do you think there are things you “ought to do”?
2. Do you think that there are things you ought to do irrespective of your desires and inclinations?
3. What are Categorical and Hypothetical Imperatives? Do you think that rules of etiquette are categorical or hypothetical?
4. How might Kant respond to the SS officer example?
5. Can you think of some examples where you might be treating someone solely as means-to-an-end?
6. Would capital punishment pass the CI-2 test?
7. How might CI-2 relate to prostitution? Do you think that Kant would say that it is morally permissible? (See also Chapter 10).
8. Why might Kant’s theory be well placed to respect people’s rights?
9. Do you think we have any moral obligations towards animals? What would Kant say?
10. What role do you think intuitions should have in assessing moral theories?
KEY TERMINOLOGY
|
A priori Categorical Imperative Deontological Duty Egalitarian Good will Hypothetical Imperative Maxim Rationalist Rights |
References
Cholbi, Michael J., ‘Kant and the Irrationality of Suicide’, History of Philosophy Quarterly , 17.2 (2000): 159–76.
Foot, Philippa, ‘Morality as a System of Hypothetical Imperatives’, The Philosophical Review , 81.3 (1972): 305–16, https://doi.org/10.2307/2184328
Kant, Immanuel, Moral Law: The Groundwork of the Metaphysics of Morals , Translated and analysed by H. J. Paton (Oxford: Routledge, 2013).
―, Lectures on Ethics (Cambridge: Cambridge University Press, 1997).
Rickman, Peter, ‘Having Trouble with Kant?’, Philosophy Now , freely available at https://philosophynow.org/issues/86/...uble_With_Kant
Velleman, J. David, ‘A Right of Self-termination?’, Ethics , 109.3 (1999): 606–28, https://doi.org/10.1086/233924
―, Beyond Price: Essays on Birth and Death (Cambridge: Open Book Publishers, 2015), https://doi.org/10.11647/OBP.0061 ; freely available at https://www . openbookpublishers.com/reader/349
|
1 H. J. Paton, ‘Preface’ in I. Kant, Moral Law, p. 7. 2 I. Kant, Moral Law, p. 40. 3 Ibid., p. 39. 4 Ibid., p. 40. 5 Ibid., p. 43. 6 Ibid., p. 15. 7 Ibid., p. 66. 8 Ibid., p. 21. 9 Ibid. 10 Ibid., p. 66. 11 Ibid., p. 21. 12 I. Kant, Lectures on Ethics, 27; 373. 13 I. Kant, Grounding for the Metaphysics of Morals, Ak IV, 422 14 P. Rickman, ‘Having Trouble with Kant?’, https://philosophynow.org/issues/86/...uble_With_Kant 15 P. Foot, ‘Morality as a System of Hypothetical Imperatives’. |


16.1: Factorial ANOVA 1- Balanced Designs, No Interactions
When we discussed analysis of variance in Chapter 14, we assumed a fairly simple experimental design: each person falls into one of several groups, and we want to know whether these groups have different means on some outcome variable. In this section, I’ll discuss a broader class of experimental designs, known as
factorial designs
, in we have more than one grouping variable. I gave one example of how this kind of design might arise above. Another example appears in Chapter 14, in which we were looking at the effect of different drugs on the
mood.gain
experienced by each person. In that chapter we did find a significant effect of drug, but at the end of the chapter we also ran an analysis to see if there was an effect of therapy. We didn’t find one, but there’s something a bit worrying about trying to run two
separate
analyses trying to predict the same outcome. Maybe there actually
is
an effect of therapy on mood gain, but we couldn’t find it because it was being “hidden” by the effect of drug? In other words, we’re going to want to run a
single
analysis that includes
both
drug
and
therapy
as predictors. For this analysis each person is cross-classified by the drug they were given (a factor with 3 levels) and what therapy they received (a factor with 2 levels). We refer to this as a 3×2 factorial design. If we cross-tabulate
drug
by
therapy
, using the
xtabs()
function (see Section 7.1), we get the following table:
230
load("./rbook-master/data/clinicaltrial.rdata")
xtabs( ~ drug + therapy, clin.trial )
## therapy
## drug no.therapy CBT
## placebo 3 3
## anxifree 3 3
## joyzepam 3 3
As you can see, not only do we have participants corresponding to all possible combinations of the two factors, indicating that our design is completely crossed , it turns out that there are an equal number of people in each group. In other words, we have a balanced design. In this section I’ll talk about how to analyse data from balanced designs, since this is the simplest case. The story for unbalanced designs is quite tedious, so we’ll put it to one side for the moment.
What hypotheses are we testing?
Like one-way ANOVA, factorial ANOVA is a tool for testing certain types of hypotheses about population means. So a sensible place to start would be to be explicit about what our hypotheses actually are. However, before we can even get to that point, it’s really useful to have some clean and simple notation to describe the population means. Because of the fact that observations are cross-classified in terms of two different factors, there are quite a lot of different means that one might be interested. To see this, let’s start by thinking about all the different sample means that we can calculate for this kind of design. Firstly, there’s the obvious idea that we might be interested in this table of group means:
aggregate( mood.gain ~ drug + therapy, clin.trial, mean )
## drug therapy mood.gain
## 1 placebo no.therapy 0.300000
## 2 anxifree no.therapy 0.400000
## 3 joyzepam no.therapy 1.466667
## 4 placebo CBT 0.600000
## 5 anxifree CBT 1.033333
## 6 joyzepam CBT 1.500000
Now, this output shows a cross-tabulation of the group means for all possible combinations of the two factors (e.g., people who received the placebo and no therapy, people who received the placebo while getting CBT, etc). However, we can also construct tables that ignore one of the two factors. That gives us output that looks like this:
aggregate( mood.gain ~ drug, clin.trial, mean )
## drug mood.gain
## 1 placebo 0.4500000
## 2 anxifree 0.7166667
## 3 joyzepam 1.4833333
aggregate( mood.gain ~ therapy, clin.trial, mean )
## therapy mood.gain
## 1 no.therapy 0.7222222
## 2 CBT 1.0444444
But of course, if we can ignore one factor we can certainly ignore both. That is, we might also be interested in calculating the average mood gain across all 18 participants, regardless of what drug or psychological therapy they were given:
mean( clin.trial$mood.gain )
## [1] 0.8833333
At this point we have 12 different sample means to keep track of! It is helpful to organise all these numbers into a single table, which would look like this:
knitr::kable(tibble::tribble(
~V1, ~V2, ~V3, ~V4,
NA, "no therapy", "CBT", "total",
"placebo", "0.30", "0.60", "0.45",
"anxifree", "0.40", "1.03", "0.72",
"joyzepam", "1.47", "1.50", "1.48",
"total", "0.72", "1.04", "0.88"
), col.names = c("", "no therapy", "CBT", "total"))
| no therapy | CBT | total | |
|---|---|---|---|
| NA | no therapy | CBT | total |
| placebo | 0.30 | 0.60 | 0.45 |
| anxifree | 0.40 | 1.03 | 0.72 |
| joyzepam | 1.47 | 1.50 | 1.48 |
| total | 0.72 | 1.04 | 0.88 |
Now, each of these different means is of course a sample statistic: it’s a quantity that pertains to the specific observations that we’ve made during our study. What we want to make inferences about are the corresponding population parameters: that is, the true means as they exist within some broader population. Those population means can also be organised into a similar table, but we’ll need a little mathematical notation to do so. As usual, I’ll use the symbol μ to denote a population mean. However, because there are lots of different means, I’ll need to use subscripts to distinguish between them.
