Some Deontic Logicians

Some Deontic LogiciansAuthor(s): Lawrence PowersSource: Noûs, Vol. 1, No. 4 (Dec., 1967), pp. 381-400Published by: WileyStable URL: http://www.jstor.org/stable/2214625 .

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Some Deontic Logicians

LAWRENCE POWERS WAYNE STATE UNIVERSITY

I shall discuss the preceding papers by Aqvist, Sellars, and Anderson, and also Castanfeda's deontic calculus.'

Many of the logical properties ascribed to deontic notions in deontic calculi are exhibited in payoff calculi with less problematic interpretations. A preliminary exposition of a payoff calculus will make certain logical points easier to grasp.

A payoff machine has a Start lever, a Stop (or Payoff) lever and an array of buttons numbered 1, 2, 3, . . . N. One puts the machine through one cycle by pulling the Start lever, pushing any of the buttons, and finally pulling the Payoff or Stop lever. Then the machine drops money out of a slot to you. The amount is determined solely by which buttons you pushed (and did not push). Each partition of the buttons into pushed and unpushed has its payoff, characteristic of the machine.

A canonical description of a payoff machine includes a specification (by numeral) of the number of buttons and a specification (similarly explicit) of the payoff function-i.e. of what payoff ac- crues to each partition.

Consider a machine with three buttons and the following payoff function:

1. Push only button 1: Receive $5.00 2. " " " 2: " 2.00 3. " " " 3: " 4.00 4. " buttons 1, 2: " 7.00 5. " " 1, 3: " 7.00 (line 5) 6. " " 2, 3: " 2.00 7. " "1, 2, 3: " 3.00

1 H.-N. Castafieda, "Ethics and Logic: Stevensonian Emotivism Revisited" The Journal of Philosophy, LXIV (1967): 671-683; also his "Acts, the Logic of Obligation, and Deontic Calculi," Critica, I (1967): 77-99.

381



382 NOdS

If someone pushes buttons 1 and 3 and doesn't push 2, he gets $7.00. (See line 5.)

Let 'M' be a canonical description of the above machine. The sentence "If John gets maximum payoff on M, he gets $7.00," is obviously true. Moreover, it expresses a necessary statement because 'M' is canonical and includes a description of M's payoff function.

The statement that John gets maximum payoff on M also entails John pushes button 1. He must push 1 together with either 2 or 3 to get the $7.00.

Let 'P1' abbreviate "John pushes button 1 (and perhaps other buttons)," and 'P2' "John pushes button 2," etc. These I call our elementary action sentences.

For any machine M, I call complete action sentences those sentences which contain N conjuncts, one for each of M's buttons, the conjunct for button i being either Pi or -,Pi. M gives a payoff corresponding to each complete action.

For any M, an action sentence is any logical function of M's elementary action sentences.

In standard deontic calculi, '0' is read "it ought to be the case that," but the logical properties ascribed to '0' suggest that it should be construed as "If everything is as it ought to be & z, then . .

where z is an exhaustive statement of the form "it ought to be that a & it ought to be that b & * * " and "if . . . then . . ." is strict implication. [This reading is basically due to Anderson; the z, suggested by Smiley.]2

In my little calculus, I introduce a sentential operator 'B'. For machine M and action sentence 'p', 'Bp' is defined as "(John gets maximum payoff on M) -> p." Here 'M' is canonical, '->' is strict implication, and I am assuming that some definite cycle on M is in question.

For the machine already considered, with maximum payoff of $7.00, we have: BP1, B[(P1 * P2 * P3)v(Pl P3 *P2)], B(PlvP3), B(P3v P3), B( -(P2 P3)), and B[(P1 P2 P3) D

(P2 P3. For short we may read 'Bp' as "Best payoff requires that p." Since 'B' is defined in terms of 'M', there is a 'B' operator for

every canonical machine specification, rather than a single inter-

2 See, e.g. Anderson's present paper. For Smiley, see: Timothy Smiley, "Relative Necessity," Journal of Symbolic Logic, XXVIII (1963): 113 ff.



SOME DEON1IC LOGICIANS 383

pretation for 'B'. But it is always understood that some one 'M' is supposed to be in question.

Two "theorems" of our "calculus" may now be stated:

(1) Bp ->B p, (2) (p->q)-->(Bp-->Bq). From (2) there follow:

(3) Bp->B(qD p), (4) Bp >B (- p Dq). Next I introduce conditional best-payoff-requirements, follow-

ing the same general line taken for conditional probability in probability theory and for conditional obligation by von Wright.3

Consider a machine with 3 buttons and payoff functions:

Push only button 1: Receive $1.00 "f " " 2: " 7.00 "f "f "-C 3: " 4.00 "f buttons 1, 2: " 2.00 "C "f 1, 3: " 3.00 "f " 2, 3: " -0- -" " 1,2, 3: " 2.00

The maximum payoff is again $7.00. John should push only button 2. He should not push button 1. We have B(, P1i).

