12

Deontic and Imperative Logic

Imperative logic studies arguments with imperatives, like “Don’t do this.” Deontic logic studies arguments whose validity depends on “ought,” “permissible,” and similar notions. We’ll take imperative logic first and then build deontic logic on it.¹

¹ I’ll mostly follow Hector-Neri Castañeda’s approach. See his “Imperative reasonings,” Philosophy and Phenomenological Research 21 (1960): pp. 21–49; “Outline of a theory on the general logical structure of the language of action,” Theoria 26 (1960): pp. 151–82; “Actions, imperatives, and obligations,” Proceedings of the Aristotelian Society 68 (1967–68): pp. 25–48; and “On the semantics of the ought-to-do,” Synthese 21 (1970): pp. 448–68.

12.1 Imperative translations

Imperative logic builds on previous systems and adds two ways to form wffs:

1. Any underlined capital letter is a wff.

2. The result of writing a capital letter and then one or more small letters, one small letter of which is underlined, is a wff.

Underlining (combined with bolding in this e-book version) turns indicatives into imperatives:

Indicative (You’re doing A)

A

Au

Imperative (Do A)

A

Au

Here are some further translations:

· Don’t do A = ∼A

· Do A and B = (A • B)

· Do A or B = (A ∨ B)

· Don’t do either A or B = ∼(A ∨ B) 0268

· Don’t combine doing A with doing B

· = ∼(A • B)

· Don’t both do A and do B

· Don’t combine doing A with not doing B

· = ∼(A • ∼B)

· Don’t do A without doing B

Underline imperative parts but not factual ones:

· You’re doing A and you’re doing B = (A • B)

· You’re doing A, but do B = (A • B)

· Do A and B = (A • B)

· If you’re doing A, then you’re doing B

· = (A ⊃ B)

· If you (in fact) are doing A, then do B

· = (A ⊃ B)

· Do A, only if you (in fact) are doing B

· = (A ⊃ B)

Since English can’t put an imperative after “if,” we can’t read “(A ⊃ B)” as “If do A, then you’re doing B.” But we can read it as the equivalent “Do A, only if you’re doing B.” This means the same as “(∼B ⊃ ∼A)”: “If you aren’t doing B, then don’t do A.”

There’s a subtle difference between these two:

· If you (in fact) are doing A, then don’t do B

· = (A ⊃ ∼B)

· Don’t combine doing A with doing B

· = ∼(A • B)

“A” is underlined in the second but not the first; otherwise, the two wffs would be equivalent. The if-then “(A ⊃ ∼B)” says that if A is done then you aren’t to do B. But the don’t-combine “∼(A • B)” just forbids a combination: doing A and B together. If you’re doing A, it doesn’t follow that you aren’t to do B; maybe you should do B and stop doing A. We’ll see more on this distinction later.

These examples underline the letter for the agent:

· X, do (or be) A = Ax

· X, do A to Y = Axy

These use quantifiers:

· Everyone does A = (x)Ax

· Let everyone do A = (x)Ax

· Let everyone who (in fact) is doing A do B

· = (x)(Ax ⊃ Bx)

· Let someone who (in fact) is doing A do B

· = (∃x)(Ax • Bx)

· Let someone both do A and do B

· = (∃x)(Ax • Bx)

Notice which letters are underlined. 0269

12.1a Exercise: LogiCola L (IM & IT)

Translate these English sentences into wffs; take each “you” as a singular “you.”

If the cocoa is about to boil, remove it from the heat

(B ⊃ R)

Our sentence also could translate as “(B ⊃ Ru)” or “(Bc ⊃ Ruc).”

1. Leave or shut up. [Use L and S.]

2. If you don’t leave, then shut up.

3. Do A, only if you want to do A. [Use A and W.]

4. Do A, only if you want to do A. [This time use Au and Wu.]

5. Don’t combine accelerating with braking.

6. If you accelerate, then don’t brake.

7. If you brake, then don’t accelerate.

8. If you believe that you ought to do A, then do A. [Use A for “You do A” and B for “You believe that you ought to do A.”]

9. Don’t combine believing that you ought to do A with not doing A.

10.If everyone does A, then do A yourself.

11.If you have a headache, then take aspirin. [Hx, Ax, u]

12.Let everyone who has a headache take aspirin.

13.Gensler, rob Jones. [Rxy, g, j]

14.If Jones hits you, then hit Jones. [Hxy, j, u]

15.If you believe that A is wrong, then don’t do A. [Use A for “You do A” and B for “You believe that A is wrong.”]

16.If you do A, then don’t believe that A is wrong.

17.Don’t combine believing that A is wrong with doing A.

18.Would that someone be sick and also be well. [Sx, Wx]

19.Would that someone who is sick be well.

20.Would that someone be sick who is well.

12.2 Imperative proofs

Imperative proofs work much like indicative ones and require no new inference rules. But we must treat “A” and “A” as different wffs. “A” and “∼A” aren’t contradictories; it’s consistent to say “You’re now doing A, but don’t.”

Here’s an imperative argument that follows an I-rule inference:

· If you’re accelerating, then don’t brake.

· You’re accelerating.

· ∴ Don’t brake.

· (A ⊃ ∼B) Valid

· A

· ∴ ∼B

While this seems valid, there’s a problem with calling it “valid.” We earlier defined “valid” using “true” and “false” (§1.2): an argument is valid if it would 0270 be contradictory to have the premises all true and conclusion false. But “Don’t brake” and other imperatives aren’t true or false. So how can the valid/invalid distinction apply to imperative arguments?

We need a broader definition of “valid” that applies equally to indicative and imperative arguments. This one (which avoids “true” and “false”) does the job:

An argument is valid if the conjunction of its premises with its conclusion’s contradictory is inconsistent.

To say that our argument is valid means that this combination is inconsistent:

“If you’re accelerating, then don’t brake; you’re accelerating; brake.”

The combination is inconsistent. So our argument is valid in this new sense.¹

¹ We could equivalently define a valid argument as one in which every set of imperatives and indicatives that’s consistent with the premises also is consistent with the conclusion.