Here’s how the notation works. Our table is defined in terms of two factors: each row corresponds to a different level of Factor A (in this case
drug
), and each column corresponds to a different level of Factor B (in this case
therapy
). If we let R denote the number of rows in the table, and C denote the number of columns, we can refer to this as an R×C factorial ANOVA. In this case R=3 and C=2. We’ll use lowercase letters to refer to specific rows and columns, so μ
rc
refers to the population mean associated with the rth level of Factor A (i.e. row number r) and the cth level of Factor B (column number c).
231
So the population means are now written like this:
knitr::kable(tibble::tribble(
~V1, ~V2, ~V3, ~V4,
"placebo", "$\\mu_{11}$", "$\\mu_{12}$", "",
"anxifree", "$\\mu_{21}$", "$\\mu_{22}$", "",
"joyzepam", "$\\mu_{31}$", "$\\mu_{32}$", "",
"total", "", "", ""
), col.names = c( "", "no therapy", "CBT", "total"))
| no therapy | CBT | total | |
|---|---|---|---|
| placebo | μ 11 | μ 12 | |
| anxifree | μ 21 | μ 22 | |
| joyzepam | μ 31 | μ 32 | |
| total |
Okay, what about the remaining entries? For instance, how should we describe the average mood gain across the entire (hypothetical) population of people who might be given Joyzepam in an experiment like this, regardless of whether they were in CBT? We use the “dot” notation to express this. In the case of Joyzepam, notice that we’re talking about the mean associated with the third row in the table. That is, we’re averaging across two cell means (i.e., μ 31 and μ 32 ). The result of this averaging is referred to as a marginal mean , and would be denoted μ 3 . in this case. The marginal mean for CBT corresponds to the population mean associated with the second column in the table, so we use the notation μ .2 to describe it. The grand mean is denoted μ .. because it is the mean obtained by averaging (marginalising 232 ) over both. So our full table of population means can be written down like this:
knitr::kable(tibble::tribble(
~V1, ~V2, ~V3, ~V4,
"placebo", "$\\mu_{11}$", "$\\mu_{12}$", "$\\mu_{1.}$",
"anxifree", "$\\mu_{21}$", "$\\mu_{22}$", "$\\mu_{2.}$",
"joyzepam", "$\\mu_{31}$", "$\\mu_{32}$", "$\\mu_{3.}$",
"total", "$\\mu_{.1}$", "$\\mu_{.2}$", "$\\mu_{..}$"
), col.names=c( NA, "no therapy", "CBT", "total"))
| NA | no therapy | CBT | total |
|---|---|---|---|
| placebo | μ 11 | μ 12 | μ 1. |
| anxifree | μ 21 | μ 22 | μ 2. |
| joyzepam | μ 31 | μ 32 | μ 3. |
| total | μ .1 | μ .2 | μ .. |
Now that we have this notation, it is straightforward to formulate and express some hypotheses. Let’s suppose that the goal is to find out two things: firstly, does the choice of drug have any effect on mood, and secondly, does CBT have any effect on mood? These aren’t the only hypotheses that we could formulate of course, and we’ll see a really important example of a different kind of hypothesis in Section 16.2, but these are the two simplest hypotheses to test, and so we’ll start there. Consider the first test. If drug has no effect, then we would expect all of the row means to be identical, right? So that’s our null hypothesis. On the other hand, if the drug does matter then we should expect these row means to be different. Formally, we write down our null and alternative hypotheses in terms of the equality of marginal means :
knitr::kable(tibble::tribble(
~V1, ~V2,
"Null hypothesis $H_0$:", "row means are the same i.e. $\\mu_{1.} = \\mu_{2.} = \\mu_{3.}$",
"Alternative hypothesis $H_1$:", "at least one row mean is different."
), col.names = c("", ""))
| Null hypothesis H 0 : | row means are the same i.e. μ 1. =μ 2. =μ 3. |
| Alternative hypothesis H 1 : | at least one row mean is different. |
It’s worth noting that these are exactly the same statistical hypotheses that we formed when we ran a one-way ANOVA on these data back in Chapter 14. Back then I used the notation μ P to refer to the mean mood gain for the placebo group, with μ A and μ J corresponding to the group means for the two drugs, and the null hypothesis was μ P =μ A =μ J . So we’re actually talking about the same hypothesis: it’s just that the more complicated ANOVA requires more careful notation due to the presence of multiple grouping variables, so we’re now referring to this hypothesis as μ 1. =μ 2. =μ 3. . However, as we’ll see shortly, although the hypothesis is identical, the test of that hypothesis is subtly different due to the fact that we’re now acknowledging the existence of the second grouping variable.
Speaking of the other grouping variable, you won’t be surprised to discover that our second hypothesis test is formulated the same way. However, since we’re talking about the psychological therapy rather than drugs, our null hypothesis now corresponds to the equality of the column means:
knitr::kable(tibble::tribble(
~V1, ~V2,
"Null hypothesis $H_0$:", "column means are the same, i.e., $\\mu_{.1} = \\mu_{.2}$",
"Alternative hypothesis $H_1$:", "column means are different, i.e., $\\mu_{.1} \\neq \\mu_{.2}"
), col.names = c("", ""))
| Null hypothesis H 0 : | column means are the same, i.e., μ .1 =μ .2 |
| Alternative hypothesis H 1 : | column means are different, i.e., $ {.1} {.2} |
Running the analysis in R
The null and alternative hypotheses that I described in the last section should seem awfully familiar: they’re basically the same as the hypotheses that we were testing in our simpler one-way ANOVAs in Chapter 14. So you’re probably expecting that the hypothesis tests that are used in factorial ANOVA will be essentially the same as the F-test from Chapter 14. You’re expecting to see references to sums of squares (SS), mean squares (MS), degrees of freedom (df), and finally an F-statistic that we can convert into a p-value, right? Well, you’re absolutely and completely right. So much so that I’m going to depart from my usual approach. Throughout this book, I’ve generally taken the approach of describing the logic (and to an extent the mathematics) that underpins a particular analysis first; and only then introducing the R commands that you’d use to produce the analysis. This time I’m going to do it the other way around, and show you the R commands first. The reason for doing this is that I want to highlight the similarities between the simple one-way ANOVA tool that we discussed in Chapter 14, and the more complicated tools that we’re going to use in this chapter.
