PHILOSOPHICAL STUDIES Edited by WILFRID SELLARS and HERBERT FEIGL with the advice and assistance of PAUL MEEHL, ~'OHN HOSPERS, MAY BRODBECK VOLUME XIII

Contents

]'anuary-February

1962

NUMBERS 1-2

Conditional Permission in Deontie Logic by Nicholas Reseher, LEHIGH UNIVERSITY

Reply to Mr. Rescher by Alan Ross Anderson, YALEUNIVERSITY Tautological Entailments by Alan Ross Anderson and Nuel D. Belnap, Jr., YALE UNIVERSITY

Infinite Analysis by William Todd, NORTHWESTERN"UNIVERSITY TWO Extralogical Uses of the Principle of Induction by Rollin W. Workman, UNIVERSITY OF CINCINNATI

Conditional Permission in Deontic Logic by NICHOLAS RESCHER LEHIGH UNIVERSITY

IN A recent study, 1 1 proposed an axiom-system for conditional deontic logic --the logical theory of the ethical concepts of permission, disallowa1, obligation, and (moral) commitment following out a suggestion of von Wright's3 The concept of conditional permission, in terms of which this system is articulated, is represented by the symbolism P(p/c), 8 to be read and interpreted as "the act p is permitted (allowed) in the circumstance (under the condition) c." In a most interesting analysis of the proposed formalization of the theory of this concept, 4 Alan Ross Anderson has suggested that the idea of conditional permission is definable in terms of the deontic concept of unconditional permission, by use of the definition P (p/c) --D~ C---) P (p) .5

2

PHILOSOPHICAL STUDIES

Such a definition of conditional in terms of unconditional permission would, in effect, achieve a corresponding reduction of conditional to unconditional deontic logic. It is the object of the present paper to argue that Anderson's proposed reduction of conditional to unconditional permission is unacceptable because it leads to a variety of deontically unacceptable consequences. CRITIQUE OF A DEONTIC THESIS

As axiomatic basis for the logical theory of conditional permission, I have proposed the following postulates: A1 - P ( p v ,~p/c) A2.1 - P(p v q/c) --~ [P(p/c) v P (q/c)] A2.2 - [P(p/c) v P(q/c)] --~ P(p v q/c) A3 - (p --~ q) --~ [P(p/e) --~ P(q/c)] A4 - P ( p & q / c ) - - ~ P ( p / c & q ) A5 - [P(p/c) &P(q/c & p)]--~ P(p & q/c) A6 - P ( p / c v , ~ c ) ~ P ( p / d ) A7 - P ( p / d ) --~ P ( p / c & ,~c) Anderson quite justly points out that A5 cannot be accepted as it stands. His argument is as follows. From A5 (letting q be ,~p) we obtain (1) ]- [P(p/c) & P(,-~p/c & p)] ~ P(p & ,~p/c). But by A3 we have (2) I- [(P& " P ) "-~ q] --+ [P(P& N p / c ) --~ P(q/c)]. Since the antecedent of (2) is a theorem in the familiar systems of strict implication, we may assert its consequent, and thus (1) and (2) yield (3) [- [P(p/c) &P(,~p/c & p)] ~ P(q/c). But (3), as Anderson rightly argues, is not an acceptable deontie thesis. For (to give his counter-example) a person is permitted to smoke in a railroad smoking car, and is also permitted not to smoke, even if in fact doing so. And this does surely not entail that "anything goes" in the smoking car. I quote Anderson's analysis of this situation, which I endorse as entirely correct: "The trouble appears to me to stem from AS, which enables us to arrive at the conclusion that a contradiction might under suitable circumstances be permitted; contradictions are not "permitted" logically, and it is doubtful we should be deontically more generous than we are logically." It is thus evident that A5 must be modified to avoid just this feature that it may lead to the "permission" of a contradiction. To this end, it is clearly adequate to add to the antecedent of A5 the requirement that the compound act per-

CONDITIONAL PERMISSION IN DEONTIC LOGIC

3

mitted by its consequent is logically possible. I therefore propose to replace A5 by A*5 I-[P(P/C) & P ( q / e & p ) & O ( p & q ) ] ~ P ( p & q / c ) . With this necessary correction in our deontie axioms accomplished, we can proceed to a consideration of Anderson's proposal for a reduction of conditional to unconditional deontic logic. ANDERSON'S REDUCTION OF CONDITIONAL DEONTIC LOGIC