This next argument uses a don’t-combine premise, which makes it invalid:

· Don’t combine accelerating with braking.

· You’re accelerating.

· ∴ Don’t brake.

· ∼(A • B) Invalid

· A

· ∴ ∼B

The first premise forbids us to accelerate and brake together. Suppose we’re accelerating. It doesn’t follow that we shouldn’t brake; maybe, to avoid hitting a car, we should brake and stop accelerating. So the argument is invalid. It’s consistent to conjoin the premises with the contradictory of the conclusion:

Don’t combine accelerating with braking – never do both together; you in fact are accelerating right now; but you’ll hit a car unless you slow down; so stop accelerating right away – and brake immediately.

Here it makes good consistent sense to endorse the premises while also adding the denial of the conclusion (“Brake”).

We’d work out the symbolic argument this way (being careful to treat “A” and “A” as different wffs, almost as if they were different letters):

· * 1 ∼(Aº • B¹) = 1 Invalid

· 2 A¹ = 1

· [ ∴ ∼B¹ = 0

· 3 asm: B

· 4 ∴ ∼A {from 1 and 3}

A, ∼ A, B

On our refutation:

A = 1

A = 0

B = 1

We quickly get a refutation – a set of assignments of 1 and 0 to the letters that make the premises 1 but conclusion 0. Our refutation says this: 0271

You’re accelerating; don’t accelerate; instead, brake.

But our refutation assigns false to the imperative “Accelerate” – even though imperatives aren’t true or false. So what does “A = 0” mean?

We can generically read “1” as “correct” and “0” as “incorrect.” Applied to indicatives, these mean “true” or “false.” Applied to imperatives, these mean that the prescribed action is “correct” or “incorrect” relative to some standard that divides actions prescribed by the imperative letters into correct and incorrect actions. The standard could be of different sorts, based on things like morality, law, or traffic safety; generally we won’t specify the standard.

Suppose we have a propositional-logic argument with imperative letters added. The argument is valid if and only if, relative to every assignment of “1” or “0” to the indicative and imperative letters, if the premises are “1,” then so is the conclusion. Equivalently, the argument is valid if and only if, relative to any possible facts and any possible consistent standards for correct actions, if all the premises are correct then so is the conclusion.

So our refutation amounts to this: we imagine certain facts being true/false and certain actions being correct/incorrect:

· A = 1 “You’re accelerating” is true.

· A = 0 Accelerating is incorrect.

· B = 1 Braking is correct.

Our argument could have all the premises correct but not the conclusion.

Compare the two imperative arguments that we’ve considered:

If you’re accelerating, then don’t brake.

· You’re accelerating.

· ∴ Don’t brake.

· (A ⊃ ∼B) Valid

· A

· ∴ ∼B

· Don’t combine accelerating with braking.

· You’re accelerating.

· ∴ Don’t brake.

· ∼(A • B) Invalid

· A

· ∴ ∼B

Both arguments are the same, except that the first uses an if-then “(A ⊃ ∼B),” while the second uses a don’t-combine “∼(A • B).” Since one argument is valid and the other isn’t, the two wffs aren’t equivalent.

Imagine that you find yourself accelerating and braking, thus wearing down your brakes and wasting energy. Then you violate all three of these imperatives:

· (A ⊃ ∼B) = If you’re accelerating, then don’t brake

· (B ⊃ ∼A) = If you’re braking, then don’t accelerate

· ∼(A • B) = Don’t combine accelerating with braking

The three differ on what to do next. The first tells you not to brake. The second tells you not to accelerate. But the third leaves it open whether you’re to stop 0272 accelerating or stop braking. Maybe you need to brake (and stop accelerating) to avoid hitting another car; or maybe you need to accelerate (and stop braking) to pass another car. The don’t-combine form doesn’t tell a person in this forbidden combination exactly what to do.

Consistency imperatives need the don’t-combine form. Suppose that you’re inconsistent if you combine doing A with doing B. Then:

· ∼(A • B) = Don’t combine doing A with doing B.

This forbids a combination but doesn’t say exactly what to do. Suppose that you’re inconsistently doing A and B together. From this we can’t conclude which you are to change; both of these are invalid:

Don’t combine doing A with doing B.

· You’re doing A.

· ∴ Don’t do B.

· ∼(A • B) Invalid

· A

· ∴ ∼B

· Don’t combine doing A with doing B.

· You’re doing B.

· ∴ Don’t do A.

· ∼(A • B) Invalid

· B

· ∴ ∼A

These inference forms are wrong, even though they may seem correct. Together they’d tell you to give up both A and B. But all you need to do is give up one of these, A or B. The “∼(A • B)” form is logically equivalent to “(∼A ∨ ∼B),” which means “Either don’t do A or don’t do B.”

Suppose that acting to do this is somehow inconsistent with believing that this is wrong. Here’s the corresponding consistency imperative:

· ∼(A • B) = Don’t combine acting to do this with believing that this is wrong

This combination always has a faulty element. If your act is correct, then your belief is wrong; if your belief is correct, then your act is wrong. If you combine this act with this belief, then your act clashes with your belief. How should you regain consistency? This depends on the situation – since either of the two could be faulty; so sometimes it’s better to change your act and sometimes it’s better to change your belief.¹ The don’t-combine form forbids an inconsistency, but it correctly doesn’t tell a person in this forbidden combination exactly what to do. For this reason, it’s important to express consistency imperatives as pure don’t-combine imperatives instead of as mixed if-then imperatives like these:

¹ Maybe your act is fine but your belief is faulty; for example, you treat dark-skinned people fairly but believe that this is wrong. More typically, your belief is fine but your act is faulty.

· (B ⊃ ∼A) = If you believe that this is wrong, then don’t act to do this

· (A ⊃ ∼B) = If you act to do this, then don’t believe that this is wrong 0273

The first wrongly assumes that your belief has to be correct in such conflict cases, while the second wrongly assumes that your act has to be correct. Since either can be faulty, both if-then imperatives can give bad advice. So it’s better to express consistency imperatives as don’t-combine forms, like “∼(A • B).”