However, though John knows the payoff function, he inadvertently pushes button 1. Noting his mistake, he considers what he ought to do next. Theorem (4) tells him that B(P1 D (-P2 . O-P3)). But he sees that P1 -.`P2 -. P3 only gives him $1.00, while buttons 1 and 2 give $2.00, and 1 and 3 give $3.00. John pushes button 3 and pulls the payoff lever. His pushing button 3 may bother us if we read 'B' as "it ought to be the case that for we have 'B( -P3)'. However, John, not being a deontic logician, pushes 3 because the maximal payoff consistent with Pi requires Pa.

(A) The proposition: (1) John gets the greatest payoff copossible on M with P1; entails the proposition: (2) John pushes button 3.

8 For probability theory of the sort in question, see for instance Irving Adler's Probability and Statistics for Everyman (New York: John Day Com- pany, 1966; also a Signet paperback). For von Wright, see Aqvist's footnote 5 and reference 11.



384 NOUS

We abbreviate this fact (A) by 'B ( P3/P1 )'. We read this as "Best payoff conditional on P1 requires P3." A reading to be avoided as somewhat misleading is "Best payoff requires P3, conditional on P1."

A reading to be especially avoided is: "If P1, then the best payoff requires P3." This reading suggests that P1 D BP3. But we have on the contrary assumed that Pi - - BP3; indeed, P1 - B( (P3 ). In saying B(P3/P1) we apply a binary operator to P1 and P3 (not on P1 and BP3). Nor is there any B* such that B(P3/P1)= P1 D B*P3, since we in fact have both B(P3/P1) and B(~-P3/ P1 * P2).

In discussing Sellars' paper, I shall have to admit that B(P3/ P1) may be expressed after a fashion as "If P1, then best payoff requires P3." However any relation can be made to sound like an implication by any speaker familiar enough with the language to speak it sloppily. For instance, "John is to the right of the tree," or R(j,t), is expressible as "If the tree (is what's in question), then John is to the right (of it)," and thus as "If the tree, John's to the right," or [t] D Rj. The symbolism '[t] D Rj' disguises the logical form of Rjt and is to be avoided.

Neither Bp nor B(p/q) is logically like "John ought to so act that: p." They are not intended to be. They are logical analogues of standard Op and O(p/q). Standard Op means roughly that the best possible world cannot obtain unless p; Bp means that the best payoff cannot result unless p. Similarly for B( p/q) and O( p/q).

Now we turn to Aqvist's paper. Aqvist sets out to treat three problems. One of his interesting insights is that the three problems are alike. I shall only discuss one of these problems: the problem of contrary-to-duty obligations.

Couched in terms of payoff machines and instrumental 'ought', the problem is as follows. John Doe ought to refrain, or ought to have refrained, from pushing button 1. However he did push button 1. These facts are reflected in our symbolism by: B( -,iP1) and P1. Ideally John ought not push button 3, either. This is reflected by: B( -P3). However the situation is not ideal, for John has pushed button 1. Moreover, though B( -P3), still the best thing John can do consistent with pushing button 1 is to push 3 also. This is caught in B(P3/P1). So, since P1, John ought now to push button 3. No formula in our system reflects this last assertion. The problem, as Aqvist sees it, is to introduce a B* such that that assertion may be represented as: B*(P3).



SOME DEONTIC LOGICIANS 385

On my payoff version of what he is saying, Aqvist suggests that we need two operators for (non-conditional) obligation. The first is our old friend B. If Bp is true, then John ought ideally, or in the primary sense, to act so that p. The second (B') operator we may write 'BR'. I now explain the meaning of 'BRp'.

BRp is true just in case p is required for the best payoff consistent with all of John's primary sins, where a primary sin is an action which John ought not to have done in the primary sense and which John nevertheless did do. Roughly, 'BRp' means that p is required to make the best of a world laden down with all of John's (primary) sinful behavior.

Let 'r' be a variable having action sentences as substituends. Then BRp is true just in case there is a proposition (an action proposition), r, such that (1) B(p/r) and (2) r is an exhaustive statement of John's primary sins and (3) r is the weakest such (i.e., is only a description of John's sins and not also of, say, the weather).

Formally:

BRp = def (Ear) B(p/r) & r & (r') ((r' * B r') D (r-> r') ) & Wr (r' ) (r' B -,r' D r" -.r') D (r" -> r)

The quantifications into the opaque contexts will provide no problem so long as the restriction to action sentence substituends is observed. The third conjunct states that r is to be a statement of all of John's sins. The fourth tells us that it is to be the "smallest" such.

We see that BRp, unlike Bp, is in general contingent, since it implies that one of those r's such that B(p/r) must be true.

'Blp' may be conveniently read as: Best payoff consistent with alt Iokzns primirt skins requuz-es p.

Aqvist regards BRp as reflecting a secondary or reparational obligatoriness of John's acting so that p.

He himself notes the following objection: John's secondary sins may leave him with tertiary obligations. And so forth. Aqvist suggests introducing a whole array of new operators, of successive levels, to counter this objection.

However, the restriction to all sins leads us to another objection.

John Doe is not primarily obligated to marry Suzy Mae. How- ever, doing what he primarily ought not to do, he gets Suzy Mae pregnant. One would think that he was secondarily obligated to



386 NOUS

marry Suzy Mae. But we have not yet taken all of John's primary sins into account, for upon hearing of Suzy's condition, John Doe shot her through the head, causing her death. So now he cannot marry her.