If the data you’re trying to analyse correspond to a balanced factorial design, then running your analysis of variance is easy. To see how easy it is, let’s start by reproducing the original analysis from Chapter 14. In case you’ve forgotten, for that analysis we were using only a single factor (i.e.,
drug
) to predict our outcome variable (i.e.,
mood.gain
), and so this was what we did:
model.1 <- aov( mood.gain ~ drug, clin.trial )
summary( model.1 )
## Df Sum Sq Mean Sq F value Pr(>F)
## drug 2 3.453 1.7267 18.61 8.65e-05 ***
## Residuals 15 1.392 0.0928
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note that this time around I’ve used the name
model.1
as the label for my
aov
object, since I’m planning on creating quite a few other models too. To start with, suppose I’m also curious to find out if
therapy
has a relationship to
mood.gain
. In light of what we’ve seen from our discussion of multiple regression in Chapter 15, you probably won’t be surprised that all we have to do is extend the formula: in other words, if we specify
mood.gain ~ drug + therapy
as our model, we’ll probably get what we’re after:
model.2 <- aov( mood.gain ~ drug + therapy, clin.trial )
This output is pretty simple to read too: the first row of the table reports a between-group sum of squares (SS) value associated with the
drug
factor, along with a corresponding between-group df value. It also calculates a mean square value (MS), and F-statistic and a p-value. There is also a row corresponding to the
therapy
factor, and a row corresponding to the residuals (i.e., the within groups variation).
Not only are all of the individual quantities pretty familiar, the relationships between these different quantities has remained unchanged: just like we saw with the original one-way ANOVA, note that the mean square value is calculated by dividing SS by the corresponding df. That is, it’s still true that
\(\ MS = {SS \over df}\)
regardless of whether we’re talking about
drug
,
therapy
or the residuals. To see this, let’s not worry about how the sums of squares values are calculated: instead, let’s take it on faith that R has calculated the SS values correctly, and try to verify that all the rest of the numbers make sense. First, note that for the
drug
factor, we divide 3.45 by 2, and end up with a mean square value of 1.73. For the
therapy
factor, there’s only 1 degree of freedom, so our calculations are even simpler: dividing 0.47 (the SS value) by 1 gives us an answer of 0.47 (the MS value).
Turning to the F statistics and the p values, notice that we have two of each: one corresponding to the
drug
factor and the other corresponding to the
therapy
factor. Regardless of which one we’re talking about, the F statistic is calculated by dividing the mean square value associated with the factor by the mean square value associated with the residuals. If we use “A” as shorthand notation to refer to the first factor (factor A; in this case
drug
) and “R” as shorthand notation to refer to the residuals, then the F statistic associated with factor A is denoted F
A
, and is calculated as follows:
\(\ F_A={MS_A \over MS_R}\)
and an equivalent formula exists for factor B (i.e.,
therapy
). Note that this use of “R” to refer to residuals is a bit awkward, since we also used the letter R to refer to the number of rows in the table, but I’m only going to use “R” to mean residuals in the context of SSR and MSR, so hopefully this shouldn’t be confusing. Anyway, to apply this formula to the
drugs
factor, we take the mean square of 1.73 and divide it by the residual mean square value of 0.07, which gives us an F-statistic of 26.15. The corresponding calculation for the
therapy
variable would be to divide 0.47 by 0.07 which gives 7.08 as the F-statistic. Not surprisingly, of course, these are the same values that R has reported in the ANOVA table above.
The last part of the ANOVA table is the calculation of the p values. Once again, there is nothing new here: for each of our two factors, what we’re trying to do is test the null hypothesis that there is no relationship between the factor and the outcome variable (I’ll be a bit more precise about this later on). To that end, we’ve (apparently) followed a similar strategy that we did in the one way ANOVA, and have calculated an F-statistic for each of these hypotheses. To convert these to p values, all we need to do is note that the that the sampling distribution for the F
statistic
under the null hypothesis (that the factor in question is irrelevant) is an F
distribution
: and that two degrees of freedom values are those corresponding to the factor, and those corresponding to the residuals. For the
drug
factor we’re talking about an F distribution with 2 and 14 degrees of freedom (I’ll discuss degrees of freedom in more detail later). In contrast, for the
therapy
factor sampling distribution is F with 1 and 14 degrees of freedom. If we really wanted to, we could calculate the p value ourselves using the
pf()
function (see Section 9.6). Just to prove that there’s nothing funny going on, here’s what that would look like for the
drug
variable:
pf( q=26.15, df1=2, df2=14, lower.tail=FALSE )
## [1] 1.871981e-05
And as you can see, this is indeed the p value reported in the ANOVA table above.
At this point, I hope you can see that the ANOVA table for this more complicated analysis corresponding to
model.2
should be read in much the same way as the ANOVA table for the simpler analysis for
model.1
. In short, it’s telling us that the factorial ANOVA for our 3×2 design found a significant effect of drug (F
2,14
=26.15,p<.001) as well as a significant effect of therapy (F
1,14
=7.08,p=.02). Or, to use the more technically correct terminology, we would say that there are two
main effects
of drug and therapy. At the moment, it probably seems a bit redundant to refer to these as “main” effects: but it actually does make sense. Later on, we’re going to want to talk about the possibility of “interactions” between the two factors, and so we generally make a distinction between main effects and interaction effects.
the sum of squares calculated?
In the previous section I had two goals: firstly, to show you that the R commands needed to do factorial ANOVA are pretty much the same ones that we used for a one way ANOVA. The only difference is the
formula
argument to the
aov()
function. Secondly, I wanted to show you what the ANOVA table looks like in this case, so that you can see from the outset that the basic logic and structure behind factorial ANOVA is the same as that which underpins one way ANOVA. Try to hold onto that feeling. It’s genuinely true, insofar as factorial ANOVA is built in more or less the same way as the simpler one-way ANOVA model. It’s just that this feeling of familiarity starts to evaporate once you start digging into the details. Traditionally, this comforting sensation is replaced by an urge to murder the the authors of statistics textbooks.
