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Modal bloopers: why believable impossibilities are necessary.

Here is a debate I cannot lose. I argue that it is possible to (at least unwittingly) believe the impossible, say, that there is a largest prime number. The impossibilist objects that I am mistaken. Wrong move! By trying to correct me, the impossibilist concedes that I believe a false proposition. The proposition in question (i.e. that impossibilities can be believed), if false, is necessarily false. Thus, the impossibilist would be conceding that an impossibility can be believed.

If believing the impossible is impossible, then so is believing one can believe the impossible, and believing one can believe one can believe the impossible, and so on. The "and so on" elevates my debating point beyond the status of a rhetorical zinger. Most transcendental arguments can be circumvented with judicious rephrasing. Claims about essential preconditions have a notorious tendency to beg the question. However, the soundness of "I believe that it is possible to believe an impossible proposition, therefore, it is possible to believe an impossible proposition" can only be challenged only in ways that also undermine its intended adversaries. For example, a skeptic about modalities might judge the premise as false on the grounds that my thesis, [diamond] [diamond]([inverted E]p)([inverted E]x)(~[diamond]p & Bxp), is meaningless. However, the impossibilist can accept skepticism about modality only by undermining his own thesis. For the negation of a modal proposition is itself a modal proposition. Or to emphasize a related symmetry, the negation of any meaningful sentence must itself be meaningful, so the impossibilist must view his adversary's position as meaningful. This prevents him from portraying me as a "believer" who lacks an object of belief. If the impossibilist can get into the belief relation with "No one can believe an impossibility," then I can get into the belief relation with its negation.


I take myself to be in a debate that philosophers normally only dream about. To win, I need only believe my thesis. (Indeed, later it will be demonstrated that I need only appear to believe it.) The reason is that belief in "Some impossibilities are believable" guarantees its own truth. Believing so makes it so. (Self-intimation is the converse where being so makes it believed so.) This logical entailment ensures that the argument does not rely on dialectical hocus pocus. Those who affirm Bp [contains] [diamond]p tend to subscribe to substantive modal logics such as the highly popular S5. Therefore, no question is begged by recasting my argument for the infallibility thesis with the help of an asocial, though heart-warmingly quick and dirty, proof:

1. Ba[[diamond]([inverted E]p)([inverted E]x)(~[diamond]p & Bxp)] Premise.

2. ~[diamond]([inverted E]p)([inverted E] x)(~[diamond]p & Bxp) Assume for the sake of reduction ad absurdum.

3. ~[diamond][diamond]([inverted E]p)([inverted E]x)(~[diamond]p & Bxp) 2, Whatever is impossible is not possibly possible.

4. ~[diamond][[diamond]([inverted E]p)([inverted E]x)(~[diamond]p & Bxp)] & Ba[[diamond]([inverted E]p) ([inverted E]x)(~ p & Bxp)] 3, 1 Conjunction.

Intermission: Notice that line 4 gives us a concrete example of believing the impossible. To make the point a matter of principle, we need only name the bracketed proposition and apply existential generalization. So let m name [[diamond]([inverted E]p)([inverted E] x)(~[diamond]p & Bxp)]. And on with the show:

5. ~[diamond]m & Bam 4, Synonymous redescription of a proposition with a name.

6. ([inverted E]x)(~[diamond]m & Bxm) 5, Existential generalization.

7. ([inverted E]p)([inverted E]x)(~[diamond]p & Bxp) 6, Existential generalization.

8. [diamond]([inverted E]p)([inverted E]x )(~[diamond]p & Bxp)

7, Whatever is actual is possible.

9 ~~[diamond]([inverted E]p)([inverted E]x)(~[diamond]p & Bxp) By reductio ad absurdum from 2 through 8.

If we venture from the modal system S4 to S5, we can go on to invoke the principle that whatever is possible is necessarily possible. This yields the modest but fertile necessary truth:

[square]~[diamond]([inverted E]p)( [inverted E]x)(~[diamond]p & Bxp).

If no one actually happened to believe [diamond]([inverted E]p)([inverted E]x )(~[diamond]p & Bxp), then my thesis would still be rendered true by a possible believer. "Some impossibilities can be believed" may look contingent because the natural demonstration is prompted by my actual reply in a real-life debate. From the contingent fact that I believe [diamond]([inverted E]p)([inverted E]x ) (~[diamond]p & Bxp) the possibility of believing in impossibilities follows by the principle that whatever is actual is possible. Since whatever is possible is necessarily possible, my contingent belief constitutes decisive evidence for a necessary truth. Or to put the point cruelly, impossibilism is itself impossible. Belief in the thesis that impossibilities cannot be believed is a subtle instance of what it precludes!