John should not have gotten Suzy pregnant and he should not have shot her. These facts are reflected in his primary obligations: 01 (John does not get Suzy pregnant) and 01 (John does not shoot Suzy). Further John is not now obligated to marry Suzy, who is dead, and this is reflected by: `02 (John marries Suzy). But, having gotten Suzy pregnant, he should have married her. This fact is not reflected by any formula, for this obligation was not primary, and, since it was brought about by some of John's sins and made irrelevant by still others, it is not secondary in the sense of BR either.

A case involving payoffs would run as follows. B( PiP1), but then P1. Now --B(-P3) but still John ought to refrain from P3, for B( -P3/P1). But P3. And then, just as we were about to charge John with secondary sinning in pushing button 3, he pushes 4 which is such that B(Q-P4) and -B(QP3/P1 * P4). The result is that though John should not have pushed 3, we have neither B(~ P3) nor BR( P3). If Saint Peter is to keep a complete record of John's sins, he will need a better deontic logic for his reckoningsl

Another difficulty for Aqvist is that one would think that some actions are both primarily and reparationally sinful. For example, if you ought primarily to supply your son with money for his school lunch and if you should also reparationally give your son back a quarter which you stole from his piggy bank yesterday, then one would think that your present dastardly refusal to give little Henry any money at all is both primarily and reparationally reprehensible. However, in fact we have: BPBR (you give Henry money). This is so because your not giving Henry money is one of your primary sins. So in order for BR (give Henry money) to be true, we require: B(you give Henry money/you don't give Henry money, and you also commit other sins). That is, we require B(H/H - Z). This means that H is required by the best payoff consistent with OH (and Z). But this is impossible.

In payoff terms, the objection runs as follows. Suppose B(~P1) but P1. Suppose B(~P2) and B(~-P3/P1), but P3 and P2. We have B(~(P3 P2)), but, precisely for that reason, we cannot have BR(--(p3 *P2)). (Note that we have B ( ~ (P2 * P3)

because we have B P2. This is the root of my next objection.)




I have now raised two objections. The first is most instructive, the second perhaps paltry. My next and last shows that Aqvists technical apparatus is wholly inadequate. I formulate it first in my B calculus.

Let Q be a primary sin committed by John. Then B (Q/Q * Z) for any Z. So BR(Q). This is not yet the objection. What we have here is a fact noted by Aqvist, namely that a primary sin is secondarily an obligation. But this only means that the best of all worlds compatible with all one's sins still has those sins in it. The secondary "obligation" to commit one's sins is an empty and trivial "obligation". Aqvist refers to it as trivial.

But my theorem 2 states that (p .- q) -* (Bp -> Bq), and it follows that (q - p) -> (B p- - B ~ q), substituting -p for p, and -q for q. So, in other words, whatever entails a sin, is a sin.

Let John run through a cycle on machine M, sinning at least once, and suppose 'C' is the complete action sentence true for that cycle. Then, if 's' describes John's sin, C -> s, since 'C' is complete. So C is a sin; B ~ C. So C, John's complete behavior (and non- behavior) was a primary sin. So BR(C) vacuously. So we have the following remarkable theorem: (T) If John sins, then (BR(p) _ p). In other words, John is reparationally obligated to do exactly and only what he did do, whatever it may have been.

On the other hand, if John does not sin, then reparational obligation is of no interest and in fact BR(p) _ Bp.

(T) is quite devastating, so I now derive it from Aqvist's own formulation.

I assume that some sin has occurred: 01 - p and p; and derive for arbitrary q that: q = 02q. It is sufficient to show q D 02q for arbitrary q; for if this is true in general, then -q D02 ? q and 02 - q D -02q by Aqvist's A2. Then -q D O02q and so 02q D q.

I therefore assume 01 - p, p, and q, and show 02q. Al-A3 and R1 holds for both 01 and 02, as Aqvist remarks on page 372. Further, corresponding to my theorem 2, we have on page 365 if HA D B. then HO1A D 01B. As in the B version, it follows that: if H B D A then H0O1- A D 01 - B.

Now Hq * p D p. So 0, - p D 01 - (q * p). But 01 - p. So 01 - (q * p). But also q p. We therefore have (01 - (q * p)) &(q * p). Now it is obvious that I- - B -B, and so Aqvist's R1 tells us that - ( ( 01 - B )&B ) D 02B. So 02(q * p). But (q * p D

q, so F-02(q p) D 02(q). So 02q, as was to be shown.



388 NOOS

Thus, if I commit a primary sin, it is logically impossible for me thereafter to violate my secondary reparational obligations! In terms of Aqvist's model-theoretic semantics (if this fits his axioms), this result means that an imperfect actual world has only one extension (ideal or not)-namely itself!

However, I have not derived my result from the Hintikka-type model Aqvist gives. If every actual world must include qv - q as a member, for arbitrary 'q', and therefore either have q as a member or ~q, so that its extensions would also have these as members, then my result would follow. But I do not have the premise that every actual world has qv ~ q, for arbitrary q. Perhaps Aqvist's model is better than his system. Perhaps Aqvist should, as it were, follow Aristotle to the sea-battle and conceive of his actual worlds as not yet completed. The suggested O(p would be true just in case p is conditional obligatory upon, not all John's sins, but only those up to the time when John either does p or fails to do p. Since I shall later discuss an operator of this sort, I shall not pursue the question further at this point.

