(1) O(a v ~a)
(2) F(a &~a)
or in general:
(3) Ot
(4) Ff
where "t" stands for a tautology and "f" a contradiction. That is to say, to realize a tautology is obligatory and it is forbidden to actualize a contradiction. This may give us some misgivings. Indeed we may even be afraid that to evaluate the tautologous act obligatory or the contradictory act forbidden, we simply stretch the deontic concepts a good way beyond recognition. However, some deontic logicians might tend to think that the situation is harmless and hence tolerable. They might suggest that we simply regard(3) and (4) above as extreme or special cases in our deontic evaluation. Indeed we get quite used to all kinds of unnatural special cases both in mathematics and in logic. For instance, we may want to treat "p" standing alone as a disjunction (a degenerate one, that is) which has only one disjunct;2 or to treat "~" as a connective which conjoins a proposition to nothing. Accordingly, we may think (3) and (4) innocuous, because every one necessarily does the tautologous act all the time,3 and no one can ever perform the contradictory act. It follows that every one automatically and logically fulfills the obligation of actualizing a tautology and no one can ever commit the immorality of bringing about a contradiction. Hence, it may be argued as intuitively unobjectionable that we treat the obligation of realization of a tautology as a special case of obligation, and the forbiddance of actualization of a contradiction a special case of forbiddance.
Our purpose now is to see whether this is only an innocent diversion from our intuitive notion of obligation, forbiddance and the like. However, we shall take the consideration of this matter only as a starting point and step into questions of a more general nature, namely, the questions of the association of deontic modalities and alethic modalities and its consequences.
SOME OBJECTIONS
It might be argued that the tautologous act and the contradictory act are trivial examples of act if they are acts at all. Indeed the triviality of them can be seen from the universality of the performance of the former and the impossibility of the performance of the latter. It might even be claimed that they are in fact not bona fide acts. They are bogus because they are so different in nature from the genuine acts such as smoking a cigarette or walking across the hall. Indeed, if we count the tautologous act as a genuine act, then we can no longer say that a certain person does nothing, namely, performs no acts, because every body at any time in any place does at least one act, the tautologous one. But we do think we say something significant when we utter "He does nothing" or something of this sort. Hence, should it not be the case that when we say "doing something" or "performing some act", what we mean is a bona fide act? But if it should be the case, then what is under deontic evaluation perhaps should also be a genuine act. It follows then that the tautologous act is not under deontic evaluation. Consequently (1) or (3) above should not be a theorem in a deontic system.
SOME OBJECTIONS
It might be argued that the tautologous act and the contradictory act are trivial examples of act if they are acts at all. Indeed the triviality of them can be seen from the universality of the performance of the former and the impossibility of the performance of the latter. It might even be claimed that they are in fact not bona fide acts. They are bogus because they are so different in nature from the genuine acts such as smoking a cigarette or walking across the hall. Indeed, if we count the tautologous act as a genuine act, then we can no longer say that a certain person does nothing, namely, performs no acts, because every body at any time in any place does at least one act, the tautologous one. But we do think we say something significant when we utter "He does nothing" or something of this sort. Hence, should it not be the case that when we say "doing something" or "performing some act", what we mean is a bona fide act? But if it should be the case, then what is under deontic evaluation perhaps should also be a genuine act. It follows then that the tautologous act is not under deontic evaluation. Consequently (1) or (3) above should not be a theorem in a deontic system.
One can refuse the theoremhood of (2) or (4) in a deontic system in a similar way, because we know that the negation-act of the tautologous act is the contradictory one.
Perhaps equally damaging is the fact that the tautologous act and the contradictory one are hardly distinguishable extensionally or exemplified ostenively.
One may also express the misgivings from a slightly different point of view. But this time let us examine the matter in a more general context. In some deontic systems which contain both deontic modalities and alethic modalities, we find the following are theorems:
(5) □a
Oa
(6) ~◇aFa
That is, of something is necessary then it is obligatory; and something is forbidden provided that it is impossible. In other words, a necessary act is obligatory, and an impossible act is forbidden.4
(5) □a
Oa(6) ~◇a
FaThat is, of something is necessary then it is obligatory; and something is forbidden provided that it is impossible. In other words, a necessary act is obligatory, and an impossible act is forbidden.4
Again one may claim the innocuousness of (5) and (6) on the ground that the obligation depicted in (5) is trivially fulfilled by every one, and the forbiddance expressed in (6) can obviously be offended by no one. However, further consideration discloses that this comfort is rather short-lived. For instance, we might find that our moral rules not only prescribe what action we should do or may not do, but also dictate what performance we should attempt or may not attempt. The following, for example, may well be one of our moral principles:
(7) What one ought to do one ought to try, and what one is forbidden to do one must not try.