Before leaving this section, let me point out problems with two alternative ways to understand imperative logic. Consider this argument:

· If you get 100 percent, then celebrate.

· Get 100 percent.

· ∴ Celebrate.

· (G ⊃ C) Invalid

· G

· ∴ C

G, ∼G, ∼C

This is intuitively invalid. Don’t celebrate yet – maybe you’ll flunk. To derive the conclusion, we need, not an imperative second premise, but rather a factual one saying that you did get 100 percent.

Two common ways to understand imperative logic would wrongly judge this argument to be valid. The obedience view says that an imperative argument is valid just if doing what the premises prescribe necessarily involves doing what the conclusion prescribes. This is fulfilled in the present argument; if you do what both premises say, you’ll get 100 percent and celebrate. So the obedience view says that our argument is valid. So the obedience view is wrong.

The threat view analyzes the imperative “Do A” as “Either you will do A or else S will happen” – where sanction “S” is some unspecified bad thing. So “A” is taken to mean “(A ∨ S).” But if we replace “C” with “(C ∨ S)” in our argument and “G” with “(G ∨ S),” then our argument becomes valid. So the threat view says that our argument is valid. So the threat view is wrong.

12.2a Exercise: LogiCola MI

Say whether valid (and give a proof) or invalid (and give a refutation).

(A ⊃ ∼ B)

(∼A ⊃ ∼C)

∴ ∼(B • C)

1. ∼A

∴ ∼(A • B)

2. ∼(A • ∼B)

∴ (A ⊃ B)

3. (A ⊃ B)

∴ (∼B ⊃ ∼A)

4. (A ⊃ B)

∴ ∼(A • ∼B)

5. ∼◇(A • B)

∼(C • ∼A)

∴ ∼(C • B) 0274

6. (x)(Fx ⊃ Gx)

Fa

∴ Ga

7. (x)∼(Fx • Gx)

(x)(Hx ⊃ Fx)

∴ (x)(Gx ⊃ ∼Hx)

8. (x)(Fx ⊃ Gx)

(x)(Gx ⊃ Hx)

∴ (x)(Fx ⊃ Hx)

9. (∼A ∨ ∼B)

∴ ∼(A • B)

10.∼(A • ∼B)

∴ (∼A ∨ B)

12.2b Exercise: LogiCola MI

First appraise intuitively. Then translate into logic (using the letters given) and say whether valid (and give a proof) or invalid (and give a refutation).

1. Make chicken for dinner or make eggplant for dinner.

Peter is a vegetarian.

If Peter is a vegetarian, then don’t make chicken for dinner.

∴ Make eggplant for dinner. [Use C, E, and V. This one is from Peter Singer.]

2. Don’t eat cake.

If you don’t eat cake, then give yourself a gold star.

∴ Give yourself a gold star. [Use E and G.]

3. If this is greasy food, then don’t eat this.

This is greasy food.

∴ Don’t eat this. [Use G and E; from Aristotle, except that he saw the conclusion of an imperative argument as an action: since you accept the premises, you don’t eat the thing. I’d prefer to say that if you accept these premises and are consistent, then you won’t eat the thing.]

4. Don’t both drive and watch the scenery.

Drive.

∴ Don’t watch the scenery. [D, W]

5. If you believe that you ought to commit mass murder, then commit mass murder.

You believe that you ought to commit mass murder.

∴ Commit mass murder. [Use B and C. Suppose we take “Follow your conscience” to mean “If you believe that you ought to do A, then do A.” Then this principle can tell us to do evil things. Would the corresponding don’t-combine form also tell us to do evil things? See the next example.]

6. Don’t combine believing that you ought to commit mass murder with not committing mass murder.

You believe that you ought to commit mass murder.

∴ Commit mass murder. [B, C]

7. Don’t combine having this end with not taking this means.

Don’t take this means.

∴ Don’t have this end. [E, M] 0275

8. Lie to your friend only if you want people to lie to you under such circumstances.

You don’t want people to lie to you under such circumstances.

∴ Don’t lie to your friend. [Use L and W. Premise 1 is based on a simplified version of Immanuel Kant’s formula of universal law; we’ll see a more sophisticated version in Chapter 14.]

9. Studying is needed to become a teacher.

“Become a teacher” entails “Do what’s needed to become a teacher.”

“Do what’s needed to become a teacher” entails “If studying is needed to become a teacher, then study.”

∴ Either study or don’t become a teacher. [Use N for “Studying is needed to become a teacher,” B for “You become a teacher,” D for “You do what’s needed to become a teacher,” and S for “You study.” This example shows that we can deduce complex ends-means imperatives from purely descriptive premises.]

10.Winn Dixie is the largest grocery store in Big Pine Key.

∴ Either go to Winn Dixie or don’t go to the largest grocery store in Big Pine Key. [w, l, Gxy, u]

11.Drink something available.

Only juice and soda are available.

∴ Drink some juice or soda. [Dxy, u, Ax, Jx, Sx]

12.If the cocoa is about to boil, remove it from the heat.

If the cocoa is steaming, it’s about to boil.

∴ If the cocoa is steaming, remove it from the heat. [B, R, S]

13.Don’t shift.

∴ Don’t combine shifting with not pedaling. [S, P]

14.If he’s in the street, wear your gun.

Don’t wear your gun.

∴ He isn’t in the street. [Use S and G. This imperative argument, from Hector-Neri Castañeda, has a factual conclusion; calling it “valid” means that it’s inconsistent to conjoin the premises with the denial of the conclusion.]

15.If you take logic, then you’ll make logic mistakes.

Take logic.

∴ Make logic mistakes. [T, M]

16.Get a soda.

If you get a soda, then pay a dollar.

∴ Pay a dollar. [G, P]

17.∴ Either do A or don’t do A. [This (vacuous) imperative tautology is analogous to the logical truth “You’re doing A or you aren’t doing A.”]