I now begin my discussion of Sellars. Sellars takes seriously, as the other writers do not, the idea that an ought statement must be asserted at some definite time ("now"), which is fixed in any given discussion, but is not of course the same for different discus- sions. Sellars then supposes, if we defer certain complications, that at any given time there is a unique set of total relevant circuin- stances, and that something ought (now) to be done iff it ought to be done, in some sense, conditionally upon the present (now prevalent) total relevant circumstances.

Let 'Onp' abbreviate "It is now obligatory for John that p." Then if C. is the now total relevant circumstance, there is presum- ably some underlying operator 0 such that Onp O( p/Cn). Before pursuing this further, I introduce a model.

A one man stage machine is like a payoff machine except that times are introduced. At to (time before beginning), John pulls the Start lever. At to.5 he rests. At ti he pushes buttons. At t1.5 he rests. At t2 he pushes buttons again, either the same ones as before or different ones, or some the same and some different. At t2.5 he rests. At t3 he pushes buttons again. At tT he pushes buttons for the last time and pulls the Payoff lever. The payoff is determined by which buttons he pushed at which times.

I write 'Pll' 1, for "John pushes button 1 at time 1," 'plat 2, for



SOME DEONTIC LOGICLANS 389

"John pushes button 1 at time 2," et cetera. The action sentences are formed from these.

Bp means that p is required for best payoff and this is defined essentially as before. B(p/q) is also defined essentially as before.

A complete ta action sentence for a time ta is an action sentence completely describing the action up to but not including ta.

The rigorous definition is much like that for 'complete action sentence'. A complete ta action sentence need not be true, it need only be complete (up to ta). The (unique) true complete ta action sentence is written: Ca.

In Sellars' theory, I said, we require an 0 and On such that, where n = now, Onp 0( p/Cn), where Cn is total relevant circumstances now. Let us suppose that this Cn is the true complete ta action sentence for whatever ta is equal to n.

Suppose that B ( p1at 1) and B ( p2at 2/p1at 1) and also B( P2at2/_Plat1). At time to05 nothing has happened and so B(Platl/C0.5) and B(P2at2/C0.5). The latter holds because best possible payoff requires plat 1 * p2at 2. But, we now suppose, John inadvertently fails to push button 1 at time t1 and so B ( p2at 2/ C1.5).

Switching from 0 to B, we have Bnp B (p/Cn). I do not know what time it is now. But if it now = n to.5, then Bn(P2at 2); whereas if it is now = n = t1.5, then Bn( Ip2at 2). Suppose we were to construct a calculus by supposing Bnp to be true if 'Bnp' was truly sayable at any time. Then we should be compelled to say: Bn (P2 at 2) . Bn( -P2at 2). In other words, we would have what all deontic logicians regard as an absurdity: Op * 0 ~ p.

In just this way, Sellars thinks, we cannot construct a satis- factory and consistent deontic logic unless we pay attention to the fact that "It is now obligatory that . . ." is not the same as "It was obligatory that . . ." or "It will be obligatory that . .

In light of our discussion of Aqvist's paper, Sellars' idea of tensing 'is obligatory' cannot but be salutary for deontic reasonings. However let us temporarily set aside the temporal grammar, as it were, of 'ought' and suppose n fixed, at say n =t3.5. Then we may- develop a logic for O = now = On = 03.5. This logic will be one aspect of Sellars total system and we may guess that this partial logic will be somewhat like other logics in the literature which do not take note of the importance of 'now'.

In fact this partial logic is usefully considered together with Castafieda's deontic logic, with which it shares two characteristic



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theorems and other important features. It will be profitable to discuss Castafieda's system at this point.

Castafieda, like Sellars, appears sometimes to take seriously the fact that there are many moral agents. This suggests the following.

A sociomachine is a sort of many-person stage machine. There are A people, each with N buttons. At to someone pulls the Start lever. At t1 each person pushes buttons on his bank of buttons. At ti.5 each person gets a note telling him what happened at ti, and some persons may also get another note which I discuss later. Similarly for t2, t2_5, et cetera. At tT everyone pushes buttons for the last time. 'p34, at 1, means that his button 3 is pushed by person 4 at time 1. The action sentences are formed from sentences of this sort. At tT the machine pays off into a box called the General-Good Box. The button pushers do not get anything. The payoff into the General-Good Box is determined by who (by number) pushed which buttons at which times.

'Bp' means: (M pays off maximally to the General-Good Box) ->p.

B ( p/q) iff Best payoff (to the GG Box) consistent with q requires p.

A complete ta action sentence completely describes the action up to ta-, if a is an integer.

Let 'p' be an action sentence which concerns only time ta. I call 'p' a stage sentence. We write SPp just in case there is a true complete ta action sentence q such that B(p/q).

SB is intended to capture the thought that an action or state of affairs may be obligatory because of the situation prevalent at the time of the action. This was suggested in our discussion of Aqvist. SB should not be thought to be just like the Bn we discussed in connection with Sellars. SB contains no 'now'; the time ta which settles the conditions for p is not the time at which 'SBp' is to be stated, but rather the time at which p is to happen.

