14

A Formalized Ethical Theory

This chapter gives a precise logical formulation of an ethical theory, one that builds on ideas from Immanuel Kant and R. M. Hare.¹ This gives an example of how to use logical systems to formalize larger philosophical views. As in the belief-logic chapter, we’ll systematize consistency norms. But here we feature the golden rule (roughly, “Treat others as you want to be treated”).

¹ For fuller accounts, see my Ethics and the Golden Rule (New York: Routledge, 2013) and Ethics and Religion (New York: Cambridge University Press, 2016), my technical Formal Ethics (New York: Routledge, 1996), or my simpler Ethics: A Contemporary Introduction, 3rd ed. (New York: Routledge, 2018). See also Immanuel Kant’s Groundwork of the Metaphysics of Morals (New York: Harper & Row, 1964) and R. M. Hare’s Freedom and Reason (New York: Oxford University Press, 1963).

We’ll first consider practical reason, highlighting the role of consistency. Then we’ll focus on the golden rule. After seeing problems with the usual wording, we’ll give a better formulation and an intuitive argument for it. Then we’ll add symbols and inference rules to formalize these ideas. We’ll end with a formal proof of the golden rule in logical symbols.

14.1 Practical reason

The most important elements of practical reason are factual understanding, imagination, and consistency. As we decide how to act on important matters, and as we form related desires or moral beliefs, we ought as far as practically possible to be vividly aware of the relevant facts, avoid falsehoods, and be consistent.

Factual understanding requires that we know the facts of the case: circumstances, alternatives, consequences, and so on. To the extent that we’re misinformed or ignorant, our moral thinking is flawed. Of course, we can never know all the facts; and often we have no time to research a problem and must act quickly. But we can act out of greater or lesser knowledge. Other things being equal, a more informed judgment is a more rational one.

We also need to understand ourselves, and how our feelings and moral beliefs originated; we can to some extent neutralize our biases if we understand their origin. Some people are hostile toward a group because they were brought up that way, especially through false stereotypes. Their attitudes might change if they understood the source of their hostility and broadened their experience 0313 and knowledge; if so, then their attitudes are less rational, since they exist because of a lack of experience and self-knowledge.

Imagination (role reversal) is a vivid and accurate awareness of what it would be like to be in the place of those affected by our actions. This differs from just knowing facts. So in dealing with poor people, besides knowing facts about them, we also need to appreciate and envision what these facts mean to their lives; movies, literature, and personal experience can help us to visualize another’s life. We also need to appreciate future consequences of our actions on ourselves; knowing that drugs would have harmful effects on us differs from being able to imagine these effects in a vivid and accurate way.

Consistency (which we’ll explore in the next section) demands, among other things, a coherence between our beliefs, our ends and means, and our moral judgments and how we live. We need all the dimensions of moral rationality working together for our practical thinking to be fully reasonable; consistency is important but isn’t everything. Appeals to consistency in ethics are often distressingly vague; my goal here is to clarify and defend consistency norms.

The most important part of practical reason is the golden rule. As we decide how to act toward others, we ought as far as practically possible to be vividly aware of the relevant facts (especially about how our action affects the other person and what it would be like to be treated that way), avoid falsehoods (about this), and be consistent (so we don’t treat another as we’re unwilling that we be treated in the same situation).

14.2 Consistency

Consistency is the basis for key elements of practical reason, including reflective equilibrium, ends-means rationality, and the golden rule. Our belief-logic chapter touched on these three consistency norms:¹

¹ We noted at the end of the last chapter that consistency duties require qualifiers, like “insofar as you’re able to be consistent in these ways and no disaster would result from so doing ….” This also applies to the golden rule. We’ll regard such qualifiers as implicit throughout.

· Logicality: Avoid inconsistency in beliefs.

· Ends-means consistency: Keep your means in harmony with your ends.

· Conscientiousness: Keep your actions, resolutions, and desires in harmony with your moral beliefs.

Our belief logic contains logicality norms forbidding inconsistent beliefs:

· (∼◇(A • B) ⊃ ∼(u:A • u:B))

· Don’t combine inconsistent beliefs.

· If A is inconsistent with B, then don’t combine believing A with believing B.0314

· (☐(A ⊃ B) ⊃ ∼(u:A • ∼u:B))

· Don’t believe something without believing what follows from it.

· If A logically entails B, then don’t combine believing A with not believing B.

Consistency pushes us toward a reflective equilibrium in our thinking between principles and concrete judgments. Suppose I accept an appealing moral principle but reject an unappealing concrete judgment that it logically entails. Then something has to give; I have to reject the principle or accept the concrete judgment. Before deciding which to do, I need to investigate the principle further. Much moral thinking follows this reflective-equilibrium pattern.

Our belief logic can prove this ends-means consistency argument (§13.4b #3):

· ☐(E ⊃ (N ⊃ M))

· ∴ ∼((u:E • u:N) • ∼u:M)

· “Attain this end” entails “If taking this means is needed to attain this end, then take this means.”

· ∴ Don’t combine wanting to attain this end, believing that taking this means is needed to attain this end, and not acting to take this means.¹

¹ If we added “[c]” for causal necessity (see Arthur Burks’s Chance, Cause, Reason (Chicago: University of Chicago Press, 1977)) to our system, then “∼((u:E • u:[c](∼M ⊃ ∼E)) • ∼u:M)” could be provable by itself: “Don’t combine wanting to attain this end, believing that taking this means is needed to attain this end, and not acting to take this means.”