18.Don’t combine believing that A is wrong with doing A.

∴ Either don’t believe that A is wrong, or don’t do A. [B, A] 0276

19.Mail this letter.

∴ Mail this letter or burn it. [Use M and B. This one was used to try to discredit imperative logic. The argument is valid, since this is inconsistent: “Mail this letter; don’t either mail this letter or burn it.” Note that “Mail this letter or burn it” doesn’t entail “You may burn it”; it’s consistent to follow “Mail this letter or burn it” with “Don’t burn it.”]

20.Let every incumbent who will be honest be endorsed.

∴ Let every incumbent who won’t be endorsed not be honest. [Use Hx, Ex, and the universe of discourse of incumbents.]

12.3 Deontic translations

Deontic logic adds two operators: “O” (for “ought”) and “R” (for “all right” or “permissible”); these attach to imperatives to form deontic wffs:

· OA = It’s obligatory that A

· OAu = You ought to do A

· RA = It’s permissible that A

· RAu = It’s all right for you to do A

“O”/“☐” (moral/logical necessity) are somewhat analogous, as are “R”/“◇” (moral/logical possibility).

“Ought” here is intended in the all-things-considered, normative sense that we often use in discussing moral issues. This sense of “ought” differs from at least two other senses that may follow different logical patterns:

· Prima facie senses of “ought” (which give a moral consideration that may be overridden in a given context): “Insofar as I promised to go with you to the movies, I ought to do this [prima facie duty]; but insofar as my wife needs me to drive her to the hospital, I ought to do this instead [prima facie duty]. Since my duty to my wife is more weighty, in the final analysis I ought to drive my wife to the hospital [all-things-considered duty].”

· Descriptive senses of “ought” (which state what’s required by conventional social rules but needn’t express one’s own positive or negative evaluation): “You ought [by company regulations] to wear a tie to the office.”

I’ll be concerned with logical connections between ought judgments, where “ought” is taken in this all-things-considered, normative sense.¹ I’ll mostly avoid metaethical issues, like how to further analyze “ought,” how to justify ethical principles, and whether moral judgments are objectively true or false. While my 0277 explanations sometimes assume that ought judgments are true or false, what I say could be rephrased to avoid this assumption.²

¹ I’m also taking imperatives in an all-things-considered (not prima facie) sense. So I don’t take “Do A” to mean “Other-things-being-equal, do A.”

² For a discussion of whether moral judgments are true-or-false (as I contend they are), see my Ethics: A Contemporary Introduction, 3rd ed. (New York: Routledge, 2018) and Ethics and Religion (New York: Cambridge, 2016).

Here are some further translations:

· Act A is obligatory (required, a duty)

· = OA

· Act A is all right (right, permissible, OK)

· = RA

· Act A is wrong

· = ∼RA = Act A isn’t all right

· = O∼A = Act A ought not to be done

· It ought to be that A and B

· = O(A • B)

· It’s all right that A or B

· = R(A ∨ B)

· If you do A, then you ought not to do B

· = (A ⊃ O∼B)

· You ought not to combine doing A with doing B

· = O∼(A • B)

The last pair are deontic if-then and don’t-combine forms. Here are translations using quantifiers:

· It’s obligatory that everyone do A = O(x)Ax

· It’s not obligatory that everyone do A = ∼O(x)Ax

· It’s obligatory that not everyone do A = O∼(x)Ax

· It’s obligatory that everyone refrain from doing A = O(x)∼Ax

These two are importantly different:

· It’s obligatory that someone answer the phone = O(∃x)Ax

· There’s someone who has the obligation to answer the phone = (∃x)OAx

The first might be true while the second is false; it might be obligatory (on the group) that someone or other in the office answer the phone – while yet no specific person has the obligation to answer it. To prevent the “Let the other person do it” mentality in such cases, we sometimes need to assign duties.

Compare these three:

· It’s obligatory that some who kill repent

· = O(∃x)(Kx • Rx)

· It’s obligatory that some kill who repent

· = O(∃x)(Kx • Rx)

· It’s obligatory that some both kill and repent

· = O(∃x)(Kx • Rx)

These three are importantly different; underlining in the wffs shows which parts are obligatory: repenting, killing, or killing-and-repenting. If we just attached “O” to indicatives, our formulas couldn’t distinguish the forms; all three would translate as “O(∃x)(Kx • Rx).” Because of such examples, we need to attach “O” 0278 to imperative wffs, not to indicative ones.¹

¹ We can’t distinguish the three as “(∃x)(Kx • ORx),” “(∃x)(OKx • Rx),” and “(∃x)O(Kx • Rx)” – since putting “(∃x)” outside “O” changes the meaning. See the previous paragraph.

Wffs in deontic logic divide broadly into descriptive, imperative, and deontic (normative). Here are examples of each:

· Descriptive (“You do A”): A, Au

· Imperative (“Do A”): A, Au

· Deontic (“ought” or “all right”): OA, OAu, RA, RAu

Such wff-types can matter for logic; for example, “O” and “R” must attach to imperative wffs. Here are rules for distinguishing these three types of wff:

· Any not-underlined capital letter not immediately followed by a small letter is a descriptive wff. Any underlined capital letter not immediately followed by a small letter is an imperative wff.

· The result of writing a not-underlined capital letter and then one or more small letters, none of which are underlined, is a descriptive wff. The result of writing a not-underlined capital letter and then one or more small letters, one small letter of which is underlined, is an imperative wff.

· The result of prefixing any wff with “∼” is a wff and is descriptive, imperative, or deontic, depending on what the original wff was.

· The result of joining any two wffs by “•” or “∨” or “⊃” or “≡” and enclosing the result in parentheses is a wff. The resulting wff is descriptive if both original wffs were descriptive; it’s imperative if at least one was imperative; it’s deontic if both were deontic or if one was deontic and the other descriptive.

· The result of writing a quantifier and then a wff is a wff and is descriptive, imperative, or deontic, depending on what the original wff was.