For an example of how SB works, if best payoff conditional upon what happened at t1 and t2 requires that either person 5 pushes button 5 at t3 or person 6 pushes button 1 at t3, then SB(P55, at 3 v p16, at 3). Also then SB(P55, at 3 v p16 at3 v p27, at 3)

Let 'p' be an action sentence involving more than one time. We write SBp if there is a stage q such that SBq * (q-ip), or if there are stages q and r such that SBq - SBr - (q - rep), and, generally, we write SBp just in case p is entailed by statements (ultimately stage statements) which already have SB before them.




If we say, for a stage p, that p is demanded at its time iff SBp, we have as a short reading for SBq generally: 'SBq' means that Stages demanded at their time require q. Notice that q need not itself be in any way demanded at any time, for the stages which require (entail) q may be temporally diverse from one another. The first such stage may be demanded at, say, t3, but not performed. Suppose the second would have been undesirable at t4 had the first been done at t3, but is demanded at t4 precisely as a result of the first's not having been done at t3. What requires q is not either of these two stages, but only their conjunction.

Now we come to those "other notes" which some people get at ti.5, t2.5, et cetera. At t1.5, the t2 action is about to commence. Suppose that some t2 act q involves only person a. Then if SBq, person a ought to do q. In this case, the machine passes a a note saying 'SBq'. For instance if SB(PP513) then at t2.5 the machine passes person 1 a note reading 'SB( P)13)'. On the other hand, if SB( P1,3 v P52,3) is true, the machine need pass no notes, for this entails nothing for any particular individual (specifically, for 1 or 2). (Notes demanding tautologous actions, like SB ( P5 3 v Prl,3) may be understood as vacuously passed for purposes of our calculus.)

We shall write 'Mq' just in case the machine at some time passes (or vacuously passes) a note reading 'SBq'. In this case q must be temporally and personally pure; it might, e.g., be (p312 v P41 2). If q is temporally and personally pure, then Mq iff SBq. Deontically a fiction like that of our previous "best possible worlds" and "best conditionally possible worlds" suggests itself. This fiction is that of the Almighty Moralist, or the Voice of Morality Itself. Our machine passing these Sbq messages represents this Voice calling upon agents to do duties which have come home to rest squarely upon these agents. Mq may be read as "Morality demands that q."%

The introduction of the fiction of the Great Moralist demanding duties was suggested by the following (unrepresentative selection of) ideas from Castanfeda. Castanfeda suggests (Critica article, page 84)4 that a distinction between 'imperatival' 'and 'non-imperatival' components is relevant to the problem of contrary-to-duty obligations. He recalls Chisholm's statement of the problem and says "Chisholm does have . . . a good point. It is more clearly brought out with non-conditional examples." I shall consider these

' See footnote 1.



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examples later. What they suggest as far as Chisholm's problem is concerned is that "If you steal, you should return the money" should be formalized as O(s D r) rather than as O(s D r) or O(s D r). (I here put aside the implausible but possible 's D Or' or 's D Or'.) Here the underlined component of O(s D r) is what Castafieda calls the imperatival component, the component with some sort of command-giving character, the target of the 0.

Now our M operator involves Morality demanding, command- ing, ordering, as it were. Let us therefore introduce a related conditional obligation operator and adopt Castafieda's style of formu- lating it as O(s D r).

Let p be only about times before t and let q be wholly about t. It may be that the machine M is such that (p D Mq) is derivable from M's payoff function alone. If so, we write Mo(q/p). We also write MO(p D q) and also MO(Q. pvq) and also MO( ~ (p- q)). The misleadingness of these last ways of writing that MO(q/p) cannot be overstated. They suggest falsely that MO is operating on some statement p D q, whose consequent is some peculiar thing expressed by 'q'.

Let 91[(r), where r need not be an action sentence (for a change), mean that r may be deduced from the structure of the machine M. Then MO (p D q) means 9M(p D Mq), MO( .ipvq) means 9M(~pvMq), MO(~(p. - q)) means SM(~-(p -Mq)). These equivalences hold providing the MO formulae are well-formed.

Roughly speaking, MO(p/q) means that Morality, from the very beginning (necessarily), demands conditionally upon q's hap- pening before p, and regardless of what else happens before p, that p take place when its time comes.

It is an easy matter to introduce an operator M' which is such (without going into details) that M' ( p/q) means that Morality, at some time t before p and q, demands conditionally upon q's hap- pening between t and p, that p take place when its time comes. In effect M'(p/q) allows for contingent conditional obligations in the same way as MO (p/q) allows for necessary ones. The misleading representation M' (q D p) could be adopted here also and be eliminated in favor of the derivability of q D Mp from the machine's structure and some appropriate Ct.

Representing this derivability by M( ), we would have M'(p D q) = RM(p D Mq). Now RM(p D Mq) is different from 9M(M(p 2 q)) and also from p D AM(Mq). The symbolism




M'(p D q) obscures the fact that both p and q are under the scope of one operator (iM) while only q is under the scope of another (M). Put otherwise, though it is not possible to get it out of the scope of every operator, as in p D 7(Mq), it is possible to get it out of the scope of one (namely M). The need for two kinds of components in M'(p D q) is an artifact resulting from the fact that we are trying to do with one-too-few one-place operators.