The modal status of the commitment to believable impossibilities exposes a quasi-Godelian tension between deductive closure and consistency. In addition to driving another nail into the coffin of doxastic logic, [square][diamond]([inverted E]p)([inverted E] x)(~[diamond]p & Bxp) ruptures theories that link belief and necessity, such as Brian Ellis' epistemic semantics and Robert Stalnaker's possible worlds analysis of propositions. This will be backed up at the end of the paper.

Another welcome feature of my thesis is that its infallibility is dialectically robust; my adversaries cannot wriggle free by rejecting discretionary background assumptions. Some might think that the strength of this result is inversely proportional to its interest. If it is certain that impossibilities can be believed, then the impossibilists already have their trousers half-way down. Am I just pulling their trousers all the way down?

Only in the fashion praised through the ages. The skeptic, the egoist, and the hard determinist are partly refuted by obvious counterexamples -- the sort volunteered by sophomores and refined by common sense philosophers such as Thomas Reid and G. E. Moore. However, our teachers showed how these (usually authentic) counterexamples beg the question against subtle characters such as the skeptic, the egoist, and so on. Since question begging is relative to an audience, the counterexamples may rightly shape the opinion of other people. Partial refutations are instructive. However, universal refutations have special value. These use premises that even one's adversary must grant.

Better yet would be an explanation of how one's adversary mix-reasoned. But all I have is my negative point. I have not found any master fallacy behind impossibilism. Indeed, impossibilist arguments strike me as careful deductions from attractive premises. The impossibilists certainly do not have their pants half down in the sense of being slovenly dialecticians. Their premises are tenets from our most promising models of belief. Their logic is state of the art. I will survey the impossibilist's basic arguments and offer some criticisms. But I cannot specify where each argument goes wrong. Some have me stumped. Nevertheless, I know that all of the impossibilist's arguments are unsound. For I have an infallible belief in the negation of each argument's conclusion.


Impossibilists grant that people often act as if they believe impossibilities. A shopper will heatedly assert that 4.39 + 9.84 = 13.23. A chess novice who only has a king and a knight will attempt to checkmate his adversary's lone king. Experimenters report that some subjects judge town A as northeast of town B but B as not southwest of town A (Moar and Bower 1983). All philosophers have felt the pain of contradiction. Brother, I've had my share! Happily, reductio ad absurdum can't hurt me now. If an impossibilist were to extract an absurd consequence from my belief in believed impossibilities, then my belief would be self-exemplifying. If I'm wrong, I'm right, therefore, I'm right.

Impossibilists try to keep up appearances by distinguishing between what people really believe and what they believe they believe. For instance, the resemblance between a contradiction and a consistent statement may lead us to mistake the believer of the consistent statement for a believer in the contradiction. But given impossibilism, believing someone believes impossibilities should be a special case of believing an impossibility. Nor can they save appearances by scaling to the higher order hypothesis that we only look as if we seem to believe impossibilities. For higher order appearances would also be precluded by the impossibilist's shackle of impossibility results:

(p)(~[diamond]p=- [diamond]Bp [contains] [diamond]BBp [contains] [diamond]BBBp [contains] ... ~[diamond][B.sup.n]p)

What goes for thought, goes for talk, the chain of impossibilities precludes (even insincere) assertion of impossibilities. One can assert only what one can give the appearance of believing. Nor can the appearance of appearances be reclaimed by distinguishing between senses of "believe." Any plausible reading of "believe" must permit disagreement over the believability of impossibilities.

Of course, one can stipulate a sense of "believe" that entails logical omniscience. It's a free country. But this liberty invites the charge of misleading advertising. For example, if the impossibilist has a technical sense of "believe" in which it means "consistently believe," then his audience will protest that they assumed he was using "believe" in an ordinary sense.

Another stratagem is to abandon impossibilism about belief in favor of another propositional attitude that resembles belief (Marcus 1990). However, an attitude can resemble belief only if it allows for opposed attitudes to the same object. One can countenance this dissonance only by tolerating disharmony about the impossibility issue. The range of what is assumable or presumable is at least as wide as the range of what is believable. This difficulty will also haunt attempts to construct an interesting sort of impossibilism with respect to a technical term such as "acceptance."