Ends and means are important to human life. We have many goals – including food, shelter, health, companionship, and meaningfulness. Practical reason has us try to understand our goals, investigate how to satisfy them, satisfy ends-means consistency, and reject ends or means that lead us to violate golden-rule consistency.

Our belief logic also can prove conscientiousness principles that prescribe a harmony between our moral beliefs and how we live. Here’s an example:

· ∼(u:OAu • ∼u:Au))

· Don’t combine believing that you ought to do A with not acting to do A.

This is a formal ethical principle – an ethical principle that can be formulated using the abstract notions of our logical systems plus variables (like “u” and “A,” which stand for any person and action). All our consistency requirements are formal in this sense.

Consistency is important in criticizing norms. Suppose I was taught to discriminate against short people and to believe shortism: “All short people ought to be beat up, just because they’re short.” Now shortism entails “If I were short, then I ought to be beat up”; so consistency in beliefs commits me to accepting this too. But then, by conscientiousness, I’m committed to desiring that if I were short then I be beat up. So consistency forbids this combination: 0315

· I believe “All short people ought to be beat up, just because they’re short.”

· I don’t desire that if I were short then I be beat up.

When I understand short people (including how it feels for them to be beat up) and how my negative attitudes about them originated (through social indoctrination), and when I vividly imagine myself being beaten up in their place, then I likely won’t desire that if I were short then I be beat up. But then I’m inconsistent in accepting shortism. The same general approach – which may remind us of the golden rule – can be used to counter other discriminatory principles (racial, religious, gender, sexual orientation, etc.).

Here are three further formal consistency requirements:

· Impartiality: Make similar evaluations about similar actions, regardless of the individuals involved.¹

· Golden rule: Treat others only as you consent to being treated in the same situation.

· Formula of universal law: Act only as you’re willing for anyone to act in the same situation – regardless of imagined variations of time or person.

¹ I defend only a weak impartiality, not a strong utilitarian impartiality that claims that we ought to promote everyone’s good equally. Weak impartiality lets us accept that we ought to have greater concern for our children than for strangers, so long as we accept that other parents also ought in similar cases to have a greater concern for their children.

We’ll add logical machinery for all three, but mostly focus on the golden rule.

14.3 The golden rule

GR says “Treat others as you want to be treated.” GR is a global standard, endorsed by nearly every religion and culture, important for professionals and families across the planet, and a key part of a growing global-ethics movement.

Here’s a story to introduce GR.¹ There once was a grandpa who lived with his family. As Grandpa grew older, he began to slobber and spill his food; so the family had him eat alone. When he dropped his bowl and broke it, they scolded him and got him a cheap wooden bowl. Grandpa was so unhappy. Now one day the young grandson was working with wood. “What are you doing?” Mom and Dad asked. “I’m making a wooden bowl,” he said, “for when you two get old and must eat alone.” Mom and Dad then looked sad and realized how they were mistreating Grandpa. So they decided to keep quiet when he spills his food and let him eat with the family.

¹ This traditional “The old man and his grandson” story was published by the Brothers Grimm in 1812 and is online (http://www.gutenberg.org/ebooks/2591).

The heart of the golden rule is switching places. You step into another’s shoes. What you do to Grandpa, you imagine being done to you. You ask, “Am I willing that if I were in the same situation then I be treated that same way?” 0316

The golden rule seems simple. But the usual loose wordings invite objections; many academics dismiss GR as a folksy proverb that self-destructs when analyzed carefully. But I think that we just need to understand GR more clearly. I put my improved wording on a shirt.² It has “the golden rule” with symbols for eight GR religions (Bahá’í, Buddhism, Christianity, Confucianism, Hinduism, Islam, Judaism, and Taoism). It also has my GR formula:

² You can get your own golden-rule shirt, in many styles and colors, from my http://www.harryhiker.com/gr GR Web page. This popular page also has GR information, videos, stories, chronology, links, and so on.

My formula is intended to help us apply GR to difficult cases.

My GR formula commands consistency. It demands a fit between my act toward another and my desire about how I’d be treated in the same situation. It doesn’t replace other moral norms or theories, or give all the answers. It doesn’t say specifically what to do (so it doesn’t command bad actions if we have flawed desires); instead, it forbids an inconsistent combination:

· I do A to another.

· I’m unwilling that if I were in the same situation then A be done to me.¹

¹ “Unwilling” here can be taken as “objecting to.” Then the forbidden combination is: (1) I do A to another and (2) I object to the idea of A being done to me in the same situation. If we’re playing chess, I object to the idea of your cheating to beat me (I’m unwilling that you do this) but I don’t object to the idea of your beating me if you do so fairly (I’m in this sense “willing” that you do this). (I thank Tom Carson for this clarification and example.)

GR, far from being vague, is a precise consistency test. Suppose I force Grandpa to eat alone. I switch places in my mind: I imagine that I am forced to eat alone in the same situation. Do I condemn this same act done to me? Then I condemn how I treat Grandpa. I condemn how I treat another, if I condemn the same act when I imagine it done to me in the same situation.

People who reject GR usually understand it crudely, often as:

Literal GR: If you want X to do A to you, then do A to X.

