Church's Paradox of Modality
Steve Bayne

One of Russell's contributions to philosophy was to impress on philosophers that an incremental approach to philosophical problems can have implications for systems of "global" significance. Nevertheless, an examination of the logicians that followed him, those at any rate who thought philosophically, reveals little temerity when it came to making bold assertions. Extravagant theories abounded and Popper's maxim that the better theory is the theory that takes more risks but resists refutation seemed to apply to logicians as well. I want to suggest that there is a common misconception about Alonzo Church's treatment of the relationship between Russell's doxastic paradox and Quine's modal paradox I will argue that Church believed that the modal paradoxes that come about by substitution (ala Quine) and the doxastic paradoxes (like Russell's) that come about by substitution involve a single paradox In other words, the modal and doxastic paradoxes are in fact the same paradoxes. This is not just to say that there is a common source of both kinds of paradox (such as would follow if both sorts were a variation on the paradox of material implication as Nathan Salmon appears to suggest). It is to say that the class of modal paradoxes is equinumerous to the class of doxastic paradoxes because they are inter derivable. This, I believe, is Church's position ("A Remark Concerning Quine's Paradox about Modality." in Propositions and Attitudes. edited by Nathan Salmon and Scott Soames. Oxford. 1988).
    What I will deny is that modal paradoxes have corresponding doxastic correlates in a couple of cases, maybe. This reflects a temerity that underestimates both Church's character and the power of his arguments. I will also suggest that even *if* it can be maintained that this is true then Church did not go far enough because in fact there is a strong argument that the modal and doxastic paradoxes are essentially one and the same in ALL their manifestations. This latter claim, if not Church's then mine, requires the license of one philosophical move, viz. that "knows" stands to "believes" as "necessary" stands to "possible." Given this, ALL modal and doxastic paradoxes reduce to paradox of a single sort, and the class of modal and doxastic paradoxes have identical extensions. I have received some resistance from friends and colleagues on this matter, but on each occasion an underlying concern by critics is a "patriotic" devotion to modal logic. Having never been a devotee of the notion that modality is the key to philosophy what I offer here is not treason but reasons. I have elected to break this posting into several, since not all Russell fans will be interested in this topic.
    I am going to offer an attenuated version of the account Church gives of Quine's modal and Russell's doxastic paradoxes. I use 'L' for 'necessarily' and 'M' for 'possibly'. Consider, then, that from

1. L(9=9)


2. The number of planets = 9

that (3) follows.

3. L(The number of major planets = 9)

(3) seems false, even though (1) and (2) seem true. (3) seems false, since I may falsely believe that the number of planets is 6 without believing an impossibility. Consider next a slight variation on Russell's doxastic paradox (from 'On Denoting'. Mind. 1905). It arises from

4. George the IV believed that 9 = 9

by substitution of 'the number of major planets' (2) for one occurrence of '9'.

5. George the IV believed that the number of major planets = 9

What is Church up to? Consider a couple of things he says. First, he clearly believes that the steps leading up to (3) and (5), respectively are exactly the same. The *only* difference is that in place of a modal operator in one case, the first, there is a doxastic operator in the other.

It may therefore be argued that there are not two genuinely different paradoxes, but only various examples illustrating what may be regarded as a single paradox (op cit. p. 59)
But it seems to me that Church wants to make a more important claim than simply that in one instance a modal and doxastic paradox have been shown to be closely related. No! Church wants to make a much bigger and important claim, one that extends to the antinomy of the name relation itself and beyond He remarks,
The paradox may arise not only in connection with 'believes that' and 'it is necessary that' but also with any of various other phrases... (op cit. p.60).
Although here he is addressing only the antinomy of the name relation, this quotation serves to emphasize the importance of the operators, the individual semantics of which I believe are essential to extending his views with the greatest generality. I will clarify this in a later part.
    My claim is that Church wants to say that all, not just a couple, of the modal and doxastic paradoxes can be shown to be basically one. This is a much bolder claim and addresses not only consequences of unrestricted substitution but the nature of modal paradox itself. Could it be that there is something paradoxical about modal systems, something more fundamental than substitution in intensional contexts? If it should turn out that Alonzo Church one of the greatest logicians of the century (and beyond) only intended to show the ultimate identity of two paradoxes without showing that other related paradoxes are also so reducible then someone else will have to take the credit for making this claim. The credit I believe belongs to Church. Now lets take a look at exactly what Church thought paradoxical about some modal statements and present his case where variables are at issue.
    We have examined two paradoxes, one modal, one doxastic. The steps involved in the derivation were identical. In fact, as Church notes, the paradoxes differ ONLY in "the replacement of 'George IV believes that' by the modal operator 'L' (Church uses box notation)." The steps involved in the derivation are the same but there are other derivations where the same "steps" are made without the ensuing paradox It is substitution of these operators that is important in making the derivations derivations of a paradox Let's take a look at "two" other paradoxes involving modality and belief and see exactly what it is that Church finds paradoxical about them. Let's start with the modal case. Here Church shows that he was not only a really bold logician but also an interesting fellow. Why do I say this? Judge for yourselves.
    The innovation is in taking possibility rather than necessity as the relevant modal operator, again it is the operators not the form the derivation takes that gives rise to the paradoxes. Taking as his point of departure the discernibility of non-identicals

1'. not-Fx   É x Fy   É y not(x=y)

and the assumption

2'. (x) not-M not-(x=x)

he derives by substitution (of F) in the first formula (3'):

3'. not-M not-[x=x]   É x M not-[x=y]   É y not-(x=y)

By modus ponens, that leaves

4'. M not-[x=y]   É  not-(x=y),

"If two things are possibly different then they *are* different" (op. cit. p.61).
Next Church show that there is a corresponding paradox which differs only in exhibiting a doxastic (i.e. belief) operator.