One of the early impossibilists, George Berkeley (1710, 273), paraphrases apparent belief in terms of what we imagine we believe. However, like most impossibilists, Berkeley also denies that we can imagine impossibilities. After all, Berkeley subscribes to David Hume's (1739, 32) principle equating conceivability with possibility. Hume's principle generates its own nesting of impossibilities: if it is impossible to imagine an impossibility, then it is also impossible to imagine imagining an impossibility, and so on. Since imagination is the minimally encumbered propositional attitude, the progression of unimaginables is maximally constraining. Therefore, we cannot appeal to propositional attitudes to explicate the impossibilist's concession that we at least seem to believe impossibilities. But if this explication fails, all fail. Hence the impossibilist cannot afford to admit that we seem to believe impossibilities. Even the illusion of disagreement is lethal! Obviously, however, there is at least the appearance of disagreement. Therefore, we should infer that it is possible to believe impossibilities and should reject Hume's principle that conceivability implies possibility. This also scotches the logical positivist's claim that logical impossibilities can be sharply distinguished from empirical impossibilities by means of the unthinkability of contradictions (Schlick 1932/33: 42).

In one respect, the impossibilist is a lonelier debater than the solipsist. Sure, the solipsist has the same problem of self-defeating disputation; if the solipsist admits that someone else disagrees, the solipsist loses. However, the belief that only I exist is compatible with its seeming as if other people exist. So the solipsist can at least invent an imaginary friend and have pretend debates about solipsism. The impossibilist is also worse off than an eliminative materialist such as Paul Churchland (1979). Those who believe there are no beliefs can sustain debate by re-describing the controversy in a vocabulary free of "folk psychology." Or at least these eliminativists could issue the promissory note that future science will furnish this redescription. The same promissory note can be waved at the problem of higher order appearances. Just as one cannot prove to the eliminativist that there are beliefs by appealing to the infallibiilty of belief in "There are beliefs," one cannot prove to the eliminativist that some impossibilities can be believed by appealing to the infallibility of belief in "Some impossibilities can be believed." However, the impossibilist believes in belief; he restricts his skepticism to belief in impossibilities. He cannot emulate the eliminativist's indiscriminate rejection of belief in the impossible. Therefore, the anti-impossibilist argument is effective against its intended adversary.


The impossibilist might try to salvage his thesis by restricting it to a special class of beliefs. Richard Purtill distinguishes between a strong and a weak sense of 'believe." "To believe in this weak sense is to do no more than to sincerely give it as one's opinion that p" (1970, 19). This sense is lenient enough to permit belief in impossibilities. The strong sense is intended to forbid belief in the impossible. An individual a "believes p in the strong sense if and only if a understands p, and a has some reason (which seems to him to be a good reason) for thinking p to be true" (20).

The essay you are now reading is jolly good evidence that I understand my thesis, [diamond][inverted E]p)([inverted E] x)(~[diamond]p & Bxp), and have "some reason" for thinking it true. So if Purtill applies his distinction to me in accordance with his stated criteria, he should count me as a believer in the strong sense. But then I will have achieved just what the distinction was intended to thwart. My second objection is to the whole idea of treating "belief" as having disparate senses. If "belief" is as ambiguous as "bank," reflective equilibrium is impossible. To consider the case at hand, just how do my strong beliefs interact with my weak beliefs? A modest degree of epistemological holism will require that I draw inferences from mixtures of strong and weak beliefs. Yet applied logic demands an underlying unity in the attitude adopted towards the premises. The Bayesian meets this requirement by treating degrees of belief as commensurable units. But Purtill has made no gesture towards finding a common element between weak and strong belief. If he were to opt for a reductionist strategy, he would need to take strong belief as the fundamental attitude. Anything less renders the deduction of further strong beliefs intractable. But then the thinker will need to be able identify which beliefs are strong beliefs. If these selectors must themselves be strong beliefs, there will be a vicious infinite regress because each strong belief must be backed with "some reason," and this reason must be a strong belief.

The interaction problem does not arise if the impossibilist merely distinguishes between species of belief. (Then "belief" is general, not ambiguous.) For example, Curtis Brown (1990: 283) maintains that one cannot directly believe an impossibility. A direct belief holds in virtue of intrinsic facts about the believer, not facts about his environment and society. So one test for directness is to imagine whether an exact replica of the believer must share the belief. Indirect beliefs can be analyzed as those that arise from the interaction of direct beliefs and extrinsic facts. Since the believer has limited knowledge of external conditions, he has an imperfect understanding of his indirect beliefs. For example, ignorance of the fact that the man called "Samuel Clemens" is identical to the man called "Mark Twain" might lead a student to (consistently) believe that the man called Twain is a better writer than the man called Clemens. Under actual conditions, a referential convergence generates the indirect belief in the impossibility that Mark Twain is a better writer than Samuel Clemens. Since direct beliefs are not vulnerable to these misfortunes, the object of direct beliefs must always be possible. A parallel line is pursued by Albert Casullo (1979). He defends Hume's principle that conceivability implies possibility by distinguishing between basic and derived conceivability.