The literal GR “(u:Axu ⊃ Aux)” has no same-situation clause and it commands a specific act (instead of forbidding an inconsistent combination). This literal GR often works well. Suppose you want Lucy to be kind to you; then you’re to be kind to her. Or suppose you want Adam not to hurt you (or rob you, or be rude to you); then you aren’t to do these things to him. These applications seem sensible. But the literal GR can lead to absurdities in two ways. 0317

First, you may be in a different situation from the other person. Consider this instance of the literal GR:

Suppose your father is hard of hearing: If you want your father not to speak more loudly to you (who hear well), then don’t speak more loudly to him.

This ignores differences between you and your father. To get around the problem, you need a same-situation qualifier: “How do I desire that I’d be treated if I were in the same situation as my father (and thus hard of hearing)?” You desire that if you were in his same situation then people would speak loudly to you; so you speak loudly to him.

We can take “same” situation here as “exactly similar” or “relevantly similar.” In the first case, I imagine myself in my father’s exact place (with all his properties). In the second, I imagine myself having those properties of my father (such as being hard of hearing) that I think are or might be relevant to deciding how to speak to him. Either approach works fine.

Here’s another case where the literal GR leads to problems:

To a patient: If you want the doctor to remove your appendix, then remove the doctor’s appendix.

Again, we need a same-situation qualifier. The patient clearly doesn’t desire that if he were in the place of his doctor (with a healthy appendix), then his appendix be removed by a sick patient ignorant of medicine. As you apply GR, ask this:

Am I willing that if I were in the same situation then this be done to me?

The other person’s situation includes likes and dislikes. So if you’re a waiter who hates broccoli, but your customer likes and orders it, then you imagine being served broccoli in a hypothetical situation where you like and order it.

GR is about our present reaction to a hypothetical situation; it isn’t about how we’d react if we were in that situation. Suppose I have a two-year-old son, little Will, who puts fingers into electrical outlets. I try to discourage him from doing this, but nothing works. Finally, I decide that I need to punish him when he does it. I want to see if I can punish him without violating GR. I should ask this:

Am I willing that if I were in the same situation as little Will then I be punished?

I’d answer yes (since punishment would likely have saved my life). I might add, “I’m thankful that my parents punished me in such cases, even though I wasn’t pleased then.” So here I can punish my child without breaking GR, since I’m willing that if I were in the same situation then I be treated the same way. 0318

I’ve been underlining “willing that if,” because this phrase guards against a common GR misunderstanding, one that would force us to do whatever the other person wants. People often ask, wrongly, “If I were in the other person’s place, how would I then want to be treated?” Now if you were in little Will’s place (not knowing about electricity and not wanting to be punished), then you wouldn’t want to be punished. Misapplying GR, we’d conclude that we shouldn’t punish Will for putting his fingers into outlets. So it’s better to apply GR as explained above. I can punish little Will (to save his life), since I’m now willing that if I were in his situation then I be punished. In asking the GR question, say “willing that if”:

Am I willing that if I were in the same situation then this be done to me?

Immanuel Kant’s 1785 objection to GR rests on this confusion. Here you’re a judge, about to sentence a dangerous criminal to jail. The criminal protests and appeals (incorrectly) to GR: “If you were in my place, you wouldn’t want to be sent to jail; so by the golden rule you can’t send me to jail.” You should respond: “I can send you to jail, because I’m now willing that if I were in your place (as a dangerous criminal) then I be sent to jail.” You could add, “If I do such things, then please send me to jail too!”¹

¹ Groundwork of the Metaphysics of Morals, trans. H. Paton (New York: Harper & Row, 1964), p. 97 footnote. GR requires wide scope, roughly, “I desire that if A happened then B be done” “i:(A ⊃ B),” instead of “If A happened then I’d desire that B be done” “(A ⊃ i:B).”

Sometimes we need to act against what others want. We may need to stop a baby who wants to put fingers into outlets, refuse a salesperson who wants to sell us overpriced products, fail a student who doesn’t work, defend ourselves against an attacker, or jail a dangerous criminal. GR lets us act against what others want, as long as we’re now willing that if we were in their situation then we be treated similarly.

Recall that the literal GR can lead to absurdities in two ways. We dealt with the different-circumstances problem by adding a same-situation clause. A second problem is that the literal GR can tell us to do bad things if we have flawed desires about how we’re to be treated. I’ll give three examples.

There once was a woman named Electra. Electra wanted to follow GR, but she got her facts wrong; she thought electrical shocks were pleasant. Since she wanted others to shock her, she applied the literal GR and shocked them:

To Electra (who thinks electrical shocks are pleasant): If you want others to give you electrical shocks, then give them electrical shocks.

Given flawed desires, the literal GR can command evil actions. 0319

We’ll use a triple strategy for dealing with flawed desires. (1) Emphasize that GR, instead of telling us specifically what to do, just forbids a combination:

· I give electrical shocks to another.

· I’m unwilling that if I were in the same situation then electrical shocks be given to me.

Since the consistency GR doesn’t say specifically what to do, it doesn’t tell Electra to do evil things (like shock others).

(2) Emphasize that GR consistency, to lead reliably to right action, needs to combine with other things, like knowledge and imagination. If we’re misinformed, then we might do evil things without violating GR consistency. Here Electra shocks others (an evil thing) but satisfies GR consistency (she’s willing that she be shocked in similar cases), since she’s misinformed and thinks these shocks are pleasurable.

(3) Use reason against flawed desires. Here we’d show Electra that electrical shocks are painful (perhaps by giving her a small one). Once she understands this, GR consistency will lead her away from shocking others.