Okay, let’s start looking at some of those details. The explanation that I gave in the last section illustrates the fact that the hypothesis tests for the main effects (of drug and therapy in this case) are F-tests, but what it doesn’t do is show you how the sum of squares (SS) values are calculated. Nor does it tell you explicitly how to calculate degrees of freedom (df values) though that’s a simple thing by comparison. Let’s assume for now that we have only two predictor variables, Factor A and Factor B. If we use Y to refer to the outcome variable, then we would use Yrci to refer to the outcome associated with the i-th member of group rc (i.e., level/row r for Factor A and level/column c for Factor B). Thus, if we use \(\ \bar{Y}\) to refer to a sample mean, we can use the same notation as before to refer to group means, marginal means and grand means: that is, \(\ \bar{Y_{rc}}\) is the sample mean associated with the rth level of Factor A and the cth level of Factor B, \(\ \bar{Y_r}\). would be the marginal mean for the rth level of Factor A, \(\ \bar{Y_{.c}}\) would be the marginal mean for the cth level of Factor B, and \(\ \bar{Y_{..}}\) is the grand mean. In other words, our sample means can be organised into the same table as the population means. For our clinical trial data, that table looks like this:
knitr::kable(tibble::tribble(
~V1, ~V2, ~V3, ~V4,
"placebo", "$\\bar{Y}_{11}$", "$\\bar{Y}_{12}$", "$\\bar{Y}_{1.}$",
"anxifree", "$\\bar{Y}_{21}$", "$\\bar{Y}_{22}$", "$\\bar{Y}_{2.}$",
"joyzepam", "$\\bar{Y}_{31}$", "$\\bar{Y}_{32}$", "$\\bar{Y}_{3.}$",
"total", "$\\bar{Y}_{.1}$", "$\\bar{Y}_{.2}$", "$\\bar{Y}_{..}$"
), col.names = c("", "no therapy", "CBT", "total"))
| no therapy | CBT | total | |
|---|---|---|---|
| placebo | \(\ \bar{Y_{11}}\) | \(\ \bar{Y_{12}}\) | \(\ \bar{Y_{1.}}\) |
| anxifree | \(\ \bar{Y_{21}}\) | \(\ \bar{Y_{22}}\) | \(\ \bar{Y_{2.}}\) |
| joyzepam | \(\ \bar{Y_{31}}\) | \(\ \bar{Y_{32}}\) | \(\ \bar{Y_{3.}}\) |
| total | \(\ \bar{Y_{.1}}\) | \(\ \bar{Y_{.2}}\) | \(\ \bar{Y_{..}}\) |
And if we look at the sample means that I showed earlier, we have \(\ \bar{Y_{11}}\) =0.30, \(\ \bar{Y_{12}}\)=0.60 etc. In our clinical trial example, the
drugs
factor has 3 levels and the
therapy
factor has 2 levels, and so what we’re trying to run is a 3×2 factorial ANOVA. However, we’ll be a little more general and say that Factor A (the row factor) has R levels and Factor B (the column factor) has C levels, and so what we’re runnning here is an R×C factorial ANOVA.
Now that we’ve got our notation straight, we can compute the sum of squares values for each of the two factors in a relatively familiar way. For Factor A, our between group sum of squares is calculated by assessing the extent to which the (row) marginal means \(\ \bar{Y_{1.}}\), \(\ \bar{Y_{2.}}\) etc, are different from the grand mean \(\ \bar{Y_{..}}\). We do this in the same way that we did for one-way ANOVA: calculate the sum of squared difference between the \(\ \bar{Y_{i.}}\) values and the \(\ \bar{Y_{..}}\) values. Specifically, if there are N people in each group, then we calculate this:
\(\ SS_A=(N \times C) \sum_{r=1}^R(\bar{Y_{r.}} - \bar{Y_{..}})^2\)
As with one-way ANOVA, the most interesting
233
part of this formula is the \(\ (\bar{Y_{r.}} - \bar{Y_{..}})^2\) bit, which corresponds to the squared deviation associated with level r. All that this formula does is calculate this squared deviation for all R levels of the factor, add them up, and then multiply the result by N×C. The reason for this last part is that there are multiple cells in our design that have level r on Factor A: in fact, there are C of them, one corresponding to each possible level of Factor B! For instance, in our toy example, there are
two
different cells in the design corresponding to the
anxifree
drug: one for people with
no.therapy
, and one for the
CBT
group. Not only that, within each of these cells there are N observations. So, if we want to convert our SS value into a quantity that calculates the between-groups sum of squares on a “per observation” basis, we have to multiply by by N×C. The formula for factor B is of course the same thing, just with some subscripts shuffled around:
\(\ SS_B = (N \times R)\sum_{c=1}^C(\bar{Y_{.c}} - \bar{Y_{..}})^2\)
Now that we have these formulas, we can check them against the R output from the earlier section. First, notice that we calculated all the marginal means (i.e., row marginal means \(\ \bar{Y_{r.}}\) and column marginal means \(\ \bar{Y_{.c}}\)) earlier using
aggregate()
, and we also calculated the grand mean. Let’s repeat those calculations, but this time we’ll save the results to varibles so that we can use them in subsequent calculations:
drug.means <- aggregate( mood.gain ~ drug, clin.trial, mean )[,2]
therapy.means <- aggregate( mood.gain ~ therapy, clin.trial, mean )[,2]
grand.mean <- mean( clin.trial$mood.gain )
Okay, now let’s calculate the sum of squares associated with the main effect of
drug
. There are a total of N=3 people in each group, and C=2 different types of therapy. Or, to put it another way, there are 3×2=6 people who received any particular drug. So our calculations are:
SS.drug <- (3*2) * sum( (drug.means - grand.mean)^2 )
SS.drug
## [1] 3.453333
Not surprisingly, this is the same number that you get when you look up the SS value for the drugs factor in the ANOVA table that I presented earlier. We can repeat the same kind of calculation for the effect of therapy. Again there are N=3 people in each group, but since there are R=3 different drugs, this time around we note that there are 3×3=9 people who received CBT, and an additional 9 people who received the placebo. So our calculation is now:
SS.therapy <- (3*3) * sum( (therapy.means - grand.mean)^2 )
SS.therapy
## [1] 0.4672222
and we are, once again, unsurprised to see that our calculations are identical to the ANOVA output.
So that’s how you calculate the SS values for the two main effects. These SS values are analogous to the between-group sum of squares values that we calculated when doing one-way ANOVA in Chapter 14. However, it’s not a good idea to think of them as between-groups SS values anymore, just because we have two different grouping variables and it’s easy to get confused. In order to construct an F test, however, we also need to calculate the within-groups sum of squares. In keeping with the terminology that we used in the regression chapter (Chapter 15) and the terminology that R uses when printing out the ANOVA table, I’ll start referring to the within-groups SS value as the residual sum of squares SS R .