Brown (1992) supplements the motivation for the direct/indirect distinction by applying it to the belief puzzles raised by Hilary Putnam, Tyler Burge, and Saul Kripke. The guiding precedent is perception. Analysis of illusion forces us to abandon naive realism in favor of representative realism. Brown's search for "immediate objects of belief" is reminiscent of the perception theorist's rummagings for sense data (and the action theorist's quest for "basic actions"). As J. L. Austin's Sense and Sensibilia documented, the direct/indirect distinction is a slippery fish. So the historical record cannot be heartening to Professor Brown.

But in addition to these general worries, the direct/indirect distinction is particularly ill adapted to the specimen under scrutiny. My belief that impossibilities can be believed enjoys a Cartesian aloofness from the extrinsic factors cited by Brown. My Twin Earth doppelganger also believes that some impossibilities can be believed. The object of my belief is a necessary truth that only uses the concepts of belief and possibility.

Brown could reply that I only indirectly believe that impossibilities can be believed. After all, Tyler Burge (1986) has shown that factors outside the speaker's psychology affect the meanings of words such as "arthritis," "sofa," and "contract." Perhaps similar examples could be constructed for "possible," "belief," and "there is."

The first problem with this strategy is that Burgean cases always involve an omissive or commissive error about the term. It is implausible that every possible believer in "It is possible to believe the impossible" has a misconception about one of the terms in the sentence. All I need to prove my thesis is the possible existence of a single conceptually well-adapted believer in believable impossibilities.

The second problem with the strategy of expanding the domain of indirect beliefs is that no direct beliefs would remain. Even Descartes' cogito uses "I," "think," and "exists." Notice that the Cartesian immediacy of my basic idea can be underscored by switching to a first person claim; "I can believe an impossibility" is just as infallibly believed as the logically weaker "Someone can believe an impossibility." My thesis has as much independence from external conditions as the cogito.


Our Cartesian meditations draw out a resemblance between my thesis and the preface paradox. Prefaces frequently contain apologies for the errors that are bound to exist in the book. Although the author believes each of his assertions, he does not believe the conjunction of these assertions. In addition to refuting (Bp & Bq) [contains] B(p & q), the preface paradox shows that rational inconsistent belief is possible. For it is impossible for a believer in "At least one of my beliefs is false" to have all of his beliefs turn out true.

This last step in the reasoning becomes problematic if we let the statement become self-referential. That is, instead of letting "At least one of my beliefs is false" cover just the beliefs in the text, let it also include itself. Now, if the beliefs in the text are all true, then "At least one of my beliefs is false" is true only if it is false and false only if true. There is kinship here with contingent variations of the liar paradox. Consider list A:

(A1) At least one of the statements on list A is false.

(A2) Bill Clinton is left-handed.

Since Clinton is indeed a lefty, the other sentence on list A is true only if it is false. And if false, it is true! The paradox is contingent in the sense that (A1) would have been unproblematically true if Clinton had been right-handed.

Notice that my proof of the infallibility of believing "It is possible that someone believes an impossibility" includes that statement in its own domain of discourse. After all, the proof is a reductio ad absurdum which points out that false belief in my thesis entails its truth. But what if all other possible beliefs are beliefs in genuine possibilities? Then "It is possible that someone believes an impossibility" would be true only if it itself is an impossibility and hence false. And if false, it would be true. Hence, if all other beliefs must be beliefs in the possible, my thesis becomes as paradoxical as the contingent liar paradox.

Fortunately, this enigma rests on a big "IF." "All other possible beliefs are beliefs in genuine possibilities" is a necessary falsehood. So my thesis is no more paradoxical than the situation arising from list B:

(B1) At least one of the statements on list B is false.

(B2) Some triangles have six sides.

However, even those who believe that impossibilities can be believed may object that I am begging the question against the impossibilist. After all, the impossibilist's thesis entails that all other possible beliefs must be beliefs in genuine possibilities. So even if my thesis is semantically unparadoxical, I haven't demonstrated its probity to the relevant audience.

This charge of question-begging is questionable because it rests on reasoning that injures my adversary more than myself. If the impossibilist thinks that the negation of his thesis is as paradoxical as the liar, then he should think that his own thesis has the same flaw. For the "negation" of a meaningless utterance is itself meaningless and so cannot be affirmed in preference to its negation. Furthermore, if the impossibilist accuses me of believing a paradoxical sentence, then he contradicts his thesis that believing p implies the possibility of p. His position would also be inductively implausible. If I can manage to believe paradoxical sentences, what stops me from believing a necessarily false proposition? Impossible propositions at least have a stable truth value. So the charge of question-begging can be sustained only by representing the impossibilist as a kamikaze dialectician whose only hope is to make his self-destruction jointly damaging.