Here’s another example. Mona hates herself and wants others to hate her; so, following the literal GR, she hates others. (1) Again, the correctly formulated GR just forbids a combination and so doesn’t prescribe that she hate others. (2) GR consistency, to lead reliably to right action, needs to combine with other things (like knowledge, imagination, and here a healthy self-love). (3) We can use reason against Mona’s flawed desires. We can try to help Mona understand why she hates herself and how to neutralize this hatred – by not fixating on her negatives, by seeing herself and her good points in a more balanced way, and, if she’s a believer, by appreciating that God loves her. Once Mona regains a healthy self-love, GR consistency will lead her more readily to love others.

Or suppose Adolph is a Nazi who so hates Jews that he kills them and desires that he be killed if he were Jewish (or found to be Jewish). The literal GR would tell Adolph to kill Jews:

To Adolph (a Jew-hating Nazi): If you want others to kill you if you were Jewish, then kill others who are Jewish.

Again, we can make three points. (1) GR, properly formulated, doesn’t command specific actions but instead just forbids an inconsistent combination:

· I kill others just because they’re Jewish.

· I’m unwilling that if I were Jewish then I’d be killed just because I’m Jewish.

Since the consistency GR doesn’t say specifically what to do, it doesn’t tell Adolph to kill Jews. (2) GR consistency, to lead reliably to right action, has to combine with other things (like knowledge, imagination, and here rational desires). (3) We can use reason against Adolph’s flawed desires. We can try to 0320 help him understand why he hates Jews so much, even desiring that he be killed if he were found out to be Jewish. His anti-Jewish hatred likely has its source in things that can be rationally criticized. Maybe Adolph thinks Aryans are superior to Jews and racially pure; we can criticize this on factual grounds. Or maybe Adolph was taught to hate Jews by his family and friends; maybe they hated Jews, called them names, and spread false stereotypes about them. And so his anti-Jewish desires likely came from false beliefs and social conditioning; such flawed desires would diminish if he understood their origin and broadened his experience and knowledge of Jews in an open and personal way. With greater rationality, Adolph wouldn’t desire that he’d be killed if found out to be Jewish – and GR would be a powerful tool against his racism.

While this example was about a Nazi, the same idea applies to those who desire that they be mistreated if they were black, female, gay, or whatever. Such desires are likely flawed (as based on a social conditioning that uses false beliefs and stereotypes) and would be given up if we expanded our knowledge and experience of the group in an open and personal way.

As you apply the golden rule, keep in mind that it doesn’t work alone. KITA (Know-Imagine-Test-Act) is an acronym to help us remember some key elements for using GR wisely:

KITA: Know Imagine Test Act

Know: “How would my action affect others?”

Imagine: “What would it be like to have this done to me in the same situation?”

Test for consistency: “Am I now willing that if I were in the same situation then this be done to me?”

Act toward others only as you’re willing to be treated in the same situation.

To lead reliably to right action, GR consistency needs to build on things like knowledge, imagination, creativity, rationalized desires, and a healthy self-love.

GR can fit many perspectives. Philosophically, GR could be a self-evident truth (or derivable from such), God’s will, a cultural convention, a social contract for mutual advantage, socially useful, reflecting our feelings, promoting self- interest (since it brings self-respect and better treatment from others), and so on. Religiously, GR is part of Bahá’í, Buddhism, Christianity, Confucianism, Hinduism, Islam, Judaism, Sikhism, Taoism, Zoroastrianism, and so on. Diverse groups share GR. The golden rule can be a point of unity in a diverse world.

14.4 Starting the GR proof

What sort of inconsistency do we have when we violate the golden rule? Clearly we don’t have an inconsistency between beliefs; what clashes here isn’t beliefs 0321 but rather actions and desires. But why is it inconsistent to violate GR?

Consistency in a broad sense includes things like ends-means consistency, conscientiousness, and impartiality. GR follows from conscientiousness and impartiality. Suppose that you’re conscientious and impartial in the required senses, and yet you want to steal Detra’s bicycle. Being conscientious, you won’t steal her bicycle unless you think this act is all right (permissible). Being impartial, you won’t think this act is all right unless you think that if you were in the same situation then it would be all right for your bike to be stolen. Being conscientiousness, you won’t think this unless you’re willing that if you were in the same situation then your bike be stolen. So if you’re conscientious and impartial, then you won’t steal Detra’s bicycle unless you’re willing that your bike be stolen in the same situation. Here’s a diagram:

You steal Detra’s bicycle

then if you’re conscientious ⇒

You believe it would be all right for you to steal her bicycle

then if you’re impartial ⇒

You believe that if you were in the same situation then it would be all right for your bicycle to be stolen

then if you’re conscientious ⇒

You’re willing that if you were in the same situation then your bicycle be stolen

So if we’re conscientious and impartial, then we’ll follow GR: we won’t do something to another unless we’re willing that it be done to us in the same situation. If we violate GR, then we violate either conscientiousness or impartiality or both. So if we assume that we ought to be conscientious and impartial, then we can deduce that we ought to follow the golden rule.

So my GR can be based on an abstract argument; similar reasoning justifies many variations. We can consider someone else we care about (maybe our daughter) on the receiving end of the action. We can give consistency conditions, not for doing something, but for wanting something to be done or for holding a moral belief. A multi-party GR has us satisfy GR toward each affected party. A future-regard form has us imagine ourselves suffering the future consequences of our present action. A self-regard form has us imagine someone we care about doing the self-destructive thing we’re doing to ourselves. My formula of universal law is a generalized GR that contains many of these: “Act only as you’re willing for anyone to act in the same situation, regardless of where or when you imagine yourself or others.”