· The result of writing a small letter and then “=a” and then a small letter is a descriptive wff.

· The result of writing “◇” or “☐,” and then a wff, is a descriptive wff.

· The result of writing “O” or “R,” and then an imperative wff, is a deontic wff.

12.3a Exercise: LogiCola L (DM & DT)

Translate these English sentences into wffs; take each “you” as a singular “you.”

“You ought to do A” entails “It’s possible that you do A.”

☐(OA ⊃ ◇A)

Here “◇A” doesn’t use underlining; “◇A” means “It’s possible that you do A” – while “◇A” means “The imperative ‘Do A’ is logically consistent.” Our sentence also could translate as “☐(OAu ⊃ ◇Au).”

0279

1. If you’re accelerating, then you ought not to brake. [Use A and B.]

2. You ought not to combine accelerating with braking.

3. If A is wrong, then don’t do A.

4. Do A, only if A is permissible.

5. “Do A” entails “A is permissible.”

6. Act A is morally indifferent (morally optional).

7. If A is permissible and B is permissible, then A-and-B is permissible.

8. It’s not your duty to do A, but it’s your duty not to do A.

9. If you believe that you ought to do A, then you ought to do A. [Use B for “You believe that you ought to do A” and A for “You do A.”]

10.You ought not to combine believing that you ought to do A with not doing A.

11.“Everyone does A” doesn’t entail “It would be all right for you to do A.” [Ax, u]

12.If it’s all right for X to do A to Y, then it’s all right for Y to do A to X. [Axy]

13.It’s your duty to do A, only if it’s possible for you to do A.

14.It’s obligatory that the state send only guilty persons to prison. [Gx, Sxy, s]

15.If it’s not possible for everyone to do A, then you ought not to do A. [Ax, u]

16.If it’s all right for someone to do A, then it’s all right for everyone to do A.

17.If it’s all right for you to do A, then it’s all right for anyone to do A.

18.It’s not all right for anyone to do A.

19.It’s permissible that everyone who isn’t sinful be thankful. [Sx, Tx]

20.It’s permissible that everyone who isn’t thankful be sinful.

12.4 Deontic proofs

We’ll now add six inference rules. The first four, following the modal and quantificational pattern, are for reversing squiggles and dropping “R” and “O.”

These reverse-squiggle rules hold regardless of what pair of contradictory imperative wffs replaces “A”/“∼A”:

Reverse squiggle RS

∼OA → R ∼ A

∼ RA → O ∼ A

These let us go from “not obligatory to do” to “permissible not to do” – and from “not permissible to do” to “obligatory not to do.” Use these rules only within the same world and only when the formula begins with “∼O” or “∼R.”

We need to expand our worlds. From now on, a possible world is a consistent set of indicatives and imperatives. And a deontic world is a possible world (in this expanded sense) in which (a) the indicative statements are all true and (b) the imperatives prescribe some jointly permissible combination of actions. So these equivalences hold:

· OA (A is obligatory) = ”Do A” is in all deontic worlds

· RA (A is permissible) = ”Do A” is in some deontic worlds 0280

Suppose I have an 8 am class (C), I ought to get up before 7 am (OG), it would be permissible for me to get up at 6:45 am (RA), and it would be permissible for me to get up at 6:30 am (RB). Then every deontic world would have “C” and “G”; but some deontic worlds would have “A” while others would have “B.”

A world prefix now is a string of zero or more instances of “W” or “D.” As before, world prefixes represent possible worlds. “D,” “DD,” and so on represent deontic worlds; we can use these in derived lines and assumptions, such as:

· D ∴ A (So A is true in deontic world D.)

· DD asm: A (Assume A is true in deontic world DD.)

We can drop deontic operators using the next two rules (which hold regardless of what imperative wff replaces “A”). Here’s the drop-“R” rule:

Drop “R” DR

RA → D ∴ A,

use a new string of D’s

Here the line with “RA” can use any world prefix – and the line with “∴ A” must use a world prefix that’s the same except that it ends with a new string (a string not occurring in earlier lines) of one or more D’s. If act A is permissible, then “Do A” is in some deontic world; we may give this world an arbitrary and hence new name – corresponding to a new string of D’s. We’ll use “D” for the first “R” we drop, “DD” for the second, and so forth. So if we drop two R’s, then we must introduce two deontic worlds:

· RA

· RB

· –––––––

· D ∴ A

· DD ∴ B

Act A is permissible, act B is permissible; so some deontic world (call it “D”) has “Do A” and another (call it “DD”) has “Do B.” It’s OK to use “D” in the first inference, since it occurs in no earlier line; but the second inference must use “DD,” since “D” has now already occurred. So permissible options need not be combinable; if it’s permissible to marry Ann and permissible to marry Beth, it needn’t be permissible to marry both Ann and Beth (bigamy).

We can drop an “R” from formulas that are more complicated, as long as “R” begins the wff; so this first inference is fine:

· R(A • B)

· ––––––––––

· D ∴ (A • B)

These next two examples are wrong (since the formula doesn’t begin with an “R” – instead, it begins with a left-hand parenthesis):

· (RA ⊃ B)

· ––––––––––

· D ∴ (A ⊃ B)

· (RA • RB)

· –––––––––

· D ∴ (A • B)

Drop only an initial “R” – and introduce a new and different deontic world whenever you drop an “R.”