That is, what is obligatory should be attempted, and what is forbidden may not be attempted. In other words, if a is obligatory, then a should be attempted; and if a is forbidden, then a may not be attempted. In symbol:
(8) OaOa
and
(9) FaFa
where the "capped a" means the attempt of a. Now (8) and (9) seem intuitively unquestionable, and indeed welcomed as principle or rule in our morality. However, (6) and (9) jointly imply
(10) ~◇aFa
That is, if a is impossible then the attempt at a is forbidden. Or, in other words, what is impossible must not be attempted. This, of course, is clearly intolerable as a deontic thesis. For (10) entails, among other things, that those mathematicians who tried to trisect an arbitrary angle with only a straightedge and compass, or those logicians who attempted to find a general decision procedure for first-order logic, committed a certain kind of moral error. And who knows how many ingenious people, while sincerely and earnestly applying their inventive minds, stepped unwittingly into a moral sin! Therefore, we can no longer take the comfort in thinking that (5) and (6) are innocuous special cases. The situation becomes even more astonishing if we allow ourselves, as we shall point out later, different kinds of possibility, necessity and contingency.
PRINCIPLE OF POSSIBILITY AND PRINCIPLE OF NON-NECESSITY
We may want to ask what then is the source of the trouble. Of course it is too hasty to suggest that deontic logic "abhors" alethic modalities. Actually this seems to be not the case. Simply consider the so-called Kantian principle, the principle usually taken to mean that ought implies can, namely
(11) Oa◇a
That is, if a is obligatory then a is possible, or, in other words, what is obligatory is possible. Examination of this principle seems to reveal that no apparent absurdities are brought about by its inclusion as a moral thesis in a deontic system. On the contrary, (11) seems totally natural and obviously defensible. For consider
(12) Oa&~◇a (a is obligatory but a is impossible)
We immediately feel an absurdity in (12). Therefore, it follows by reductio ad absurdum the soundness of the Kantian principle, namely (11). The question arises: Why does (11) look so natural but (6) sound so counter-intuitive? It might be suggested that this is so because in one case deontic modalities appear in the antecedent and alethic modalities in the consequent of a conditional such as (6) or (11), while in the other case, it is the other way around. Or, if we like, in one case the alethic concepts are used to specify the necessary condition, but not the sufficient condition, of what is expressed in a deontic formula; while in the other case the opposite is true. But this is hardly the reason. For the Kantian principle entails, among other things, the following:
(13) □aPa (If a is necessary, then a is permissible)
provided that we agree with the ordinary understanding concerning the relationships between necessity and possibility and between obligation and permission. Of course, here we may ask: But does (13)look as intuitive and unquestionable as (11)? The answer seems to be in the positive. We need only consider
(14)□a&Pa
that is
(15)□a&Fa
and the absurdity comes to the surface. Hence the soundness of (13).
(7) What one ought to do one ought to try, and what one is forbidden to do one must not try.
That is, what is obligatory should be attempted, and what is forbidden may not be attempted. In other words, if a is obligatory, then a should be attempted; and if a is forbidden, then a may not be attempted. In symbol:
(8) Oa
Oaand
(9) Fa
Fawhere the "capped a" means the attempt of a. Now (8) and (9) seem intuitively unquestionable, and indeed welcomed as principle or rule in our morality. However, (6) and (9) jointly imply
(10) ~◇a
FaThat is, if a is impossible then the attempt at a is forbidden. Or, in other words, what is impossible must not be attempted. This, of course, is clearly intolerable as a deontic thesis. For (10) entails, among other things, that those mathematicians who tried to trisect an arbitrary angle with only a straightedge and compass, or those logicians who attempted to find a general decision procedure for first-order logic, committed a certain kind of moral error. And who knows how many ingenious people, while sincerely and earnestly applying their inventive minds, stepped unwittingly into a moral sin! Therefore, we can no longer take the comfort in thinking that (5) and (6) are innocuous special cases. The situation becomes even more astonishing if we allow ourselves, as we shall point out later, different kinds of possibility, necessity and contingency.