The M, MO, M' operators are subject to much improvement, but I forego this, for it adds nothing essential to the logical point.

In ordinary English, the point is as follows. "If you steal, you ought to return the money" does not mean either s D Or or O(s D r). Castafieda suggests O(s D r). One way to explain the offending sentence which differentiates it from s D Or and O(s D r) alike and which does not involve peculiar new kinds of components, is to suppose it means "Ethical considerations alone allow us to see that s D Or," or symbolically 'E(s D Or),' using two operators instead of one.

Actually Castafieda's case for the two kinds of components is even weaker than I have suggested, for he does not have O(p D q) different from p D Oq. I shall call his theorem that O(p D q) p D Oq the Characteristic Implication Theorem, or CIT. The sug- gestion that a contrary-to-duty obligation to return money if you steal it is adequately represented by 'steal D O(return)' or ',-,steal v O(return)' seems highly implausible. Castafieda gives no reason for CIT that I can see. However CIT is the first of the two theorems that Sellars and Castafieda share, and Sellars does give a persuasive reason for it. It is as follows.

In ordinary English, the statement: (1) If John smokes, he ought to use the ashtray, is expressible no matter how construed (e.g., let 'if . . . then . . .' be material implication) as: (2) John ought, if he smokes, to use the ashtray. Expressing (2) as O(p D q), we have (p D Oq) = O(p D q). If we prefer, we can phrase the argument in terms of the equivalence of (1') If John smokes, it ought to be that he use the ashtray and (2') It ought to be, if John smokes, that he use the ashtray.

This persuasive reasoning unfortunately does not stand up under examination. No doubt it is important in clearing away con- fusions in deontic logic to point out that 2' sometimes is simply a way of saying 1' and so p D Oq, and that 2 sometimes means only 1 and so p D Oq. But the importance of this point is precisely that



394 NOOS

if we do not notice it we shall be tempted to formalize 2 and 2' as O(p D q).

Consider the following sentences.

(1") If Fred is bald, John is tall. (2") John, if Fred is bald, is tall.

Let John = j, Fred is bald = p, is tall = T.

(1*) pD Tj (2*) T(p D j). (Tallness, if Fred is bald, applies to John.)

The notation of 2* misrepresents the scope of T; nor is j actually the consequent of any conditional.

According to this analysis, the theorem CIT is a result of adopting a notation so unperspicuous as to logical form that it would produce as a theorem of standard quantification theory cum propositional calculus the result: p D Pa 3 P(p D a). Thus Sellars' reason 'for' CIT is actually a reason against it. Not, of course, that CIT is false; rather it will evaporate away when the notation in which it is written is rejected.

I now turn to the second theorem shared by Sellars' partial logic and Castafieda's deontic calculus. This I call the Characteristic Conjunction Theorem, or CCT. Sellars and Castafieda both give the same reason for this theorem. The theorem is: O(a * b) = a * Ob. In order to see why Castanieda (and also Sellars) thinks CCT should obtain, we need to consider examples like some of those Castanieda gives to support his distinction between imperatival and non-imperatival components. In ordinary English "It ought to be that John, who has a sister, go to the moon," is the same statement as "John has a sister and it ought to be that John go to the moon." Thus O(a .b)= a Gb.

A first indication that all is not well with CCT lies in the fact that, if Fp = it is forbidden that p, then the reasoning behind CCT leads us to affirm: F(a * b) = it is forbidden that John, who has a sister, go to the moon = John has a sister and it is forbidden that John go to the moon = a * Fb. Now Sellars does not introduce an F operator in his paper. But Castafieda, who does, does not write F(a * b) = a Fb. For F = On. And Castanfeda does not write O (p * q) =p *O q, but instead has O- (p q) =(p*

O q) (p Pq) = -pvFq = p D O q). (See theorem T2 in Critica paper.)

To see what is wrong with the reasoning behind CCT, let us




consider an example I owe essentially to a graduate student, Richard Sharvy. "It is false that John, who has a sister, is on the moon" means "John has a sister and it is false that John is on the moon." So our previous reasoning would lead to: (a * b) = a* nab. Ap- parently, then, the symbolism '~(a - b)' simply misrepresents the scope of 'a'. And we may conclude that CCT is of a piece with CIT.

This point about negation also explains how I can get F(a * b) = a * Fb while Castanieda gets F(a * b) = a D Fb. Letting a bottom line represent the scope of 'a' and an upper represent that of 'O', my reasoning was: F(a b)= 0 (a * b) = O(a nb) = a O(b) = a* Fb. Castanfeda's was: F(a - b) = O (a b)=O(avb) = O(a D ~b) =a D O~b =a D Fb. Despite doubling lines, there are still scope ambiguities in the above reasoning. These are due to the lamentable nature of the notation being used.