So GR follows from the requirements to be conscientious and impartial. But why be conscientious and impartial? Why care about consistency at all?

Different views could answer differently. Maybe we ought to be consistent because this is inherently right; our minds see consistency as the first duty of a 0322 rational being. Or maybe we accept consistency because it’s commanded by God, useful to social life, accords with how we want to live, or promotes our self-interest (since inconsistency brings lowered self-respect, painful “cognitive dissonance,” and social sanctions). I’ll abstract from such issues here and assume only that there’s some strong reason to be consistent, in a broad sense that includes being conscientious and impartial. I won’t worry about the details. I’m trying to develop consistency norms that appeal to a wide range of approaches – even though these approaches may explain and justify the norms differently.

To incorporate GR into our logical framework, we need to add requirements to be conscientious and impartial. Our belief logic already has part of the conscientiousness requirement. We already can prove the imperative analogue of the first step of our GR argument – “Don’t act to do A to X without believing that it’s all right for you to do A to X”:¹

¹ See my “Acting commits one to ethical beliefs,” Analysis 42 (1983), pp. 40–3.

However, we can’t yet prove the imperative analogue of our GR argument’s third step – which also deals with conscientiousness:

Don’t believe that if you were in the same situation then it would be all right for X to do A to you, without being willing that if you were in the same situation then X do A to you.

The hard part here is to symbolize “in the same situation.” If we ignore this for the moment, then what we need is this: “Don’t believe that it would be all right for X to do A to you without being willing that X do A to you.” We’ll interpret “being willing that A be done” as “accepting ‘A may be done.’” The permissive “A may be done” here isn’t another way to say “A is all right.” Instead, it’s a member of the imperative family, but weaker than “Do A,” expressing only one’s consent to the action. We’ll symbolize “A may be done” as “MA.” Then we can symbolize the imperative mentioned above as follows:

· ∼(u:RAxu • ∼u:MAxu)

· Don’t combine believing “It would be all right for X to do A to me” with not accepting “X may do A to me.”

0323 To prove this, we need a principle like “☐(RA ⊃ MA)” – which says that a permissibility judgment entails the corresponding permissive. This is like the prescriptivity principle (“Hare’s Law”) discussed in §12.4, which says that an ought judgment entails the corresponding imperative: “☐(OA ⊃ A).”¹

¹ On “☐(RA ⊃ MA),” see my “How incomplete is prescriptivism?” Mind 93 (1984), pp. 103–7. “☐(RA ⊃ MA)” and “☐(OA ⊃ A)” affirm that violating conscientiousness is logically inconsistent. One who rejected this but still thought that violating conscientiousness was objectionable, could endorse the weaker “(RA ⊃ MA)” and “(OA ⊃ A)” – and this would suffice for the GR proof at the end of this chapter.

Our biggest task is to symbolize and prove the impartiality requirement and the imperative analogue of our GR argument’s second step:

· ∼(u:RAux • ∼u:…)

· Don’t combine believing that it would all right for you to do A to X with not believing that it if you were in the same situation then it would be all right for X to do A to you.

We need to replace “…” with a formula that means “it would be all right for X to do A to you in the same situation.” And we need an inference rule to reflect universalizability – which is one of the few principles on whose truth almost all moral philosophers agree.

The universalizability principle (U) says that whatever is right (wrong, good, bad, etc.) in one case would also be right (wrong, good, bad, etc.) in any exactly or relevantly similar case, regardless of the individuals involved. Here are three equivalent formulations for “all right” (similar forms work for “ought”):

Universalizability (U)

If it’s all right for X to do A, then it would be all right for anyone else to do A in the same situation.

If act A is permissible, then there is some universal property (or conjunction of such properties) F, such that: (1) act A is F, and (2) in any actual or hypothetical case every act that is F is permissible.

(RA ⊃ (∃F)(FA • ■(X)(FX ⊃ RX)))

The second phrasing, which is more technically precise, uses the notion of a “universal property.” A universal property is any non-evaluative property describable without proper names (like “Gensler” or “Chicago”) or pointer terms (like “I” or “this”). Suppose that I’m tempted to steal Pat’s new computer. This possible act has several properties; for example, it’s:

· wrong (evaluative term),

· an act of stealing Pat’s computer (proper name), and

· something I would be doing (pointer word).

0324 These aren’t universal, since they use evaluative terms, proper names, or pointer words. But the act also has universal properties; for example, it is:

· an act of stealing a new computer from one’s neighbor,

· an act whose agent has blue eyes, and

· an act that would greatly distress the computer’s owner.

U says that the morality of an act depends on its universal properties (like those of the second group), properties expressible without evaluative terms, proper names, or pointer words. Two acts with the same universal properties must have the same moral status, regardless of the individuals involved.

Here’s an important corollary of universalizability:

· U* If it’s all right for you to do A to X, then it would be all right for X to do A to you in the exact same situation.

· If it’s all right for you to do A to X, then, for some universal property F, F is the complete description of your-doing-A-to-X in universal terms, and, in any actual or hypothetical case, if X’s-doing-A-to-you is F, then it would be all right for X to do A to you.

· (RAux ⊃ (∃F)(F*Aux • ■(FAxu ⊃ RAxu)))

U* relates closely to the second step in our argument for GR.