PRINCIPLE OF POSSIBILITY AND PRINCIPLE OF NON-NECESSITY
We may want to ask what then is the source of the trouble. Of course it is too hasty to suggest that deontic logic "abhors" alethic modalities. Actually this seems to be not the case. Simply consider the so-called Kantian principle, the principle usually taken to mean that ought implies can, namely
(11) Oa
◇aThat is, if a is obligatory then a is possible, or, in other words, what is obligatory is possible. Examination of this principle seems to reveal that no apparent absurdities are brought about by its inclusion as a moral thesis in a deontic system. On the contrary, (11) seems totally natural and obviously defensible. For consider
(12) Oa&~
◇a (a is obligatory but a is impossible)We immediately feel an absurdity in (12). Therefore, it follows by reductio ad absurdum the soundness of the Kantian principle, namely (11). The question arises: Why does (11) look so natural but (6) sound so counter-intuitive? It might be suggested that this is so because in one case deontic modalities appear in the antecedent and alethic modalities in the consequent of a conditional such as (6) or (11), while in the other case, it is the other way around. Or, if we like, in one case the alethic concepts are used to specify the necessary condition, but not the sufficient condition, of what is expressed in a deontic formula; while in the other case the opposite is true. But this is hardly the reason. For the Kantian principle entails, among other things, the following:
(13) □a
Pa (If a is necessary, then a is permissible)provided that we agree with the ordinary understanding concerning the relationships between necessity and possibility and between obligation and permission. Of course, here we may ask: But does (13)look as intuitive and unquestionable as (11)? The answer seems to be in the positive. We need only consider
(14)□a&Pa
that is
(15)□a&Fa
and the absurdity comes to the surface. Hence the soundness of (13).
Now it seems not only does (12) ruffle our intuition of moral concepts, but so does a whole series of others. Specifically speaking, the following seem to be no less unproblematic:
(16)Pa &~◇a
(17) Fa &◇a
(18) Ia &◇a
(16)Pa &~◇a
(17) Fa &◇a
(18) Ia &◇a
For instance, to permit someone to do something impossible is a vacuous and void permission, hence (16) seems quite unnatural. Again to prohibit something impossible is no more than an empty forbiddance, and so the rejection of (17). (18) is likewise undesirable on a similar reason, because it implies, among other things, (16) above.
If this observation is correct, then we may establish the following as three theses in a deontic system:
(19) Pa◇a
(20) Fa◇a
(21) Ia◇a
(19) Pa
◇a(20) Fa
◇a(21) Ia
◇aLet us telescope these three propositions together with (11) in one, and put it in general terms thus:
(22) △A「◇ A
This may be rendered as: If a is deontically evaluated, the a is possible. That is, only possible acts are morally judged. We may call this generalized Kantian principle.5 But we would prefer to call it principle of possibility.
(22) △A
「◇ AThis may be rendered as: If a is deontically evaluated, the a is possible. That is, only possible acts are morally judged. We may call this generalized Kantian principle.5 But we would prefer to call it principle of possibility.
However, this is hardly the end of the story. Let us ask if the following are by any means less counter-intuitive:
(23) Oa &~◇~a
(24) Pa &~◇~a
(25) Fa &~◇~a
(26) Ia &~◇~a
First, consider (23). One has the obligation to do a, but it is impossible that he does not do it. How trivial is this kind of obligation! Again, the permission depicted in (24) is so weak that one does not obtain any real license to do the act in question. And if (25) is tolerable, that is, one is forbidden not to do what is impossible, then every one is born a sinner not religiously but deontically ! Similar considerations applies to (26). Therefore, we may want to reject (23)-(26). From their rejection, we have four conditionals. We again telescope them in one and express it in a general way thus:
(27) △A「◇~A
Let us call this principle of non-necessity, because it is equivalent to
(28) △A「~□A
PRINCIPLE OF CONTINGENCY
Of course, from the principle of possibility and the principle of non-necessity, it follows
(29) △A「◇A&◇~A
We shall call this principle of contingency. Roughly speaking, it states that only contingent acts, that is, neither necessary acts nor impossible acts, are subject to moral judgement. In other words, if a is morally evaluated, then it is possible to do a and it is also possible not to do a . This principle of contingency would no doubt be welcomed by many people, especially by those who claim that moral responsibility entails free choice, regardless of how they choose to construe free choice. For how can we say that we have free choice if we cannot choose either to do an act or not to do it; and how can we choose either to do an act or not to do it if either the act or its negation-act is impossible ?!