The easiest way to think about the residual SS values in this context, I think, is to think of it as the leftover variation in the outcome variable after you take into account the differences in the marginal means (i.e., after you remove SSA and SSB). What I mean by that is we can start by calculating the total sum of squares, which I’ll label SST. The formula for this is pretty much the same as it was for one-way ANOVA: we take the difference between each observation Y rci and the grand mean \(\ \bar{Y_{..}}\), square the differences, and add them all up
\(\ SS_T=\sum_{r=1}^R\sum_{c=1}^C\sum_{i=1}^N(Y_{rci} - \bar{Y_{..}})^2\)
The “triple summation” here looks more complicated than it is. In the first two summations, we’re summing across all levels of Factor A (i.e., over all possible rows r in our table), across all levels of Factor B (i.e., all possible columns c). Each rc combination corresponds to a single group, and each group contains N people: so we have to sum across all those people (i.e., all i values) too. In other words, all we’re doing here is summing across all observations in the data set (i.e., all possible rci combinations).
At this point, we know the total variability of the outcome variable SS T , and we know how much of that variability can be attributed to Factor A (SS A ) and how much of it can be attributed to Factor B (SS B ). The residual sum of squares is thus defined to be the variability in Y that can’t be attributed to either of our two factors. In other words:
SS R =SS T − (SS A +SS B )
Of course, there is a formula that you can use to calculate the residual SS directly, but I think that it makes more conceptual sense to think of it like this. The whole point of calling it a residual is that it’s the leftover variation, and the formula above makes that clear. I should also note that, in keeping with the terminology used in the regression chapter, it is commonplace to refer to SS A +SS B as the variance attributable to the “ANOVA model”, denoted SS M , and so we often say that the total sum of squares is equal to the model sum of squares plus the residual sum of squares. Later on in this chapter we’ll see that this isn’t just a surface similarity: ANOVA and regression are actually the same thing under the hood.
In any case, it’s probably worth taking a moment to check that we can calculate SS R using this formula, and verify that we do obtain the same answer that R produces in its ANOVA table. The calculations are pretty straightforward. First we calculate the total sum of squares:
SS.tot <- sum( (clin.trial$mood.gain - grand.mean)^2 )
SS.tot
## [1] 4.845
and then we use it to calculate the residual sum of squares:
SS.res <- SS.tot - (SS.drug + SS.therapy)
SS.res
## [1] 0.9244444
Yet again, we get the same answer.
What are our degrees of freedom?
The degrees of freedom are calculated in much the same way as for one-way ANOVA. For any given factor, the degrees of freedom is equal to the number of levels minus 1 (i.e., R−1 for the row variable, Factor A, and C−1 for the column variable, Factor B). So, for the
drugs
factor we obtain df=2, and for the
therapy
factor we obtain df=1. Later on on, when we discuss the interpretation of ANOVA as a regression model (see Section 16.6) I’ll give a clearer statement of how we arrive at this number, but for the moment we can use the simple definition of degrees of freedom, namely that the degrees of freedom equals the number of quantities that are observed, minus the number of constraints. So, for the
drugs
factor, we observe 3 separate group means, but these are constrained by 1 grand mean; and therefore the degrees of freedom is 2. For the residuals, the logic is similar, but not quite the same. The total number of observations in our experiment is 18. The constraints correspond to the 1 grand mean, the 2 additional group means that the
drug
factor introduces, and the 1 additional group mean that the the
therapy
factor introduces, and so our degrees of freedom is 14. As a formula, this is N−1−(R−1)−(C−1), which simplifies to N−R−C+1.
Factorial ANOVA versus one-way ANOVAs
Now that we’ve seen
how
a factorial ANOVA works, it’s worth taking a moment to compare it to the results of the one way analyses, because this will give us a really good sense of
why
it’s a good idea to run the factorial ANOVA. In Chapter 14 that I ran a one-way ANOVA that looked to see if there are any differences between drugs, and a second one-way ANOVA to see if there were any differences between therapies. As we saw in Section 16.1.1, the null and alternative hypotheses tested by the one-way ANOVAs are in fact identical to the hypotheses tested by the factorial ANOVA. Looking even more carefully at the ANOVA tables, we can see that the sum of squares associated with the factors are identical in the two different analyses (3.45 for
drug
and 0.92 for
therapy
), as are the degrees of freedom (2 for
drug
, 1 for
therapy
). But they don’t give the same answers! Most notably, when we ran the one-way ANOVA for
therapy
in Section 14.11 we didn’t find a significant effect (the p-value was 0.21). However, when we look at the main effect of
therapy
within the context of the two-way ANOVA, we do get a significant effect (p=.019). The two analyses are clearly not the same.
Why does that happen? The answer lies in understanding how the
residuals
are calculated. Recall that the whole idea behind an F-test is to compare the variability that can be attributed to a particular factor with the variability that cannot be accounted for (the residuals). If you run a one-way ANOVA for
therapy
, and therefore ignore the effect of
drug
, the ANOVA will end up dumping all of the drug-induced variability into the residuals! This has the effect of making the data look more noisy than they really are, and the effect of
therapy
which is correctly found to be significant in the two-way ANOVA now becomes non-significant. If we ignore something that actually matters (e.g.,
drug
) when trying to assess the contribution of something else (e.g.,
therapy
) then our analysis will be distorted. Of course, it’s perfectly okay to ignore variables that are genuinely irrelevant to the phenomenon of interest: if we had recorded the colour of the walls, and that turned out to be non-significant in a three-way ANOVA (i.e.
mood.gain ~ drug + therapy + wall.colour
), it would be perfectly okay to disregard it and just report the simpler two-way ANOVA that doesn’t include this irrelevant factor. What you shouldn’t do is drop variables that actually make a difference!
What kinds of outcomes does this analysis capture?
The ANOVA model that we’ve been talking about so far covers a range of different patterns that we might observe in our data. For instance, in a two-way ANOVA design, there are four possibilities: (a) only Factor A matters, (b) only Factor B matters, (c) both A and B matter, and (d) neither A nor B matters. An example of each of these four possibilities is plotted in Figure ?? .


1.2: Biological Treatment Overview
Learning Outcomes
- Understand which environments the different types of bacteria thrive
- Explain how activated sludge treatment process is more efficient than other methods
- Describe the nitrogen cycle and how it’s used to remove nitrogen from wastewater
Aerobic, Facultative, and Anaerobic Organisms
Aerobic bacteria require an environment that has free dissolved oxygen. The bacteria use that oxygen for respiration to live. They feed on the organic matter and other nutrients in the wastewater. As the bacteria consume these materials, they are removed from the wastewater, making it less contaminated. The byproduct of the aerobic decomposition of organic matter is carbon dioxide (CO 2 ).