Mercifully, we can side-step a convoluted controversy about circularity by mounting an alternative challenge to the paradox-monger's supposition that all other beliefs are beliefs in the possible. In the course of my proof that some people can believe the impossible, I first derived the lemma that I can believe the impossible (line 4). This shows the inconsistency of the supposition that all of my beliefs are beliefs in genuine possibilities except for "It is possible for someone to believe an impossibility." The impossibilist will admit that if I believe that I can believe an impossibility, others can believe that they can believe impossibilities. Moreover, they can believe I can believe impossibilities, I can believe they can believe that, and so on. In short, the impossibilist should admit that if one impossible belief is possible, then indefinitely many impossible beliefs are possible.


Can the impossibilist avert dialectical overkill by distinguishing between kinds of impossibility? He might say that it is only metaphysically impossible to believe a contradiction. "Something is red all over and blue all over" seems impossible even though it is not a contradiction (because "red" and "blue" are primitive terms). Many impossibilists follow Saul Kripke (1972, 1980: 3) in thinking that the indiscernibility of identicals is as "self evident as the law of contradiction" and hence maintain that false identity statements are necessarily false. Indeed, Kripke thinks you cannot imagine that pain is c-fiber stimulation. He says you only think you can imagine you were born from different parents and that you only think you can imagine his lectern is actually made of ice.

However, distinctions between kinds of impossibilities will be relevant only if one reading opens a possibility for the believer that is not available for his object of belief. Contradictions are stronger than metaphysical impossibilities in the sense that whatever is contradictory is metaphysically impossible but not vice versa. But this derivational difference does not reveal an opportunity to believe an impossibility. If belief in an impossibility is only metaphysically impossible, then there is still no possible world in which someone believes an impossibility.

On the other hand, if belief in an impossibility is only physically impossible, then there are other possible worlds with different physical laws that contain believers in impossibilities. Although this would not secure an actual opponent to impossibilism, the argument would open the possibility that an actual person could seem opposed. Pseudo-adversaries could also be recruited with appeals to biological or technical impossibility. But none of these weakenings harmonize with the arguments deployed by actual impossibilists.


Antecedents of the main arguments against relations with impossibilities can be found in Plato's dialogues. In the Theaetetus and Sophist it is argued that we cannot believe what is false because what is false is not the case and what is not the case does not exist. This is the absent relatum argument: To believe is to believe something, so where there can be no object of belief, there can be no belief. Subsequent philosophers flushed out plausible relate for consistent false beliefs. However, they still have the same problem with belief in the impossible. Causal theories of representation, for example, take the object of belief to be what causes it when the belief is formed under optimal conditions. But there are no optimal conditions under which an impossibility is true. Others take sets of possible worlds as the objects of belief. However, there is no possible world in which an impossibility holds. Sentences purporting to attribute belief in the impossible appear to be meaningless danglers.

Plato's discussion of false belief also touches upon the appeal to understanding. The human predilection for viewing words as names led Plato to view all errors as misidentifications. For example, in a fog Socrates mistakes Theodorus for Theaetetus. However, Plato realized that this mental switch model is implausible because no one is irrational enough to judge that Theodorus is Theaetetus. As soon as the alleged object of belief becomes intelligible, it becomes unbelievable.

As philosophers gradually freed themselves from the assumption that all words are names, they were able to focus on more tractable cases of false beliefs. Most beliefs use the "is" of predication rather than the "is" of identity. Generalizations really use variables rather than names of abstract entities. Names were increasingly viewed as minority members of language. Bertrand Russell's theory of descriptions showed how ordinary names might be analyzed as definite descriptions. W. V. Quine modified this theory and recommended that scientific language avoid all names in favor of definite descriptions. However, Kripke's attack on the description theory of names has dis-interred the Platonic problem of mix-identification.

Prominent philosophers of language in the twentieth century have equated understanding with knowledge of truth conditions (Heidelberger 1980). This link has strengthened the reasoning against the possibility of believing contradictions. Their opening premise is that one can believe only what one understands and regards as true. Understanding requires knowledge of truth conditions. A contradiction has no conditions under which it can be true, so understanding a contradiction precludes belief. (Indeed, understanding a contradiction automatically produces disbelief.) Therefore, belief in a contradiction is itself a contradiction (Foley 1986, 350). Although each of the three premises have been challenged, the argument has considerable influence.