Here’s the drop-“O” rule: 0281

Drop “O” DO

OA → D ∴ A,

use a blank or any string of D’s

Here the line with “OA” can use any world prefix, and the line with “∴ A” must use a world prefix which is either the same or else the same except that it adds one or more D’s at the end. If act A is obligatory, then “Do A” is in all deontic worlds. So if we have “OA” in the actual world, then we can derive “∴ A,” “D ∴ A,” “DD ∴ A,” and so on; but it’s good strategy to stay in old deontic worlds when dropping “O” (and to use the actual world if there are no world with D’s). As before, we can drop an “O” from formulas that are more complicated, as long as “O” begins the wff. So this next inference is fine:

· O(A ⊃ B)

· ––––––––––

· D ∴ (A ⊃ B)

These next two example are wrong (since the formula doesn’t begin with “O” – instead it begins with a left-hand parenthesis – drop only initial operators):

· (OA ⊃ B)

· ––––––––––

· D ∴ (A ⊃ B)

· (OA ⊃ OB)

· ––––––––––

· D ∴ (A ⊃ B)

“(OA ⊃ B)” and “(OA ⊃ OB)” are if-then forms and follow the if-then rules: if we have the first part true, we can get the second true; if we have the second part false, we can get the first false; and if we get stuck, we’ll need to make another assumption.

Rule DO lets us go from “OA” in a world to “A” in the same world. This accords with “Hare’s Law” (named after R. M. Hare):

Hare’s Law

☐ (OA ⊃ A)

An ought judgment entails the corresponding imperative: “You ought to do A” entails “Do A.”

Hare’s Law (also called “prescriptivity”) equivalently claims that “You ought to do it, but don’t” is inconsistent. This law fails for some weaker prima facie or descriptive senses of “ought”; there’s no inconsistency in this: “You ought (according to company policy) to do it, but don’t do it.” The law seems to hold for the all-things-considered, normative sense of “ought”; this seems inconsistent: “All things considered, you ought to do it; but don’t do it.” However, some philosophers reject Hare’s Law; those who reject it would want to specify that in applying rule DO the world prefix of the derived line has to end in a “D” (and so we can’t use a blank world prefix in the derived line).

Here’s a deontic proof using these rules: 0282

Reverse a squiggle (line 4). Drop an initial “R,” using a new deontic world (line 5). Drop each initial “O,” using the same old deontic world (lines 6 and 7). This all works like a modal proof, except for underlining and having “O,” “R,” and “D” in place of “☐,” “◇,” and “W.” As with modal logic, we can star (and then ignore) a line when we use a reverse-squiggle or “R”-dropping rule on it.

Things get more complicated if we use the rules for dropping “R” and “O” on a formula in some other possible world. Here’s a simple case. Formulas “RA” and “OB” are in the actual world (using the blank world prefix); and so we put the corresponding imperatives in a deontic world “D.”

· RA

· OB

· –––––

· D ∴ A

· D ∴ B

In the next example, formulas “RA” and “OB” are in world W; so here we keep “W” and just add “D” (the rules for dropping “R” and “O” allow these moves):

· W ∴ RA

· W ∴ OB

· –––––––

· WD ∴ A

· WD ∴ B

Here world WD is a deontic world that depends on possible world W; this means that (a) the indicative statements in WD are those of world W, and (b) the imperatives of WD prescribe some set of actions that are jointly permissible according to the deontic judgments of world W. The following proof uses world prefix “WD” in lines 7 to 9:

0283 When we drop the “R” from line 6 (“W ∴ R∼A”), we add a new deontic world D to world W, so we get “WD ∴ ∼A” in line 7. The next two chapters will often use complex world prefixes like “WD.”

We have two more rules. The indicative-transfer rule lets us transfer indicatives freely between a deontic world and whatever world it depends on; we can do this because these two worlds have the same indicative (descriptive or deontic) wffs. IT holds regardless of what descriptive or deontic wff replaces “A”:

Indicative transfer IT

D ∴ A → A

The world prefixes of the derived and deriving lines must be identical except that one ends in one or more additional D’s. Here are some correct uses:

· A

· ––––––

· D ∴ A

· D ∴ A

· –––––––

· ∴ A

· D ∴ A

· –––––––

· DD ∴ A

· OA

· –––––––

· D ∴ OA

This next inference is wong, since IT is to be used only with indicatives (including deontic judgments):

· A

· –––––

· D ∴ A

It can be useful to move an indicative between deontic worlds when we need to do so to get a contradiction or apply an I-rule. Here’s an example:

· It’s obligatory that all teachers prepare classes.

· You’re a teacher.

· ∴ You ought to prepare classes.

Instead of moving the indicative “Tu” from the actual world to D in line 9, we could have moved “∼Tu” from D to the actual world.

Our final inference rule, Kant’s Law, is named for Immanuel Kant:

Kant’s Law KL

OA → ◇A

“Ought” implies “can”: “You ought to do A” entails “It’s possible for you to do A.”

This holds regardless of what imperative wff replaces “A” and what indicative wff replaces “A,” if the former is like the latter except for underlining, and every 0284 wff out of which the former is constructed is an imperative.¹ Kant’s Law is often useful with arguments having both deontic (“O” or “R”) and modal operators (“☐” or “◇”); note that you infer “◇A” (“It’s possible for you to do A”) and not “◇A” (“The imperative ‘Do A’ is consistent”).

¹ The proviso outlaws “O(∃x)(Lx • ∼Lx) ∴ ◇(∃x)(Lx • ∼Lx)” (“It’s obligatory that someone who is lying not lie ∴ It’s possible that someone both lie and not lie”). Since “Lx” in the premise isn’t an imperative wff, this (incorrect) derivation doesn’t satisfy KL.

Kant’s Law equivalently claims that “You ought to do it, but it’s impossible” is inconsistent. This law fails for some weaker prima facie or descriptive senses of “ought”; since company policy may require impossible things, this is consistent: “You ought (according to company policy) to do it, but it’s impossible.” The law seems to hold for the all-things-considered, normative sense of “ought”; this seems inconsistent: “All things considered, you ought to do it; but it’s impossible to do it.” We can’t have an all-things-considered moral obligation to do the impossible.

KL is a weak form of Kant’s Law. Kant thought that what we ought to do is not just logically possible, but also what we’re capable of doing (physically and psychologically). Our rule KL expresses only the “logically possible” part; but, even so, it’s still useful for many arguments. And it won’t hurt if sometimes we informally interpret “◇” in terms of what we’re capable of doing.