Anaerobic bacteria require an environment that has no free or combined oxygen. Free oxygen is when there is excess oxygen dissolved in the water and is available as O 2 . Combined oxygen is when the oxygen molecule is bound to another element. A common example of combined oxygen in wastewater is nitrate (NO 3 ). In anaerobic conditions, there is absolutely no oxygen available to the bacteria for respiration. The byproduct of anaerobic decomposition is methane gas.
Facultative bacteria have the ability to thrive in either aerobic or anaerobic conditions. While they prefer aerobic conditions, they have the ability to adapt when no oxygen is available and survive in anaerobic conditions.
Activated Sludge Treatment
The activated sludge process was developed by two scientists, Edward Arden and William Lockett, in England in 1914. Their experiments showed that by taking the microorganisms that were already established from aerobic decomposition and introducing them to fresh wastewater, it would speed up the decomposition of the new organic wastes. Instead of relying on the 30 to 45 days it would normally take to breakdown these organic wastes, the activated sludge process can achieve the same level of treatment in less than one day. This means that an activated sludge wastewater treatment system will be able to handle higher flow rates and higher organic loading than pond treatment or fixed film systems.
Activated sludge treatment systems consist of an aeration tank followed by a secondary clarifier. These two tanks each provide a unique function but also work together to reduce the BOD 5 in the wastewater. In the aeration tanks, oxygen is diffused into the water so the bacteria can survive in aerobic conditions. The air diffusers also keep the bacteria in suspension so it can continually be in contact with the incoming wastewater. In the secondary clarifiers, the bacteria will be separated from the treated wastewater. The treated wastewater will continue on to the tertiary treatment step. A majority of the settled bacteria will be sent back to the beginning of the aeration tanks. In the clarifier, the bacteria were stressed due to a lack of food and oxygen. They begin to undergo endogenous respiration and are so starved they begin to breakdown their own cells to survive. When the stressed bacteria are reintroduced into the aeration tanks they are now in an environment with plenty of oxygen and food. These bacteria will now quickly begin to breakdown the organic wastes in the wastewater much faster than before.
Return Activated Sludge
The return activated sludge, or RAS, are the bacteria that have settled in the secondary clarifier and are being sent back to the aeration tank. The rate of speed at which the microorganisms are returned is something that the operator can control.
Waste Activated Sludge
Like the other biological process discussed, the bacteria, overtime will grow and their population will increase. To control the amount of bacteria in the system, a portion of the RAS will not be returned to the aeration tank but instead is directed to a separate solids handling treatment unit. Typically, these solids are combined with the settled solids from the primary sedimentation tanks and sent to an anaerobic digester.
Food to Microorganism Ratio
A key parameter to determine the effectiveness of activated sludge systems is the food to microorganism ratio, or F/M. The food is determined by the amount of BOD 5 in the incoming wastewater and the amount of microorganisms available to consume that food is determined by the amount of MLSS in the aeration tanks. Since it’s a ratio, the units of these two laboratory results must be the same. The mg/L concentration of the laboratory results is converted into a mass with the units of lbs. While each treatment plant will determine which level of F/M has historically given effective treatment, a common range is around 0.2 to 0.5.
Mean Cell Residence Time
How much bacteria remains in the system and how much is wasted can be determined by the mean cell residence time or MCRT. The MCRT is a theoretical calculation of the average time a single bacteria will stay within the activated sludge system before being wasted. To calculate the MCRT an operator would determine how many pounds of MLSS there is in the system and divide that by how many pounds were removed from the system; typically over a 24 hour period.
Review of Basic Principles of Operation
The F/M ratio, MCRT, RAS, WAS, and dissolved oxygen concentrations can all be manipulated by the operators of a wastewater treatment facility to optimize the effectiveness of treatment. In fact, those parameters are the only things that can be easily controlled. The amount of incoming wastewater is going to be what it is and it will fluctuate throughout a 24 hour period as well as vary by season. The incoming BOD 5 loading is what it’s going to be and the operators can’t control it.
If the F/M is too low, it means that there are more bacteria than what is needed to consume the available food. This is inefficient because supplying oxygen to the bacteria requires a significant amount of energy. If there is too much bacteria in the system and not enough food, the bacteria will still be consuming oxygen but the BOD 5 won’t be further reduced. To increase the F/M ratio operators can only control the “M” portion of the equation. By increasing the wasting rate, the MLSS will be reduced causing the F/M ratio to increase.
If the F/M ratio is high, then there is not enough bacteria to consume the large amounts of incoming BOD 5 . This will lead to poor treatment and the effluent will have a high BOD 5 concentration. Operators cannot decrease the amount of BOD 5 coming into the plant so they will have to increase the amount of MLSS in the system. They can do this by decreasing or stopping the wasting rate. Decreasing the wasting rate will cause the MLSS to increase and the F/M ratio will be reduced.
The MCRT is another process control tool that is used to determine the wasting rate by manipulating the equation to calculate the MCRT. Often the desired MCRT rate is determined by the design of the treatment facility or from historical data. By taking the pounds of MLSS in the system and dividing it by the desired MCRT, you will determine what the wasting rate needs to be to achieve that MCRT. However, the MCRT and F/M methods for determining the wasting rate can often conflict with each other. Operators need to look at the changes in MCRT and F/M over time and make minor adjustments to the process so the bacteria aren’t “shocked”.
Toxic Substances
Toxic substances can be detrimental to the efficacy of activated sludge systems. Activated sludge treatment relies on living bacteria to feed off the organic wastes, thus removing it from the wastewater. If toxic substances are introduced to the activated sludge system, the bacteria can die off. Without a significant population of bacteria, the organic wastes will not be consumed and the effluent of the treatment plant will have high amounts of BOD 5 .
Toxic substances can include heavy metals, pesticides, high concentrations of salts, cyanide, PCBs, and other chemicals. These toxic substances, if introduced in the wastewater collection system, will be sent to the wastewater treatment facility and can destroy the population of bacteria that treats organic wastes. To prevent toxic substances from entering the collection system, municipalities will have pre-treatment programs. These programs will have a public outreach component to inform the community of its wastewater infrastructure and the harm it can cause if chemicals and other toxic materials are dumped down the drain. In addition to household waste, pre-treatment programs will also work closely with manufacturing and processing facilities to ensure they are not discharging toxic materials into the wastewater collection system.
Pure Oxygen Treatment
Many treatment plants utilize air blowers or surface aerators to provide oxygen to the bacteria in the aeration tanks. These systems provide oxygen that’s in the atmosphere and diffuses it into the wastewater in the tank. One drawback of these methods is that our atmosphere is only around 21% oxygen. A majority of the atmosphere is nitrogen. Having a pure oxygen system is more efficient because the oxygen concentrations can be as high as 99% pure oxygen. However, there are many drawbacks to pure oxygen plants. First off, extra equipment will be needed to create pure oxygen. This equipment will have electrical, maintenance, and operational costs associated with it. Secondly, pure oxygen can be extremely hazardous to deal with. At higher purities oxygen can be explosive. Extra caution must be taken to ensure no sparks, oil, or other contaminants are near the pure oxygen generators.