To get the third argument against the possibility of believing the impossible, the appeal to charity, switch attention from the interpretee's need for understanding to the interpreter's need for understanding (Davidson 1967,605). The point of assigning beliefs and desires is to make sense of the agent's actions by setting up applications of belief-desire psychology. If we ascribe conflicting beliefs, then no predictive progress has been made. (So even instrumentalists are hostile to belief in contradiction (Dennett 1987, chapter 4).) Charity is forced upon the interpreter because his project requires selection of the most rational portrayal of the agent's attitudes. Therefore, the appearance of incoherence is just a projection of the interpreter's confusion. This confusion reflects a lack of ingenuity on the part of the listener rather than the presence of contradictory beliefs in the speaker. The interpreter is free to admit that he is stumped. He is also free to doubt whether the preconditions for interpretation obtain. (Perhaps some forms of insanity amount to resistance to belief attribution.) However, the interpreter is never free to infer that the speaker is irrational. More positively, "In our need to make him make sense, we will try for a theory that finds him consistent, a believer of truths, and a lover of the good (all by our own lights, it goes without saying)" (Davidson 1970, 97).

Appeals to understanding frequently fuse with the defeasibility of belief. There is some temptation to think that we have immediate and infallible access to our own beliefs. However, defeasibilists note that we often describe ourselves as having been mistaken about our beliefs once an absurdity is exposed. Indeed, magnanimous practitioners of reductio ad absurdum describe themselves as showing that people do not really mean what they said. Patronizing? Not if thorough understanding is a precondition of belief. Just as knowledge must be a relation with actual states of affairs, belief must be a relation with possible states of affairs (Marcus 1983, 324). One can then go on to construe "conceive" as a success verb and dismiss the possibility of imagining an impossibility in the same way we dismiss the possibility of perceiving an impossible situation (Hart 1988: 28).

Defeasibilists admit that there are other defeaters of belief ascriptions. However, they underrepresent the wide generality of the phenomenon. For the epicenter, go back to Plato's dialogues. In Protagoras 358 and Meno 77-8, Socrates contends that we cannot desire what is bad. If a man is unaware that something is evil, then he might say he wants it and act as if he wants it. But once he recognizes it as evil, he realizes that he really never wanted it. Socrates concludes that virtue is knowledge.

Defeasibility undergirds paternalism in medical ethics. A physician once illustrated how much baldness bothers men with an anecdote. A young patient kept insisting that there must be some cure for his baldness. The exasperated physician finally declared that the only way to stop baldness was castration. The young man thought a bit. And said "Okay, whatever it takes." But the story actually shows how little weight we assign baldness. For our reaction is that the young man hasn't thought the matter through; he doesn't understand what is entailed by castration. The paternalist insists that the patient should get the treatment he really wants -- not the treatment he thinks he wants. What the patient really wants is determined by what he would want if he had rationally considered all the relevant information.

The political implications of defeasibility were taken up by some nineteenth century idealists. They reasoned that real freedom lies in the satisfaction of our ideally rational and well-informed desires. Thus the state was duty bound to ignore the apparent wishes of its citizens in order to make them free. Totalitarian Marxists incorporated this theme and concluded that they were the real democrats.

Ruth Marcus (1983) bases her impossibilism on the principle that a belief attribution is defeated when it is discovered that no state of affairs could make the belief true. Just as knowledge requires truth, belief requires possibility. In addition to applying this defeasibilism to reductions and false identity statements, Marcus (1981) extends it to the Pierre puzzle. Pierre is a monolingual Frenchman who hears that "Londres est joke" and so believes it. He moves to a place he knows as London and begins to learn English by immersion. Since he does not realize that Londres is London and happens to be confined to its shabby parts, he sincerely says that London is not pretty. So it appears that Pierre has rationally acquired contradictory beliefs. For when we translate his sincere French utterance and conjoin it with his sincere English utterance, we obtain "London is pretty and London is not pretty." Marcus' solution is that the contradiction cancels the belief reports. Upon learning that London is Londres, Pierre would disavow his assertions about London.

Pierre may well recant after learning of the impossibility. But if defeasibilist intuitions sufficed, then a vitiatingly broader program of re-description would be in order. For people retract reports of beliefs and desires upon the discovery of almost any kind of unwelcome consequence. (They also frequently just admit error.) Some repudiations may be interpreted as a reaction to absent relate. A biology student may retract his claim to have believed that the largest single organism is a mushroom in Michigan after learning that "the mushroom" is composed of clones and so is only a borderline case of a "single organism." Perhaps a statement is borderline when there is no state of affairs that would satisfy it or its negation. This might explain why we are tempted to view vagueness as a kind of meaninglessness.