(23) Oa &~◇~a
(24) Pa &~◇~a
(25) Fa &~◇~a
(26) Ia &~◇~a
First, consider (23). One has the obligation to do a, but it is impossible that he does not do it. How trivial is this kind of obligation! Again, the permission depicted in (24) is so weak that one does not obtain any real license to do the act in question. And if (25) is tolerable, that is, one is forbidden not to do what is impossible, then every one is born a sinner not religiously but deontically ! Similar considerations applies to (26). Therefore, we may want to reject (23)-(26). From their rejection, we have four conditionals. We again telescope them in one and express it in a general way thus:
(27) △A
「◇~ALet us call this principle of non-necessity, because it is equivalent to
(28) △A
「~□APRINCIPLE OF CONTINGENCY
Of course, from the principle of possibility and the principle of non-necessity, it follows
(29) △A
「◇A&◇~AWe shall call this principle of contingency. Roughly speaking, it states that only contingent acts, that is, neither necessary acts nor impossible acts, are subject to moral judgement. In other words, if a is morally evaluated, then it is possible to do a and it is also possible not to do a . This principle of contingency would no doubt be welcomed by many people, especially by those who claim that moral responsibility entails free choice, regardless of how they choose to construe free choice. For how can we say that we have free choice if we cannot choose either to do an act or not to do it; and how can we choose either to do an act or not to do it if either the act or its negation-act is impossible ?!
However, this principle of contingency is not as innocent as it may at first look to be. This can be seen as follows. From (29), indeed from (27), we may infer
(30) □a~Pa
This and (13), which is a consequence of the Kantian principle, jointly imply
(31) □a (Pa &~Pa)
That is , if something is necessary, then it is permissible to do it and it is also not permissible to do it. That means whenever we have a necessary act at hand ,i.e.,
(32) □a
we also have a predicament in sight, namely
(33) Pa &~Pa
A contradiction.
(30) □a
~PaThis and (13), which is a consequence of the Kantian principle, jointly imply
(31) □a
(Pa &~Pa)That is , if something is necessary, then it is permissible to do it and it is also not permissible to do it. That means whenever we have a necessary act at hand ,i.e.,
(32) □a
we also have a predicament in sight, namely
(33) Pa &~Pa
A contradiction.
This difficulty is not easily overcome by simply refusing to count a necessary act as a genuine act, and hence blocking it from entering into our deontic evaluation. For apart from the logical sense of alethic concepts, it is desirable in ethical context to consider other senses of alethic concepts. It is useful, for instance, to have the following quadrification of possibility: logical possibility, empirical possibility, human possibility, and personal possibility. And the corresponding necessities and contingencies. Now if an act is humanly or personally, but not logically, necessary, we cannot avoid the above-mentioned logical absurdity by saying that it is not a genuine act.
1For example, system FA of Fisher and Aqvist is an exception.
2Recall, for example, the usual definition of complete disjunctive normal form. And we may say "p&q" is a special case of that form.
3It is not quite clear whether a dead man also performs the tautologous act.
4We say that this is a more general context, because the tautologous act is a special case of necessary acts, and a contradictory act a special case of impossible acts. Moreover, it can be shown easily that
(5) tOt
and
(6') fFf
are theorems of a deontic system containing standard propositional logic provided that (3) and (4) are theorems and expressions such as (5') and (6') are well-formed in that system.
5.We take (22) as a telescoped version of (19)-(21) and (11). However, we can also propose to define △A as「OA v PA & ~FA & ~IA.From either treatment, (22') below follows from(22):
(22')「~◇A「~OA&~PA&~FA&~IA
We may observe that the consequent of (22') implies a contradiction. Therefore(22') and hence (22), the generalized Kantian principle, admit of two intuitive renderings. Either the impossible acts are not morally evaluated, or that moral evaluation of impossible acts involves a contradiction.
1For example, system FA of Fisher and Aqvist is an exception.
2Recall, for example, the usual definition of complete disjunctive normal form. And we may say "p&q" is a special case of that form.
3It is not quite clear whether a dead man also performs the tautologous act.
4We say that this is a more general context, because the tautologous act is a special case of necessary acts, and a contradictory act a special case of impossible acts. Moreover, it can be shown easily that
(5) t
Otand
(6') f
Ffare theorems of a deontic system containing standard propositional logic provided that (3) and (4) are theorems and expressions such as (5') and (6') are well-formed in that system.
5.We take (22) as a telescoped version of (19)-(21) and (11). However, we can also propose to define △A as「OA v PA & ~FA & ~IA
.From either treatment, (22') below follows from(22):(22')「~◇A
「~OA&~PA&~FA&~IAWe may observe that the consequent of (22') implies a contradiction. Therefore(22') and hence (22), the generalized Kantian principle, admit of two intuitive renderings. Either the impossible acts are not morally evaluated, or that moral evaluation of impossible acts involves a contradiction.