My conclusions with respect to what Sellars and Castanieda share are as follows. (1) In distinguishing two kinds of components in connection with Chisholim's problem and in supposing this distinction to be represented by the underlining, Castanieda confuses ethically irrelevant descriptive circumstances with ethically relevant conditions of obligation. (2) CIT and CCT do not provide good reasons for distinguishing two kinds of components. Rather CIT and CCT are the result of inadequate notation and should be rejected. (3) Remaining difficulties should, as in my E(s D Or) example, not be solved by introducing mysterious new kinds of components, but should be approached by considering whether we are trying to get by with fewer operators than are really needed. (4) The points brought out by theorems CIT and CCT, while they should not be encompassed in deontic calculi, are important and should be handled in the informal explanations surrounding such calculi.

I now revert to discussing Sellars alone. First, some small points.

(A) Sellars does not ever distinguish O(p p p D q) from O(p q) as one might think from his discussion of the parachute jumper. In fact his PPO-1 entails that O(p * p D q) = O(p * q). Perhaps he means that Onp * O(q/C. * p) or On(p * (q/p)) are different from On(p * q).

(B) Sellars efforts to represent O(q/p) as some sort of conditional lead him to write "O(s is [C1 * does not do] -> s not to



396 NOTS

tell)" for "O(s does not tell/C1( s) * s does not go)." Apparently, 'O( [p] -> q)' means 'O(If p is what's in question, then q)' and Sellars is availing himself of a combination of the CIT trick and the if-the-tree-then-John's-to-the-right trick. In particular, it is confusing that (on page 328) 's is C1' does not mean Cl(s), but rather means that C1 ( s ) and s is under no further relevant conditions. It is further rather startling that 's is [C1 * does not go]' does not entail 's is C,' but rather contradicts it. For reasons like those discussed earlier, I doubt whether this sort of representation is desirable.

(C) Note that while Onp is defined as a conditional ought, O(p/C..), nonetheless we write 'Onp' without conditionalization to anything, rather than writing, say, 'On(p/Cn).' It is in this way that Sellars solves the detachment problem mentioned in connection with his footnote 7 (which see).

Now a major point. Sellars himself sees that the idea of simply tensing 0 is not sufficient. The conditions relevant to a given ought statement are not fully determined by the time at which it is spoken, or even by the particular discussion in which it occurs. A speaker may, by whatever are the usual pragmatic ways of settling relevant contexts, make it clear that one of his statements has one context of relevant conditions while another has a different one.

Aqvist, who proposed a whole hierarchy of senses of 0, re- ceives a sort of vindication here, for if the locution "It ought to be that p" has a use which is such that its sense is determined on a given occasion by selection of a relevant context, then its use is such that it has a multitude of senses.

But though Aqvist should take heart at this new complication, it is not clear to me whether Sellars' own theory survives it. Con- sider the hoary example of the man who ought to go to a meeting on August 5 and who ought to send, on August 2, a note explaining his absence, if and only if he is in fact going to be absent. August 2 arrives and, though he is able to attend the meeting, he has no intention of doing so. He argues: "I ought to change my mind, forbear note-writing, and attend the meeting. So I am obligated not to write a note. My present fulfillment of this obligation will help make up for my sinfully staying home on the fifth!" In the face of this sophistry, it is worse than useless to suggest that there must be a total relevant circumstance, for we must account both for his obligation to change his mind and go and also for the unvirtuousness of his present willful failure to write the required note. An adequate solution of Chisholm's puzzle must contain an explanation precisely




of the fact that there appears to be more than one total relevant circumstance. Sellars theory fails this test, as far as I can see, though it certainly appears to contain much material for a theory that would pass it.

I now turn to Anderson. His paper concerns that interpretation of 'Op' which I have been calling "the standard interpretation" and which is due to Anderson.

He affirms that (1) Op= (if Up then V), where V= the bad thing. I shall suppose the bad thing = Morality is disobeyed. Sometimes Anderson appears to suggest that some form of utilitarianism is expressed by (1), that when morality (strictly, whatever rules are in question) is not obeyed, something else bad must result. But he wants to argue that (1), unlike utilitarianism, is trivially true, and further he actually asserts that morality's being disobeyed need not have any bad causal consequence, for "we have all gotten away with something, even if not murder."

Substituting for V, and transposing (1), we have (1') Op= (If Morality is obeyed, then p). Now, since Op means roughly that Morality tells us to do p, (1') is trivial (problem-begging) in the sense that it can hardly be an explication of Op. But it should not be supposed that (1') is therefore trivial logically. First, (1') entails (2) Op -> (if Morality is obeyed, then p). The arrow is strict implication. (2) is logically trivial if 'if . . . then . . .' means ' D', but not if 'if . . . then . . .' is strict implication, for (Op - M) -> p does not entail Op -> (M -> p). But second, (1') entails (3) (If Morality is obeyed, then p) -> Op. (3) means that everything "if . . . thened" by morality's being obeyed is obligatory. This is certainly not true if "if . . .then' is 'D'; for morality is not (always) obeyed, yet not everything is obligatory.

Anderson's paper purposes to find a sense of 'if . . . then' such that (1'), and so (2) and (3), will be true. We have just seen that neither 'D' nor '->' can supply such a sense. The above reason against '->' would be obviated by instead supposing (1") Op=

(Morality is obeyed p), where 'p q' means that (p * RA) ->q, and 'RA' is an exhaustive statement of what the rules are (of what morality commands). Anderson does not consider this approach (due to Smiley) but does raise against '-' one, not very compelling, reason which also applies tom', namely, that it seems implausible that O(pv -- p).