Enhanced Biological Treatment
If the treated wastewater leaving a treatment facility is being discharged into an impaired waterbody, the NPDES permit will most likely have limits on ammonia, nitrogen, and phosphorus. These nutrients at high concentrations can cause eutrophication in a waterbody. Eutrophication occurs when there is an excess of nutrients in a waterbody that spurs algae growth. The algae population will become out of control and consume dissolved oxygen in the waterbody to the point where fish and other aquatic life can not survive. Nitrogen and phosphorus can be removed from the treatment plant by utilizing biological nutrient removal (BNR) processes.
Nitrogen Removal
Nitrogen can be removed from the wastewater by making slight modifications to the activated sludge treatment process. Nitrogen enters the treatment plant as ammonia (NH 3 ). In the aeration tank, aerobic bacteria will nitrify this ammonia to create nitrite (NO 2 ) and nitrate (NO 3 ). In a well-operated plant, a majority of the nitrogen formed is nitrate. The wastewater is then conveyed to an anoxic tank. Anoxic means that the environment where the bacteria are living contains no free dissolved oxygen but there is combined oxygen. The combined oxygen is due to the NO 3 sent to the tank from the aeration tanks. In this anoxic condition facultative bacteria break apart the NO 3 bond and use the oxygen for respiration. The nitrogen molecules combine to form nitrogen gas (N 2 ) which is vented back to the atmosphere. Recall that 78% of the Earth’s atmosphere is N 2 gas.
The anoxic tanks in a BNR system can be placed in a number of different configurations. Since the nitrification process is required first to create the NO 3 one would expect to have the anoxic tanks following the aeration tanks. While there are treatment plants that operate in this manner, it may require additional chemicals as the denitrification process requires a certain amount of carbon for the bacteria to feed off of. The amount of carbon available in the wastewater can be measured by the CBOD 5 or carbonaceous biochemical oxygen demand. At the end of the aeration tank, the aerobic bacteria have significantly reduced the amount of CBOD 5 in the wastewater. A process alternative is having the anoxic tank prior to the aeration tank. The wastewater at the end of the aeration tank that has been nitrified is then recycled back to the anoxic tank. Here, fresh wastewater with high amounts of CBOD 5 is mixed with the NO 3 and the conditions will be just right for denitrification to occur. This will also lessen the aeration demands since some of the CBOD 5 will be reduced in the anoxic tanks before it enters the aeration tank. One drawback is that not all of the nitrate will be captured and sent back to the anoxic tank. Therefore, another process set up is to have alternating zones of anoxic, aerobic, anoxic, aerobic.
Phosphorus
Phosphorus can also be reduced in a well-operated wastewater treatment facility. The amount of phosphorus coming into the facility will determine the level of treatment required. Much of the phosphorus can be removed during the primary and secondary sedimentation processes. If more phosphorus removal is required then treatment plants can utilize enhanced biological phosphorus removal (EBPR). EBPR uses polyphosphate accumulating bacteria (PAO) that, under anaerobic conditions, will accumulate phosphorus in their cells and remove it from the wastewater. A typical configuration of an EBPR process is the RAS and influent wastewater will be mixed in an anaerobic tank where phosphorus reduction occurs. Then the wastewater enters the anoxic tank where the RAS, influent wastewater, and nitrified effluent from the aeration tanks create the ideal environment for denitrification. Then everything is conveyed to the aeration tanks where free dissolved air is added. The incoming ammonia is nitrified to NO 3 and the remaining BOD 5 is reduced.


6: Earth's Interior
Geologists cannot see directly into the interior of the Earth. They have to rely on various techniques and methods to infer the appearance and physical characteristics of earth’s interior. In this section, we will see how the Earth is structured, what the physical characteristics are, and just how this impacts us living on the Earth. The Earth’s interior is the basis for geology. If you recall from the Plate Tectonics section, earth exists as we see it today because of plate tectonics. We also learned how plate tectonics is important in the formation of rock, mountains, volcanoes and earthquakes. Studying the interior of the Earth helps learn about all of these and the processes that helped create the Earth and currently drive plate tectonics .
Thumbnail: Earth cutaway from core to crust, the asthenosphere lying between the upper mantle and the lithospheric mantle. (Public Domain;
).Contributors and Attributions
-
Original content from Kimberly Schulte (Columbia Basin College) and supplemented by Lumen Learning . The content on this page is copyrighted under a Creative Commons Attribution 4.0 International license.


6.5: Interval
- The distance between two pitches is the interval between them. The name of an interval depends both on how the notes are written and the actual distance between the notes as measured in half steps.
The Distance Between Pitches
The interval between two notes is the distance between the two pitches - in other words, how much higher or lower one note is than the other. This concept is so important that it is almost impossible to talk about scales, chords, harmonic progression, cadence, or dissonance without referring to intervals. So if you want to learn music theory, it would be a good idea to spend some time getting comfortable with the concepts below and practicing identifying intervals.
Scientists usually describe the distance between two pitches in terms of the difference between their frequencies. Musicians find it more useful to talk about interval. Intervals can be described using half steps and whole steps. For example, you can say "B natural is a half step below C natural", or "E flat is a step and a half above C natural". But when we talk about larger intervals in the major/minor system, there is a more convenient and descriptive way to name them.
Naming Intervals
The first step in naming the interval is to find the distance between the notes as they are written on the staff . Count every line and every space in between the notes, as well as the lines or spaces that the notes are on. This gives you the number for the interval.
Counting Intervals
To find the interval, count the lines or spaces that the two notes are on as well as all the lines or spaces in between. The interval between B and D is a third. The interval between A and F is a sixth. Note that, at this stage, key signature, clef, and accidentals do not matter at all.
The simple intervals are one octave or smaller.
Simple Intervals
If you like you can listen to each interval as written in Figure 4: prime, second, third, fourth, fifth, sixth, seventh, octave.
Compound intervals are larger than an octave.
Compound Intervals
Listen to the compound intervals in Figure 3: ninth, tenth, eleventh.
Name the intervals.
Solution
Write a note that will give the named interval.
Solution
Classifying Intervals
So far, the actual distance, in half-steps, between the two notes has not mattered. But a third made up of three half-steps sounds different from a third made up of four half-steps. And a fifth made up of seven half-steps sounds very different from one of only six half-steps. So in the second step of identifying an interval, clef, key signature, and accidentals become important.
Listen to the differences in the thirds and the fifths in Figure 8.