However, we also retract when the object of belief has defects other than nonexistence. If a lecturer tells a student, "Although mental events have bodily effects, you are an epiphenomenalist," the student might believe it on authority. Once the student learns that epiphenomenalists deny that causes run from mind to body, he can recognize the resemblance between the teacher's remark and the Moorean variant "It is raining but you do not believe it." So the student denies that he ever really believed the lecturer's remark but not on the grounds that there is no state of affairs under which the remark could be true.

Many flaws provoke disavowals: triviality, empirical bizarreness, heresy, rudeness, etc. When President Gerald Ford assured Americans in the midst of the cold war that Eastern Europe would never be dominated by the Soviet Union, his aides swiftly discounted the statement as a verbal slip. A central duty of Ronald Reagan's spokesman was to furnish reporters with official interpretations of presidential remarks. Unless we embrace the vintage real selves of the nineteenth century Idealists, we will need to rein in the implications of disavowals.

VII. Propositional Guises of the Problem

Impossibilists are inconsistent about inconsistency. Plato depicts Socrates as embarrassing his interlocutors by exposing hidden inconsistencies and thereby compelling them to change their minds. But if Socrates' adversaries never really had the conflicting beliefs, what are they embarrassed about? How could Socrates be the gadfly of Athens?

In the Meno, Plato partly agrees that the Socratic method is uninformative. The concession is prompted by the paradox of inquiry: If Socrates really knows nothing, how can he hope to acquire knowledge by asking others? Recognition of the correct answer requires some background knowledge. Socrates responds by conceding that in one sense, learning is impossible. Not because we know nothing, but because we know everything! In particular, his doctrine of recollection credits us with atrophied omniscience. In a pre-existent state, we dwelt amongst the forms and so knew all. The trauma of birth causes us to forget. The role of Socratic interrogation is to revive these heavenly memories.

The recollection model (minus the fairy tale about pre-existence) has some prospect of accounting for conflicting beliefs about contingent propositions. If I latently believe p but become persuaded that ~p, then I might seem to believe p & ~p. The problem of finding an object for this apparent single belief can be resolved by analyzing it as two beliefs. However, this divide and conquer strategy fails for belief that there is a largest prime. This belief is not a conjunction of opposed propositions. Thus the doctrine of recollection has no advantage in explaining how inquiry about necessities can be informative. This kind of cognitive advance cannot be reduced to the elimination of one member of a conflicting pair of beliefs. (Indivisible contradictions are also troublesome for Peter Strawson's (1952, 3 and 21) contention that contradictory beliefs cancel each other out.)

The contemporary paradox of analysis focuses on this problematic form of investigation. If a conceptual analysis states exactly what the original statement says, then the analysis is trivial. If it says something different from the original, then the analysis is mistaken. Hence, all analyses are either trivial or false. The case for "Sameness of meaning = triviality" meshes with the case against believing contradictions. Since disbelieving p is just believing ~p, the incredibility of contradictions precludes disbelief in tautologies. Could one be an agnostic about a tautology? If one understands a proposition, one understands the conditions under which it is true and false. But in the case of a tautology, this entails recognition that the proposition is true and hence, knowledge that it is true. Knowledge implies belief. Consequently, to understand a tautology is to believe it. In addition to precluding agnosticism about tautologies, the connection between understanding an analytic truth and believing it "demonstrates" that conceptual analysis is never informative.

The self-intimating nature of tautologies would also preclude unwitting belief in them, that is, belief unaccompanied by the recognition of their tautologous nature. For example, some philosophers believe that "All causes precede their effects" but are not sure whether the proposition is a tautology. That would be impossible if belief requires understanding of the nature of the proposition. Moreover, belief in a tautology would always have proper grounds. Fallacious proofs of mathematical theorems would be impossible because anyone who understood the tautology would automatically have cogent proof of it.

Contingencies would be partially self-intimating; although their truth-values could still be hidden, their modal status would always make itself known. For understanding a contingent proposition means understanding some conditions under which it is true and some conditions under which it is false. Moreover, contingencies would always look like contingencies. No one could believe they believe of a contingency that is non-contingent.

Propositions bear necessary relations with other propositions, so understanding a proposition would mean understanding what entails it and what it entails. But the entailers and entailees must themselves be understood. This holistic slippery slope is reminiscent of the idealist's arguments that to know anything one must know everything. But it is even more overbearing. At least the idealists could concede that we appear to know some things without knowing everything. The purveyor of logical omniscience cannot admit that we seem to understand some propositions without understanding their logical relations with all others.