Anderson's proposal is put forward in the following terms: "We can define conditional permission P(p/c) as c---> P(p), with the result that Rescher's axioms A1-4 and A6--7 are provable. A5, however, turns out not to be provable, which is as one would hope, in the light of the foregoing argument. The resulting concept of conditional permission appears to be stronger than that characterized [only] by Rescher's A1--4 and A6--7 . . ." One emendation is necessary here. On Anderson's definition of P(p/c), A2.1 becomes (4) I-[c---> (pvq)]---> [ ( c - + p ) v (c---> q)] and this is not a theorem in the usual systems of strict implication. (Thus, when ---> is construed as entailment, we have that "x is a positive integer" entails "x is odd or x is even"; but we have neither "x is a positive integer" entails "x is odd," nor "x is a positive integer" entails "x is even.") Therefore, in the face of Anderson's proposed definition of P(p/c), not only A5 (and A*5), but also A2.1 is not forthcoming. It is consequently a matter of some interest to determine the acceptability of this definition by criteria quite independent of the proposed axiomatization of conditional permission. A CRITERION OF A D E Q U A C Y F O R ANDERSON'S DEFINITION

I propose now to develop de novo several specific objections to the definition (D) P(p/e) --D~c-> P(p). The rationale which underlies these objections is as follows. Given (D), we can transpose certain theorems, governing the relationship " ~ " in the usual systems of strict implication, into theses regarding conditional permission. These theses can then be examined with a view to their deontic acceptability in the light of our informal conception of deontic ideas. To the extent that these derived theses prove to be deontically unacceptable, we have developed considerations which militate against the acceptance of (D). Objection 1. The following is an acceptable assertion in the usual systems of strict implication:

4

PHILOSOPHICAL STUDIES

(5) [- [(c-+ b) & (d-+ b)]-+ [(c&d) -+b]. We can apply (D) to transform (5) into a thesis of conditional deontic logic: (T1) [- [P(p/c) & P(p/d)]-+ P ( p / c & d ) . This asserts that if an act is permitted in each of two circumstances realized simpliciter, it is permitted when these two circumstances are realized together. But consider the following example. A man is "permitted" to keep his hat on in the presence of ladies (e.g., out of doors), and he is "permitted" to keep his hat on in an elevator (e.g., when alone, or when only men are present), but he is not "permitted" to keep his hat on in the presence of ladies on an elevator. (Here "permitted" is used in the sense of current social etiquette.) Thus (T1) is not acceptable as a deontic thesis. Obiect/on 2. Another acceptable assertion in the usual systems of strict implication is (6) l-(c-+b)-+ [ ( c & d ) - + b ] . In view of (D) this leads to the deontic theses (T2) [-P(p/c) - + P ( p / c & d ) . Now in view of the generally accepted definition of conditional obligation, viz. O(p/e) ---vf ~ P ( ~ p / c ) , (T2) yields (7) I - O ( p / c & d ) -+ O(p/c). But (7), which asserts that if an act is obligatory in a complex (compound) circumstance it is obligatory when only a part of this circumstance is realized, is clearly unacceptable as a deontic principle. For example, I am indeed (morally) obligated to pay Smith five dollars if (1) I am able to do so and (2) I owe him the money; but I am surely not obligated to pay him this sum when the former circumstance alone is realized. Thus (T2) is unacceptable as a deontic principle. Obiection 3. Again, it is an acceptable assertion in the usual systems of strict implication that (8) ]- (c-+ d) -+ [ ( d - + b ) -+ (c-+ b)]. On this basis, (D) yields the deontic thesis (T3) [- (c ~ d) -+ [P(p/d) -+ P(p/c)]. But consider the following counter-example to (T3) as a deontic principle: "X is a monk" entails "X is an unmarried male"; unmarried males are permitted to marry; therefore, monks are permitted to marry. (Here "permitted" is used in the sense of "countenanced by canon law.") This inference, though warranted by (T3), is clearly not valid deontically.6