So the second step to naming an interval is to classify it based on the number of half steps in the interval. Familiarity with the chromatic scale is necessary to do this accurately.
Perfect Intervals
Primes, octaves, fourths, and fifths can be perfect intervals.
These intervals are never classified as major or minor , although they can be augmented or diminished (see below).
What makes these particular intervals perfect? The physics of sound waves (acoustics) shows us that the notes of a perfect interval are very closely related to each other. (For more information on this, see Frequency, Wavelength, and Pitch and Harmonic Series.) Because they are so closely related, they sound particularly good together, a fact that has been noticed since at least the times of classical Greece, and probably even longer. (Both the octave and the perfect fifth have prominent positions in most of the world's musical traditions.) Because they sound so closely related to each other, they have been given the name "perfect" intervals.
Actually, modern equal temperament tuning does not give the harmonic-series-based pure perfect fourths and fifths. For the music-theory purpose of identifying intervals, this does not matter. To learn more about how tuning affects intervals as they are actually played, see Tuning Systems.
A perfect prime is also called a unison. It is two notes that are the same pitch. A perfect octave is the "same" note an octave - 12 half-steps - higher or lower. A perfect 5th is 7 half-steps. A perfect fourth is 5 half-steps.
Perfect Intervals
Listen to the octave, perfect fourth, and perfect fifth.
Major and Minor Intervals
Seconds, thirds, sixths, and sevenths can be major intervals or minor intervals. The minor interval is always a half-step smaller than the major interval.
Major and Minor Intervals
- 1 half-step = minor second (m2)
- 2 half-steps = major second (M2)
- 3 half-steps = minor third (m3)
- 4 half-steps = major third (M3)
- 8 half-steps = minor sixth (m6)
- 9 half-steps = major sixth (M6)
- 10 half-steps = minor seventh (m7)
- 11 half-steps = major seventh (M7)
Major and Minor Intervals
Listen to the minor second, major second, minor third, major third, minor sixth, major sixth, minor seventh, and major seventh.
Give the complete name for each interval.
Solution
Fill in the second note of the interval given.
Solution
Augmented and Diminished Intervals
If an interval is a half-step larger than a perfect or a major interval, it is called augmented. An interval that is a half-step smaller than a perfect or a minor interval is called diminished. A double sharp or double flat is sometimes needed to write an augmented or diminished interval correctly. Always remember, though, that it is the actual distance in half steps between the notes that determines the type of interval, not whether the notes are written as natural, sharp, or double-sharp.
Some Diminished and Augmented Intervals
Listen to the augmented prime, diminished second, augmented third, diminished sixth, augmented seventh, diminished octave, augmented fourth, and diminished fifth. Are you surprised that the augmented fourth and diminished fifth sound the same?
Write a note that will give the named interval.
Solution
As mentioned above, the diminished fifth and augmented fourth sound the same. Both are six half-steps, or three whole tones , so another term for this interval is a tritone. In Western Music, this unique interval, which cannot be spelled as a major, minor, or perfect interval, is considered unusually dissonant and unstable (tending to want to resolve to another interval).
You have probably noticed by now that the tritone is not the only interval that can be "spelled" in more than one way. In fact, because of enharmonic spellings, the interval for any two pitches can be written in various ways. A major third could be written as a diminished fourth, for example, or a minor second as an augmented prime. Always classify the interval as it is written; the composer had a reason for writing it that way. That reason sometimes has to do with subtle differences in the way different written notes will be interpreted by performers, but it is mostly a matter of placing the notes correctly in the context of the key, the chord, and the evolving harmony. (Please see Beginning Harmonic Analysisfor more on that subject.)
Enharmonic Intervals
Inverting Intervals
To invert any interval, simply imagine that one of the notes has moved one octave, so that the higher note has become the lower and vice-versa. Because inverting an interval only involves moving one note by an octave (it is still essentially the "same" note in the tonal system), intervals that are inversions of each other have a very close relationship in the tonal system.
Inverting Intervals
To find the inversion of an interval
- To name the new interval, subtract the name of the old interval from 9.
- The inversion of a perfect interval is still perfect.
- The inversion of a major interval is minor, and of a minor interval is major.
- The inversion of an augmented interval is diminished and of a diminished interval is augmented.
What are the inversions of the following intervals?
- Augmented third
- Perfect fifth
- Diminished fifth
- Major seventh
- Minor sixth
- Diminished sixth
- Perfect fourth
- Augmented fourth
- Minor second
- Major third
Summary
Here is a quick summary of the above information, for reference.
| Number of half steps | Common Spelling | Example, from C | Alternate Spelling | Example, from C | Inversion |
| 0 | Perfect Unison (P1) | C | Diminished Second | D double flat | Octave (P8) |
| 1 | Minor Second (m2) | D flat | Augmented Unison | C sharp | Major Seventh (M7) |
| 2 | Major Second (M2) | D | Diminished Third | E double flat | Minor Seventh (m7) |
| 3 | Minor Third (m3) | E flat | Augmented Second | D sharp | Major Sixth (M6) |
| 4 | Major Third (M3) | E | Diminished Fourth | F flat | Minor Sixth (m6) |
| 5 | Perfect Fourth (P4) | F | Augmented Third | E sharp | Perfect Fifth (P5) |
| 6 | Tritone (TT) | F sharp or G flat | Augmented Fourth or Diminished Fifth | F sharp or G flat | Tritone (TT) |
| 7 | Perfect Fifth (P5) | G | Diminished Sixth | A double flat | Perfect Fourth (P4) |
| 8 | Minor Sixth (m6) | A flat | Augmented Fifth | G sharp | Major Third (M3) |
| 9 | Major Sixth (M6) | A | Diminished Seventh | B double flat | Minor Third (m3) |
| 10 | Minor Seventh (m7) | B flat | Augmented Sixth | A sharp | Major Second (M2) |
| 11 | Major Seventh (M7) | B | Diminished Octave | C' flat | Minor Second (m2) |
| 12 | Perfect Octave (P8) | C' | Augmented Seventh | B sharp | Perfect Unison (P1) |
Summary Notes: Perfect Intervals
- A perfect prime is often called a unison. It is two notes of the same pitch.
- A perfect octave is often simply called an octave. It is the next "note with the same name".
- Perfect intervals - unison, fourth, fifth, and octave - are never called major or minor
Summary Notes: Augmented and Diminished Intervals
- An augmented interval is one half step larger than the perfect or major interval.
- A diminished interval is one half step smaller than the perfect or minor interval.
Summary Notes: Inversions of Intervals
- To find the inversion's number name, subtract the interval number name from 9.
- Inversions of perfect intervals are perfect.
- Inversions of major intervals are minor, and inversions of minor intervals are major.
- Inversions of augmented intervals are diminished, and inversions of diminished intervals are augmented.