By substitution into (1') we get

5'. For every x and every y, if George IV does not believe that not(x=x), if George IV believes that not(x=y), then not(x=y)

Analogous to (2') we have

6'. For every x, George IV does not believe that not-(x=x).

Similarly, we can derived (7') from (5') and (6'):

7'. For every x and every y, if George IV believes that not-(x=y), then not-(x=y).

What is paradoxical here is that beliefs "control the actual facts" just as in the modal case mere possibility controls actuality.

Either Church's intent was to show that the modal and doxastic paradoxes are *all* basically one; or he wanted to show that only a couple reduce to one. I believe he wanted to show that they were all basically one but only considered those for which derivations were available, viz. by substitution etc. I will propose that there are other derivable modal paradoxes which, strictly speaking, have no correlative doxastic paradox This will show that the classes of doxastic and modal paradoxes differ. But I will *also* argue that if we allow, as some have that 'know' stands in relation to 'believe' as 'necessary' to 'possible' the ALL modal and doxastic paradoxes have a common root or are in fact one paradox, thereby extending Church's results - assuming that his claim was not as modest as some of our timid young logicians would have it.

In the sense that Church found it paradoxical that possibility should "control" actuality, modal system S5 is paradoxical. The paradox that issues from S5, in this sense, has no strict doxastic correlate. Therefore, the class of modal paradoxes and the class of doxastic paradoxes will never be shown to be identical. However, it will be shown that there is a way of arguing to the contrary, but only by assuming a semantical relation between doxastic operators that is not without controversy. Let us begin by deriving the paradox of S5.

It is common knowledge that in S5 the following is derivable (vide An Introduction to Modal Logic by Hughes and Cresswell. London. Methuen. 1968. p.190).

1''. not-(x=y)   É xy     L(not-(x=y))

What has never been observed in the literature is that from

2''. M-(not-(x=y))   É xy     not-(x=y)

and (1'') (3'') follows.

3''. M-not-(x=y)   É xy     L (not-(x=y)).

Now the main point: if we accept Church's belief that possibility controlling (in some sense) actuality is paradoxical, then (3'') is ipso facto paradoxical. But here is the problem for the generalized Church thesis (the thesis that modal and doxastic paradoxes are essentially one): there is no corresponding doxastic paradox, i.e., no paradoxical sentence corresponding to (3''). We might attempt using the interdefinability of the relevant modal operators to reduce the number of required corresponding doxastic operators to one in the formula

4''. M-not-[x=y]   É xy     not M-not-(not (x=y))

which by double negation is the same as

4'''. M-not-[x=y]   É xy     not M-(x=y),

but the move is without appreciable effect. The reason is clear, if we take the doxastic operator and substitute it for the modal operator there is still no sentence paradoxical in the respect that makes the *conclusion* of the standard modal paradox (Quine's) paradoxical. Remember, it is not the derivation that makes a paradox; it is the conclusion. So if we do substitute the doxastic operator (in 4''') we get the truism

5''. George IV believes that not-(x=y) then not-George believes (x=y).

But now recall that I promised that I would consider the possibility that Church's claim only wanted to show a couple of paradoxes reduce to one. I would then make the following claim, attempting to be as bold (ala Popper/Peirce) as possible: all modal and doxastic paradoxes reduce to one if 'knows' stands to 'believes' as 'necessary' to 'possible'. For now if I substitute in (3'') 'knows' for 'L' and 'believes' for 'M' I arrive at another paradoxical sentence

5'''. George believes not (x=y)   É xy     not George knows (x=y)

Believing something "controls" that about which I am ignorant.Ultimately, since 'know' is factive, if I believe such and such is false, I cannot know it to be true. In the *relevant* sense of 'believe' this does not follow. I suspect, but leave it up to critics to find a case in S5 where the paradox I've discussed cannot be resolved on this view of the relation of believing and knowing. Most criticisms I suspect will be attempts to save modal logic from paradox The dominance of modal logic in philosophy, given the paucity of its philosophical rewards should make us all think twice whether pursuing it is "a game worth the candle." As for these worlds themselves, I have little to say about them - never having visited one. Perhaps someone can tell me what they are like before returning such "silly" matters as psychology as its philosophical foundations.
    One final point, Nathan Salmon suggests somewhere that the paradoxes reduce in some sense to the "paradox" of material implication. I have in the passed dismissed this out of hand, and continue to do so; but notice that the modal and doxastic paradoxes need not be equinumerous if he is right and at the same time a reduction will have been effected. At least this is how some have understood him.
    I would like to thank Mikhail Zeleny for his encouragement on these matters and his enlivening of my own interest in Alonzo Church. All errors are my own.