14.5 GR logical machinery

Now we add symbols for formulating GR:

· letters for universal properties and for actions,

· “M” (“may”) for permissives,

· “■” (“in every actual or hypothetical case”) for hypothetical cases, and

· “*” for the complete description of an act in universal terms.

We also add inference rules. This section will get complicated; you may need to read it a couple of times to follow what’s happening.

First, we’ll use letters of two new sorts (both can be used in quantifiers):

· “F,” “G,” “H,” and these with primes stand for universal properties of actions (including conjunctions of such properties).

· “X,” “Y,” “Z,” and these with primes stand for actions.

These examples use letters for universal properties:

· 0325

· FA = Act A is F (e.g., act A is an act of stealing)

· Act A has universal property F

· (FA ⊃ ∼RA) = If act A is an act of stealing, then act A is wrong

· GA = Act A is an act of a blue-eyed philosophy teacher stealing a bicycle from an impoverished student

We translate “FA” as “Act A is F” (not as “Imperative ‘Do A’ is F”). This next example uses a universal-property quantifier:

· (F)(FA ≡ FB) = Acts A and B have all the same universal properties

· For every universal property F, act A has property F if and only if act B has property F

These examples use action quantifiers:

· (∃X)FX = Some act has universal property F

· For some act X, X has universal property F

· (X)(FX ⊃ OX) = Every act that is F ought to be done

· For every act X, if act X is F, then act X ought to be done

· (X)(∃F)FX = Every act has some universal property

· For every act X there’s some universal property F, such that act X is F

These new symbols require new formation rules:

1. The result of writing “F,” “G,” “H,” or one of these with primes, and then an imperative wff is itself a descriptive wff.

2. The result of writing “(x” or “(∃,” and then “F,” “G,” “H,” “X,” “Y,” “Z,” or one of these with primes, and then “ ) ” is a quantifier.

Assume expanded versions of our quantifier rules for the new quantifiers. We have to substitute the right sort of thing for the quantified letter:

1. For individual variables: x, y, z, x’, …, substitute individual constants: a, b, c, d, …

2. For universal-property variables: F, G, H, F’, …, substitute universal-property letters not bound to quantifiers: F, G, H, F’, ….

3. For action variables: X, Y, Z, X’, …, substitute imperative wffs: Aa, B, Axy, ….¹ 0326

¹ The last case requires two technical provisos. Suppose that we drop a quantifier containing an action variable and substitute an imperative wff for the variable. Then we must be sure that (1) this imperative wff contains no free variable that also occurs in a quantifier in the derived wff, and (2) if we dropped an existential quantifier, this substituted imperative wff must be an underlined capital letter that isn’t an action variable and that hasn’t occurred before in the proof.

When “M” is prefixed to an imperative wff, we’ll translate it as “may”:²

² Capital letters have various uses, depending on context. In “((M • Ma) ⊃ (Mbc • MA)),” for example, “M” is used first for a statement, then for a property of an individual, then for a relation between individuals, and finally for “may.” It’s usually clearer to use different letters.

3. The result of prefixing an imperative wff with “M” is a wff.

· MA = Act A may be done

· MAxu = X may do A to you

· u:MAxu

· = You accept “X may do A to me”

· You consent to X’s doing A to you

· You’re willing that X do A to you

Permissives like “MA” are weaker members of the imperative family. They express our consent to the act, but not necessarily our positive desire that the act take place. We can consistently consent both to the act and to its omission – saying “You may do A and you may omit A.” Here are further wffs:

· ∼M∼A = Act A may not be omitted

· u:∼M∼Axu

· = You accept “X may not omit doing A to me”

· You demand that X do A to you

“MA” is weaker and “∼M∼A” is stronger than “A.”¹

¹ For more on permissives, see my Formal Ethics (London: Routledge, 1996), pp. 185–6, and my “How incomplete is prescriptivism?” Mind 93 (1984), pp. 103–7.

Inference rule G1 is the principle that “A is all right” entails “A may be done.” G1 holds regardless of what imperative wff replaces “A”:²

² Thinking that an act is all right commits one to consenting to the idea of it being done (being willing that it be done). We also could use words like “approve,” “accept,” “condone,” or “tolerate” – in one sense of these terms. The sense of “consent” that I have in mind refers to an inner attitude incompatible with inwardly objecting to (condemning, disapproving, forbidding, protesting, objecting to) the act. Consenting here is a minimal attitude and needn’t involve favoring or advocating or welcoming the act. It’s consistent to both consent to the idea of A being done and also consent to the idea of A not being done.

G1

RA → MA

Given this and the rules for “M,” “O,” and “R,” we also can prove the reverse entailment from “MA” to “RA.” Then either logically entails the other; so accepting one commits us to accepting the other. But the distinction between the two doesn’t vanish. “RA” is true or false; to accept “RA” is to believe that something is true. But “MA” isn’t true or false; to accept “MA” isn’t to believe something but to will something, to consent to the idea of something being done.

Some of our inference rules for “M” (and later “■”) involve new kinds of world. A world prefix is now any string of zero-or-more instances of letters from 0327 the set – where is the set of small letters. Here “P,” “PP,” “PPP,” and so on are “permission worlds,” much like deontic worlds. A permission world that depends on a given world W1 is a possible world that contains the indicative judgments of W1 and some set of imperatives prescribing actions jointly permitted by the permissives of W1.