Having eliminated 'D' and '-I' from consideration, Anderson proposes the main thesis of his paper, namely: the 'if . . . then'



398 NOOS

should be the '-c' of his favorite contingent-implication system R. I shall continue to use 'if . . . then' and 'implies' in the sense of 'D' and '-'. If p -< q, I shall say p canonically implies q.

Anderson's thesis is that (P1) Op= (M -< ). M means that Morality is always obeyed, or in other words that everything that is obligatory is done. Consequently (10 0) (p is obligatory) iff (Every- thing that is obligatory is done -' p is done) seems a fair statement of Anderson's thesis. This 'iff' may be replaced by I', for this 'if' is actually '=' and Anderson's axiom 1 tells us that A -( A. Now the thesis (200) p is obligatory -< (Everything obligatory is done-- p is done ) is an extremely plausible one if '-<' is a relevant contingent implication. For what is more natural than (20)'s generalization (300) x is A-- (all A's are B -x is B) which, by R's rules, is equivalent to (all A's are B's-c (x is A -cx is B) ) ? Moreover if 300 were given, we would have solid reason for 200 and so some reason for half of 100 and thus some reason to think that 10 might be true.

I now argue that 300 is false. All possible universes in which p f..pvq are universes in which q. This universe is certainly a possible one. 300 gives: [(this actual universe is one in which p - pvq) -' (this actual universe is one in which q)]. The premises were all necessary and therefore the conclusion was also. But surely ((This actual universe is one in which q) -c q) is as well-motivated as 1.. Now everything that's true is true in this universe, so a is true -< (this U is such that a). Thus a < (this U is such that a). So (p * (rpvq)) this universe is such that (p * (Iapvq)). Transitivity gives:. (p ( ~pvq) ) X q. All the premises having been necessary, this last is also. So (p * ( pvq) ) -' q should be a theorem of R if 300 is true. But it is not a theorem of R, and is not at all in the spirit of R, either. So 300 is incorrect.

The only unsupported assumption in the above argument is that a is true >-< a. I leave it to Anderson to surprise me by re- jecting this, if he wants to.

Though we are thus deprived of a reason for thinking 10 might be true, luck is still with us, for we have an excellent reason, provided by Anderson himself, for thinking 10 is false. On page 358, Anderson raises the problem that his theory leads to: "r -< OPr; i.e., if r happens, then it ought to be permitted." Faced with this crushing result, he asks "Does the formula really bear this interpretation?" The correct answer to this question is: of course it must,




since that is what 'O' and 'P' are supposed to mean on Anderson's theory. If the formula does not bear that interpretation, Anderson's theory is wrong.

But, argues Anderson, we should unpack the definitions and then we see that 'r - OPr' really says. 'r-' ((r- V) V)', or "'if r happens, then if r is not permitted, we are in trouble.'" And this latter "looks unexceptionable."

Indeed. Is this Anderson speaking?! Suppose it is said that "If Eisenhower is still President, the Democrats hold that office," which sounds false, cannot be correctly symbolized by the true 'E D D'. But, it is replied, if we unpack the definition, we see that we are only saying "Either it is false that Eisenhower is President, or the Democrats hold that office, or both," which sounds unexceptionable. Belnap and Anderson, in their forthcoming (we hope) book5 on R and E, fulsomely attack the reply, arguing rightly that it is not a defense of the view that 'v' means 'if . . then'. Anderson's defense of his 'r -' OPr' misses the same boat. So we are left with a refutation of 1*.

In proposing 1P Anderson was trying to clear up a slight puzzle by employing a deep mystery. For (M - p) is there intended to be a contingent canonical implication whose antecedent and consequent do not in turn involve '. Of the truth conditions of such statements, practically nothing is known-a fact which would have made 3** exciting, had it been true.

If we take 'p is [contingently and not necessarily relevantly] sufficient for q' to mean 'p D q', Anderson's reading of '-' suggests that (1) (pa-- q) = [(p v q) * (p is relevant to q)]. This reading is misleading if one does not know the subproof formulation of R, for it is natural to suppose that p is relevant to q iff p is a relevant point to bring up when it is being considered whether q is true or not. But the question whether q = the question whether q. So we would have (2) pRLVNTq pRLVNT q. Axiom 2 strongly suggests that (3) relevance is transitive. From these three assump- tions and Anderson's system R, it could be shown that (p - q) _ (p D q), which is absurd. So Anderson's reading of '-' should be taken with a grain of salt.

I now conclude this paper. Save for what I have learned from

' For opportunities to see circulating copies of parts of this very exciting book, my thanks to Professor Castafieda and to Mr. O'Connell. For helpful discussion of same, my thanks to J. Michael Dunn.



400 NOtJS

von Wright6, most of what I know of deontic logic I have learned from the papers I've discussed. So I hope I have not been overly unfair to their contents

' Principally: George Henrik von Wright, Norm and Action: A Logical Inquiry (London: Routledge and Kegan Paul, 1963).



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