CONDITIONAL PERMISSION IN DEONTIC LOGIC

5

THE ROOT OF THE DIFFICULTY

I come, finally, to what I regard as the most crucial and fundamental reason for urging the deontically unpalatable character of (D). This can readily be exhibited. Since (9) 1-(c-~ b) ~ [(c&d) -*b] is a principle inevitable in the usual systems of strict implication, (D) entails (as above) (T2) 1- P(p/c) --~ P ( p / c & d). But this thesis has the gravest consequences for deontic logic. It has the result of rendering the concept of conditional permission effectively inapplicable. For (T2) leads to the requirement that be/ore we can assert that an action is permitted in some particular circumstance, we must specify this circumstance in such a way as to exclude all imaginable countervailing conditions. The implication of (T2) for the theory of conditional permission is that it establishes the prerequisite that, in stating the circumstance c in which an act p is permitted, our description must be so compIete that no conceivable (conjunctive) modification of the circumstance c could possibly preclude the permittedness of p. But this is a task that is hopelessly cumbersome. In common life we frequently are in a position to say that an act is "permitted" in certain circumstances, for example, that smoking is permitted in a railroad smoking car, but we are seldom (if ever) in a position to specify all the conceivable further modifications of these circumstances which would, if also realized, preclude the "permitted" status of the act. In the interest of having a concept of conditional permission that is applicable in practice, it is thus necessary to exclude (T2) as a thesis of deontic logic. And a rejection of it necessitates a rejection of (D) as well. CONCLUSION

In the face of the considerations set forth above, I submit that Anderson's proposed definition (D) does not represent an acceptable construction of the concept of conditional permission. This, in turn, suggests that conditional permission must be viewed as a viable deontic relationship in its own right, and is not definable in terms of unconditional deontie concepts. It appears, then, that a reduction of conditional to unconditional deontic logic is not warranted. Received May 15, 1959 NOTES 1 N. Rescher, "An Axiom System for Deontic Logic," (1958).

Philosophical Studies, 9:24-30

6

PHILOSOPHICAL STUDIES

G. H. yon Wright, "A Note on Deontic Logic and Derived Obligation," M i n d , 65: 507-9 (1956). 8For convenience, the practice of autonomous use of symbols is adopted when confusion cannot result. ' A. R. Anderson, "On the Logic of 'Commitment,' " Philosophical Studies, 10:23-27 (1959). 5The articulation of deontic logic (in both its conditional and unconditional versions), presupposes a system of modal logic, i.e., strict implication, to whose symbolism a .primitive predicate (or relation) P is added, subject to a suitable set of additional axioms. Note that if we have (p &q) ~ p---as we do in the usual systems of strict implication-then (8) entails (6), and (6) entails (5). Thus Objection 1 is, as it were, the most fundamental in that, if sustained, it would serve also for the work done by Objections 2 and 3. However, any one of the objections suffices to invalidate the definition (D).

Reply to Mr. Rescher by ALAN ROSS ANDERSON YALE UNIVERSITY IN THE preceding paper, Mr. Rescher corrects an error of mine, and offers intuitive counter-examples to three theorems of a system of deontic logic which arises if we add to one of the familiar systems of alethic modal logic a definition of conditional permission: P ( p / c ) ----o~c---~ P ( p ) . I no longer wish to defend this proposal, since I no longer believe that the familiar systems of alethic modal logic can be regarded as providing an analysis of " i f . . . then . . .-1 But if we take the arrow as e n t a i l m e n t in the sense of the system E, 2 then it still seems to me that conditional permission so defined captures an important intuitive notion. It is clear that when we say " p is permitted in the circumstance c," we never (or at least very rarely) mean that c is a n e c e s s a r y condition for p's being permitted. But surely we sometimes mean that c is a sufficient condition for p's being permitted, and this is the notion I was trying to capture in the proposal above. In consulting a lawyer, for example, as to whether p is permitted to a person in our circumstances c, we expect him to give more than the prima-hcie answer, and to go on to consider possible relevant special circumstances; and if he does tell us to go ahead with the project, we would feel that we were badly advised if it should turn out that by reason of special overriding considerations p was not permitted to us after all.