Inference rules G2 to G4 (which won’t be used in our GR proof) govern permissions and are much like the deontic rules. G2 and G3 hold regardless of what pair of contradictory imperative wffs replaces “A” / “∼A”:

G2

∼ MA → P ∴ ∼A,

use a blank or any string of P’s

In G2, the world prefix of the derived line must be either the same as that of the earlier line or else the same except that it adds one or more P’s at the end.

G3

MA → P ∴ A,

use a new string of P’s

In G3, the world prefix of the derived line must be the same as that of the earlier line except that it adds a new string (a string not occurring in earlier lines) of one or more P’s at the end. G4 mirrors the deontic indicative transfer rule; it holds regardless of what descriptive or deontic wff replaces “A”:

G4

P ∴ A → A

In G4, the world prefixes in the derived and deriving lines must be identical except that one ends in one or more additional P’s.

“■” is a modal operator somewhat like “☐”:

4. The result of prefixing any wff with “■ ” is a wff.

“■” translates as “in every actual or hypothetical case” or “in every possible world having the same basic moral principles as those true in the actual world.” Here’s a wff using “■”:

· ■(FA ⊃ OA)

· = If act A is or were F, then act A ought to be done

· In every actual or hypothetical case, if act A is F, then act A ought to be done

Suppose that, while act A may or may not have property F (e.g., it may or may not maximize pleasure), still, if it did, then it would be what ought to be done. We’ll use “■(FA ⊃ OA)” for this idea. “(FA ⊃ OA)” is too weak to express thi 0328 (since this wff is trivially true if “FA” is false); “☐(FA ⊃ OA)” is too strong (because there’s no such entailment). So we’ll use “■” to formulate claims about what would be right or wrong in hypothetical situations (such as imagined exactly reversed situations).

We can now symbolize the universalizability principle:

U If act A is permissible, then there’s some universal property (or conjunction of such properties) F, such that: (1) act A is F, and (2) in any actual or hypothetical case every act that is F is permissible.

(RA ⊃ (∃F)(FA • ■ (X)(FX ⊃ RX)))

G5 and G6 are the “all right” and “ought” forms of the corresponding inference rules. These hold regardless of what imperative wff replaces “A,” what universal-property variable replaces “F,” and what action variable replaces “X”:

G5 & G6

RA → (∃F)(FA • ■ (X)(FX ⊃ RX))

OA → (∃F)(FA • ■ (X)(FX ⊃ OX))

In G5 and G6, the world prefix of the derived and deriving lines must be identical and must contain no “W.” The proviso prevents us from being able to prove the controversial idea that violations of universalizability are self-contradictory.

The rules for “■” resemble those for “☐.” Recall that our expanded world prefixes can use “H,” “HH,” and “HHH”; these represent hypothetical situation worlds, which are possible worlds having the same basic moral principles as those of the actual world (or whatever world the H-world depends on). G7 and G8 hold regardless of what pair of contradictory wffs replaces “A” / “∼A”:

G7

■A → H ∴ A,

use a blank or any string of H’s

In G7, the world prefixes in the derived and deriving lines must either be the same or be the same except that one adds one or more H’s at the end.

G8

∼■A → H ∴ ∼ A,

use a new string of H’s

In G8, the derived line’s world prefix must be the same as that of the earlier line except that it adds a new string (a string not occurring in earlier lines) of one or more H’s at the end. Rule G9 (which won’t be used in our GR proof) says that 0329 “☐” and “■” are equivalent when prefixed to descriptive wffs; this holds regardless of what descriptive wff replaces “A”:

G9

■A ↔ ☐A

Our final symbol is “*”; this is used with universal-property letters to represent the complete description of an action in universal terms. Here’s the rule for constructing wffs with “*,” with an example:

5. The result of writing “F,” “G,” “H,” or these with primes, then “*,” and then an imperative wff is itself a descriptive wff.

· F*A = F is the complete description of act A in universal terms

· F is the description of act A in universal terms which includes all the universal properties of act A

“F*A” means the same as this longer wff:

· (FA • (G)(GA ⊃ ☐(X)(FX ⊃ GX)))

· Act A is F, and every universal property G that A has is included as part of F

· Act A is F, and, for every universal property G that A has, it’s logically necessary that every act that’s F also is G

We adopt the corresponding inference rule G10, which lets us go back and forth between “F*A” and this longer wff. G10 holds regardless of what distinct universal-property letters replace “F” and “G,” what imperative wff replaces “A,” and what action variable replaces “X”:

G10

F*A ↔ (FA • (G)(GA ⊃ ☐(X)(FX ⊃ GX)))

Rule G11, our final inference rule, says that every act has a complete description in universal terms (even though it may be too long to write down). G11 is an axiom; it lets us put wff “(X)(∃F)F*X” on any line of a proof:

G11

(X)(∃F)F*X

We’ll use “*” in symbolizing “exactly similar situation.” Let “Amx” represent the act of my attacking X and “F” be its complete description:

· F*Amx = My-attacking-X has complete universal description F

Let’s flesh this out. Let “G,” “G’,” … be my universal properties; these include 0330 properties like being a logician. Let “H,” “H’,” … be X’s universal properties; these might include being an impoverished student. Let “R,” “R’,” … be the relationships between X and me; these might include X’s being my student. Now property F would look like this, which describes the actual situation:

· FAmx = My-attacking-X is an act of someone who is G, G’, … attacking someone who is H, H’, … and related to me in ways R, R’, ….