We’ve already mentioned the first two of these four “laws”:¹

¹ The word “law,” although traditional here, is really too strong, since all four are controversial and subject to qualifications.

· Hare’s Law: An “ought” entails the corresponding imperative.

· Kant’s Law: “Ought” implies “can.”

· Hume’s Law: We can’t deduce an “ought” from an “is.”

· Poincaré’s Law: We can’t deduce an imperative from an “is.”

Now we’ll briefly consider the last two.

Hume’s Law (named for David Hume) claims that we can’t validly deduce what we ought to do from premises that don’t contain “ought” or similar notions.² So getting a moral conclusion requires having a moral premise. Hume’s Law fails for some weak senses of “ought”; given descriptions of company policy and the situation, we can sometimes validly deduce what ought (according to company policy) to be done. Hume’s Law seems to hold for the all-things-considered, normative sense of “ought.” A more careful wording would say: “If B is a consistent non-evaluative statement and A a simple contingent action, then B doesn’t entail ‘Act A ought to be done.’” This wording sidesteps some counterexamples (§12.4a) where we clearly can deduce an “ought” from an “is.”

² Some philosophers disagree and claim we can deduce moral conclusions using only premises about social conventions, personal feelings, God’s will, or something similar. For views on both sides, see my Ethics: A Contemporary Introduction, 3rd ed. (New York: Routledge, 2018).

Poincaré’s Law (named for the mathematician Jules Henri Poincaré) similarly claims that we can’t validly deduce an imperative from indicative premises that 0285 don’t contain “ought” or similar notions. A more careful wording would say: “If B is a consistent non-evaluative statement and A a simple contingent action, then B doesn’t entail the imperative ‘Do act A.’” Again, the qualifications block objections (like problems 9 and 10 of §12.2b). We won’t build Hume’s or Poincaré’s Law into our system.

Our deontic proof strategy is much like the modal strategy. First we reverse squiggles to put “O” and “R” at the beginning of a formula. Then we drop each initial “R,” putting each permissible thing into a new deontic world. Lastly we drop each initial “O,” putting each obligatory thing into each old deontic world. Drop obligatory things into the actual world just if:

· the premises or conclusion have an instance of an underlined letter that isn’t part of some wff beginning with “O” or “R”; or

· you’ve done everything else possible (including further assumptions if needed) and still have no old deontic worlds.

Use the indicative transfer rule if you need to move an indicative between the actual world and a deontic world (or vice versa). Consider using Kant’s Law if you see a letter that occurs underlined in a deontic wff and not-underlined in a modal wff; some proofs that use Kant’s Law get tricky.

From now on, we won’t do refutations for invalid arguments in the book (LogiCola keeps doing them), since refutations get too messy when we mix various kinds of world.

12.4a Exercise: LogiCola M (D & M)

Say whether valid (and give a proof) or invalid (no refutation necessary).

∴ ∼◇(OA • O∼A)

This wff says “It’s not logically possible that you ought to do A and also ought not to do A”; this is correct if we take “ought” in the all-things-considered, normative sense. Morality can’t make impossible demands on us; if we think otherwise, our lives will likely be filled with irrational guilt for not fulfilling impossible demands. “∼◇(OA • O∼A)” would be incorrect if we took “O” in it to mean something like “ought according to company policy” or “prima facie ought.” Inconsistent company policies may require that we do A and also require that we not do A; and we can have a prima facie duty to do A and another to omit doing A.

0286

1. O∼A

∴ O∼(A • B)

2. (∃x)OAx

∴ O(∃x)Ax

3. b=c

∴ (OFab ⊃ OFac)

4. ∴ O(OA ⊃ A)

5. ∴ O(A ⊃ OA)

6. ∴ O(A ⊃ RA)

7. OA

OB

∴ O(A • B)

8. (x)OFx

∴ O(x)Fx

9. O(A ∨ B)

∴ (∼◇A ⊃ RB)

10.(A ⊃ OB)

∴ O(A ⊃ B)

11.☐(A ⊃ B)

OA

∴ OB

12.OA

RB

∴ R(A • B)

13.A

∴ O(B ∨ ∼B)

14.(x)RAx

∴ R(x)Ax

15.OA

OB

∴ ◇(A • B)

16.∴ (RA ∨ R∼A)

17.(OA ⊃ B)

∴ R(A • B)

18.∼◇A

∴ R∼A

19.A

∼A

∴ OB

20.O(x)(Fx ⊃ Gx)

OFa

∴ OGa

21.O(A ⊃ B)

∴ (A ⊃ OB)

22.O(x)Ax

∴ (x)OAx

23.∴ O(∼RA ⊃ ∼A)

24.A

∴ (A ∨ OB)

25.(A ∨ OB)

∼A

∴ OB

Problems 3, 13, and 19 deduce an “ought” from an “is.” If “(A ∨ OB)” is an “ought,” then 24 is another example; if it’s an “is,” then 25 is another example. 20 of §12.4b is another example. We formulated Hume’s Law so that these examples don’t refute it.

12.4b Exercise: LogiCola M (D & M)

First appraise intuitively. Then translate into logic (using the letters given) and say whether valid (and give a proof) or invalid (no refutation necessary).

1. It’s not all right for you to combine texting with driving. You ought to drive.

∴ Don’t text. [Use T and D.]

2. ∴ Either it’s your duty to do A or it’s your duty not to do A. [The conclusion, if taken to apply to every action A, is rigorism, the view that there are no morally neutral acts (acts permissible to do and also permissible not to do).]

3. I did A.

I ought not to have done A.

If I did A and it was possible for me not to have done A, then I have free will.

∴ I have free will. [Use A and F. Immanuel Kant thus argued that ethics requires free will.]

4. ∴ If you ought to do A, then do A.

5. ∴ If you ought to do A, then you’ll in fact do A. 0287

6. It’s not possible for you to be perfect.

∴ It’s not your duty to be perfect. [Use “P” for “You are perfect.”]

7. You ought not to combine texting with driving.

You don’t have a duty to drive.