REPLY TO MR. RESCHER

7

To consider Reseher's examples in particular, then: Reply to Objection 1. The statement that a man is "permitted" (by current etiquette) to wear a hat in the presence of ladies seems to me simply false--at least this is the moral I should wish to draw from Rescher's example. Surely the proper answer to the question whether this is permitted is not "Yes," but rather "Well, that depends . . . sometimes it's all right, and sometimes it isn't." Reply to Objection 2. The argument concerning

O(p/c &d)

O(p/c)

seems in fact an argument against O ( p / c & d) --->O(p/c & Nd). "For example, I am indeed (morally) obligated to pay Smith five dollars if (1) I am able to do so and (2) I owe him the money; but I am surely not obligated to pay him this sum when the former circumstance alone is realized." And of course the latter is not a theorem of any of the systems under con sideration. But it might be easier to see the point if we look at the equivalent formula. V(p/c) --, P(p/c &d). Under the interpretation "c is a sufficient condition for p's being permitted," it surely follows that c & d is also a sufficient condition for p's being permitted. " 0 (p/c)" then has the interpretation "c is not a sufficient condition for permitting ,~p,,,a and again it would seem to follow that if c & d is not a sufficient condition for permitting ,-,p, then c is not a sufficient condition either. Reply to Objection 3. The proposed counter-example to (c ~ d) ~ [P(p/d) ~ P ( p / c ) ] seems to me to fail to be a counter-example; if the major premise is taken to mean "some unmarried males are permitted to marry," then the syllogism is formally invalid, and if it is taken to mean "all unmarried males are permitted to marry," then it is (under the hypotheses of the example) simply false. This is not to say, however, that the notion of conditional permission as defined above is the only one, or even the most important one. It might also be of interest to understand the logic of Rescher's concept, which it seems to m e fair to characterize as doubly conditional permission: "P (p/c)" having the interpretation "p is permitted to those circumstances c, other things

8

P H I L O S O P H I C A L STUDIES

being equal (or provided there are no overriding considerations to the contrary) ." T h e difficulty, of course, is that this amounts to saying that p is permitted to those in circumstances c--unless it isn't. And while m a n y of our blanket moral injunctions have these loophole clauses ( " T h o u shalt not kill--unless it is all right t o " ) , they d o n ' t provide us with an awful lot of guidance. T h e r e is also another notion of conditional permission P', defined as P ' ( p / c ) = e l ,~ (c--> --,Pp), i.e., c is not a sufficient condition for p's being forbidden. For this notion we have the following theorems in O E 4 (corresponding to Rescher's axioms): AI.' ALl' A2.2' A3.'

c--> P ' ( p v ~ p / c ) P ' ( p v q / c ) --> [ P ' ( p / c ) v P ' ( q / c ) l [ P ' ( p / c ) v P ' ( q / c ) ] --> P ' ( p v q / c ) (p-->q) ~.P'(p/c) -->P'(q/c)

Analogues of A4-A7 are, however, not provable in O E . A6, for example, comes to d--> N p p --~. c v ,-,e --> N p p , which is unacceptable if we subscribe to the principle that anything entailed by a necessary truth is necessary, since it leads to d --> ---Pp -->. I-l,-~Pp (where

" [ ] " is logical necessity). which of these various reconstructions of the intuitive ideas is the most satisfactory seems a m o o t point; it is likely that various contexts require various interpretations--Rescher's being as adequate a candidate as any of the others.

lust

Received M a y 20,

1960

NOTES 1 This is not the place to defend this wildly heterodox claim, but it seems evident that neither material nor strict "implication" represents the intuitive "if . . . then . . ." if we reflect on the meaning of the denial of such a statement, i don't in fact think that GSdel's completeness theorem follows from the fact that snow is white, and I accordingly reject as false the statement "if snow is white then the first order functional calculus is complete." But when I say that that statement is false, I do not mean that snow is white and the first order functional calculus is incomplete, or even that that is possible. journal of S y m b o l i c Logic, 23:45%58 (1958). In E, the denial of an "if . . . then . . ." assertion p--> q does not have as a consequence that p & ~q, or even <>(p & ---q). I.e., the system arising from E if we add an uninterpreted propositional constant "b," defining "Pp" as "N(p--> (b & O ~ b ) ) , " and other deontic modalities in the obvious way. (See Mind, n.s., 67:100-4 (1958), and for motivation Logique et analyse, n.s., 1:84-91 (1958).)