Now we imagine an exactly similar situation if we imagine the situation where X’s-attacking-me has this same description F:

· FAxm = X’s-attacking-me is an act of someone who is G, G’, … attacking someone who is H, H’, … and related to X in ways R, R’, ….

In this imagined exactly similar situation, X is in my exact place – and I am in X’s exact place. All our universal properties and relationships are switched.

We can now symbolize the reversed-situation corollary of universalizability:

· U*. If it’s all right for you to do A to X, then it would be all right for X to do A to you in the exact same situation.

· If it’s all right for you to do A to X, then, for some universal property F, F is the complete description of your-doing-A-to-X in universal terms, and, in any actual or hypothetical case, if X’s-doing-A-to-you is F, then it would be all right for X to do A to you.

· (RAux ⊃ (∃F)(F*Aux • ■(FAxu ⊃ RAxu)))

Also, and most importantly, we can symbolize the golden rule:

· GR. Treat others only as you consent to being treated in the same situation.

· Don’t combine acting to do A to X with being unwilling that if you were in the same situation then A be done to you.

· Don’t combine (1) accepting “Do A to X” with (2) not accepting “For some universal property F, F is the complete description in universal terms of my-doing-A-to-X, and, in any actual or hypothetical situation, if X’s-doing-A-to-me is F, then X may do A to me.”

· ∼(u:Aux • ∼u:(∃F)(F*Aux • ◼(FAxu ⊃ MAxu)))

Here are symbolizations of two related ideas (§14.2): 0331

· Impartiality: Make similar evaluations about similar actions, regardless of the individuals involved.

· Don’t accept “Act A is permissible” without accepting “Any act exactly or relevantly similar to act A is permissible.”

· Don’t accept “Act A is permissible” without accepting “For some universal property F, act A is F and, in any actual or hypothetical situation, any act that is F is permissible.”

· ∼(u:RA • ∼u:(∃F)(FA • ◼(X)(FX ⊃ RX)))

· Formula of universal law: Act only as you’re willing for anyone to act in the same situation – regardless of imagined variations of time or person.¹

· Don’t combine acting to do A with not being willing that any similar action be done in the same situation.

· Don’t combine (1) accepting “Do A” with (2) not accepting “For some universal property F, F is the complete description in universal terms of my doing A, and, in any actual or hypothetical situation, any act that is F may be done.”

¹ My “formula of universal law” resembles Immanuel Kant’s principle. His wording went, “Act only on that maxim through which you can at the same time will that it should be a universal law.” I’m not claiming that Kant explicitly intended his principle in exactly my sense.

· ∼(u:Au • ∼u:(∃F)(F*Au • ◼(X)(FX ⊃ MX)))

This “formula of universal law” is a generalized GR. It applies, for example, to multi-party cases or to cases where my present action can harm my future self.

14.6 The symbolic GR proof

Before we do our GR proof, let’s review the larger picture.

We began this chapter by sketching various dimensions of ethical rationality. Then we narrowed our focus, first to consistency, and then to a single consistency principle – the golden rule. We had to formulate GR carefully to avoid absurd implications. We defended this wording:

Golden rule

Treat others only as you consent to being treated in the same situation.

GR forbids this combination:

· I do A to another.

· I’m unwilling that if I were in the same situation then A be done to me.

We sketched an intuitive GR proof, using the example of stealing Detra’s bicycle. 0332 Then we noted that incorporating GR and its proof into our logical framework requires adding impartiality and strengthening conscientiousness. And so now we’re ready to give a formal proof of the golden rule.

Our proof goes as follows (where justifications that use our new inference rules are in bold type):

While this is a difficult proof, you should be able to follow the individual lines and see that everything follows correctly.

Our proof begins as usual; we assume the opposite of what we want to prove and then try to derive a contradiction. Soon we get lines 4 and 5 (where 5 is 0333 addressed to yourself):

· 4 X may not do A to me in an exactly similar situation.

· 5 Do A to X.

Using line 4, we get these key lines:

· 16 Let H be the complete description of my doing A to X.

· 26 In our imagined situation, X’s-doing-A-to-me is H.

· 27 In our imagined situation, X may not do A to me.

We use line 5 to get “It’s all right for me to do A to X”:

Then we use universalizability on “It’s all right for me to do A to X” to get “Any act relevantly or exactly similar to my-doing-A-to-X would be all right.” We specify that G is the morally relevant complex of properties here; so:

· 12 My-doing-A-to-X has property G.

· 13 Any act that has property G would be all right.

We get a contradiction between lines 27 and 34:

· 16 H is the complete description of my doing A to X. {above}

· 12 My-doing-A to-X has property G. {above}

· 21 ∴ G is part of H – so every act that is H is G. {from 16 & 12}

· 26 In our imagined situation, X’s-doing-A-to-me is H. {above}

· 30 ∴ In our imagined situation, X’s-doing-A-to-me is G. {from 21 & 26}

· 13 Any act that has property G would be all right. {above}

· 33 ∴ In our imagined situation, X’s-doing-A-to-me is all right. {from 30 & 13}

· 34 ∴ In our imagined situation, X may do A to me. {from 33}

Thus ends our proof of the golden rule:¹

¹ For a challenging exercise, prove the impartiality and universal law formulas, as formulated at the end of the previous section. Answers are in the back of the book.

Always treat others as you want to be treated; that is the summary of the Law and the Prophets. (Mt 7:12)