∴ It’s all right for you to text. [T, D]

8. ∴ Do A, only if it would be all right for you to do A.

9. If it’s all right for you to insult Jones, then it’s all right for Jones to insult you.

∴ If Jones ought not to insult you, then don’t you insult Jones. [Use Ixy, u, and j. The premise follows from the universalizability principle (“What’s right for one person is right for anyone else in similar circumstances”) plus the claim that the cases are similar. The conclusion is a distant relative of the golden rule.]

10.It’s all right for someone to do A.

∴ It’s all right for anyone to do A. [Can you think of an example where the premise would be true and conclusion false?]

11.If fatalism (the view that whatever happens couldn’t have been otherwise) is true and I do A, then my doing A (taken by itself) is necessary.

∴ If fatalism is true and I do A, then it’s all right for me to do A. [F, A]

12.If it’s all right for you to complain, then you ought to take action.

∴ You ought to either take action or else not complain. [Use C and T. This is the “Put up or shut up” argument.]

13.I ought to stay with my brother while he’s sick in bed.

It’s impossible for me to combine these two things: staying with my brother while he’s sick in bed and driving you to the airport.

∴ It’s all right for me not to drive you to the airport. [S, D]

14.Jones ought to be happy in proportion to his moral virtue.

Necessarily, if Jones is happy in proportion to his moral virtue, then Jones will be rewarded either in the present life or in an afterlife.

It’s not possible for Jones to be rewarded in the present life.

If it’s possible for Jones to be rewarded in an afterlife, then there is a God.

∴ There is a God. [Use H for “Jones is happy in proportion to his moral virtue,” P for “Jones will be rewarded in the present life,” A for “Jones will be rewarded in an afterlife,” and G for “There is a God.” This is Kant’s moral argument for the existence of God. To make premise 3 plausible, we must take “possible” as “factually possible” (instead of “logically possible”). But does “ought to be” (premise 1 uses this – and not “ought to do”) entail “is factually possible”?]

15.If killing the innocent is wrong, then one ought not to intend to kill the innocent.

If it’s permissible to have a nuclear retaliation policy, then intending to kill the innocent is permissible.

∴ If killing the innocent is wrong, then it’s wrong to have a nuclear retaliation policy. [K, I, N] 0288

16.If it’s all right for you to do A, then you ought to do A.

If you ought to do A, then it’s obligatory that everyone do A.

∴ If it’s impossible that everyone do A, then you ought not to do A. [Use Ax and u. The premises and conclusion are doubtful; the conclusion entails “If it’s impossible that everyone become the first woman president, then you ought not to become the first woman president.” The conclusion is a relative of Kant’s formula of universal law; it’s also a faulty “formal ethical principle” – an ethical principle that we can formulate using abstract logical notions but leaving unspecified the meaning of the individual, property, relational, and statement letters.]

17.It’s obligatory that Smith help someone or other whom Jones is beating up.

∴ It’s obligatory that Jones beat up someone. [Use Hxy, Bxy, s, and j. This “good Samaritan paradox” is provable in most deontic systems that attach “O” to indicatives. There are similar examples where the evil deed happens after the good one. It may be obligatory that Smith warn someone or other whom Jones will try to beat up; this doesn’t entail that Jones ought to try to beat up someone.]

18.If it’s not right to do A, then it’s not right to promise to do A.

∴ Promise to do A, only if it’s all right to do A. [A, P]

19.It’s obligatory that someone answer the phone.

∴ There’s someone who has the obligation to answer the phone. [Ax]

20.Studying is needed to become a teacher.

“Become a teacher” entails “Do what’s needed to become a teacher.”

“Do what’s needed to become a teacher” entails “If studying is needed to become a teacher, then study.”

∴ You ought to either study or not become a teacher. [Use N for “Studying is needed to become a teacher,” B for “You become a teacher,” D for “You do what’s needed to become a teacher,” and S for “You study.” This is an ought-version of §12.2b #9. It shows that we can deduce a complex ought judgment from purely descriptive premises.]

21.If it’s right for you to litter, then it’s wrong for you to preach concern for the environment.

∴ It’s not right for you to combine preaching concern for the environment with littering. [L, P]

22.If you ought to be better than everyone else, then it’s obligatory that everyone be better than everyone else.

“Everyone is better than everyone else” is self-contradictory.

∴ It’s all right for you not to be better than everyone else. [Use Bx (for “x is better than everyone else”) and u.]

23.You ought not to combine braking with accelerating.

You ought to brake.

∴ You ought to brake and not accelerate. [B, A] 0289

24.“Everyone breaks promises” is impossible.

∴ It’s all right for there to be someone who doesn’t break promises. [Use Bx. Kant thought universal promise-breaking would be impossible, since no one would make promises if everyone broke them. But he wanted to draw the stronger conclusion that it’s always wrong to break promises. See problem 16.]

25.It’s all right for you to punish Judy for the accident, only if Judy ought to have stopped her car more quickly.

Judy couldn’t have stopped her car more quickly.

∴ You ought not to punish Judy for the accident. [P, S]

26.You ought to pay by check or pay by MasterCard.

If your MasterCard is expired, then you ought not to pay by MasterCard.

∴ If your MasterCard is expired, then pay by check. [C, M, E]

27.You ought to help your neighbor.

It ought to be that, if you (in fact) help your neighbor, then you say you’ll help him.

You don’t help your neighbor.

If you don’t help your neighbor, then you ought not to say you’ll help him.

∴ You ought to say you’ll help him, and you ought not to say you’ll help him. [Use H and S. Roderick Chisholm pointed out that this clearly invalid argument was provable in many systems of deontic logic. Is it provable in our system?]

28.If you take logic, then you’ll make mistakes.

You ought not to make mistakes.

∴ You ought not to take logic. [T, M]

29.If I ought to name you acting mayor because you served on the city council, then I ought to name Jennifer acting mayor because she served on the city council. I can’t name both you and Jennifer acting mayor.

∴ It’s false that I ought to name you acting mayor because you served on the city council. [U, J]