Some philosophers accept this dismal conclusion and cite the failure of reductive programs as vindication. But most philosophers view the paradox of analysis as unsound or misleading. For example, my impression had been that the paradox goes through if one decides to set the high standard for "belief" implicit in the principle that belief requires understanding. The background idea is that "understand" is an absolute term like "flat." Just as a flat surface cannot be flatter than another flat surface, someone who understands a proposition cannot understand it better than another person who understands it. This follows from an absolutist definition of "understand:" A understands p iff A can discriminate p from all other propositions. This definition quantifies over propositions, thus a person who can discriminate p from foils drawn from one domain of propositions might not be able to discriminate p from the alternatives drawn from another set. When our quantifiers are "wide open," there are so many look-alikes that one understands very few propositions. As with "flat," I concluded that although one can consistently relativize "understand" to high standards, it is impractical to do so. Thus I agreed that the paradox of analysis is correct for the limit case in which we quantify over all propositions but not for the ordinary cases. The philosopher's preoccupation with the paradox could then be explained as an artifact of idealization. Usually, we simplify a problem by considering extreme cases. But when it comes to understanding the informativeness of analysis, the strategy eliminates the resources needed for the solution.

Although I still think that the paradox of analysis is partly due to context shifting shenanigans, I now think that there are semantic restrictions on how high one can set the standards. To put the matter another way, the paradox of analysis rests on two premises that are open to a context-independent refutation. Recall that the paradox proceeds by eliminating the two ways in which an analysis might enlighten a thinker. The first path is by revealing that a belief is actually contradictory. The anti-analyst blocks this possibility by appealing to the principle that belief requires understanding. However, my belief that "Belief does not require understanding" is infallible. I could only be wrong if belief requires understanding. But then I would understand "Belief does not require understanding" and so know that there are no conditions under which it could be true. This knowledge that the proposition is false would prevent me from believing it--contradicting the original assumption that I believed it. The proponent of "Belief requires understanding" faces the same dialectical trap as his impossibilist ancestors. If he concedes that I believe "Belief does not require understanding," then he loses. Nor can he admit that I believe that I believe my thesis, because my belief that I believe "Belief does not require understanding" would itself be a belief in a contradiction. Like the impossibilist, he must deny even the appearance of disagreement. He must insist that the infallibility claim,"If someone believes that belief does not require understanding, then he must be right," is only vacuously true. The proponent of "Belief requires understanding" must say the antecedent of the conditional expresses a logical impossibility--and he must say the same for all the higher order infallibility theses. And he must say it for all contexts.

The paradox of analysis also attacks the second way analysis can be informative--revealing tautologies to those who had not previously believed them. The anti-analyst attacks the possibility of being agnostic about tautologies by arguing that understanding a tautology entails believing it. However, "Understanding a tautology entails believing it" would then itself be a tautology. So if I understand the principle, then I must believe it. However, I understand the principle and do not believe it. Therefore, some tautologies can be understood without being believed.

This proof that one can understand a tautology without believing it might be challenged on the grounds that I only appear to understand "Understanding a tautology entails believing it." The idea would be that I mistakenly believe that I understand the principle. But this is impossible given the anti-analyst's other principle that belief requires understanding and his allegiance to principle that understanding is divisional (one can understand a complex proposition only if one understands its component propositions). For then my belief that I understand the proposition that p would be infallible. For "I believe that I understand p" would entail "I understand I understand p." But I can understand "I understand p" only by understanding all of the components of the complex proposition--which includes the p component.

VIII. Deductive Guises of the problem

The puzzles of deduction distend in similar fashion. If we cannot believe impossibilities, we cannot mistake valid arguments as invalid. To believe that an argument is invalid is to believe it is possible for the premises to be true while the conclusion is false. But if the argument is valid after all, this would be belief in an impossibility. The possibility of error in the opposite direction can be "refuted" by adapting the argument against agnosticism about tautologies. For uncertainty about the validity of an argument requires uncertainty about the status of the corresponding tautology. This challenge to the informativeness of deduction is best known in the form of John Stuart Mill's (1843,120) thesis that all valid arguments are circular. As with tautologies, the informativeness of a deduction seems to evaporate under the light of reason. The more one can be said to understand, the more trivial the deduction appears.

Let's sharpen the deductive dilemma with a slippery slope argument. For openers, nothing can be learned from an invalid deduction. And even some sound deductions are epistemically impotent. To argue "Caesar's wife must be above suspicion,therefore Caesar's wife must be above suspicion" is to just run in place. Ditto for any argument of the form "P therefore P." Conjoining P with another proposition brings no improvement: P and Q, therefore, P' is just as question-begging. Separate assertion of the premises does not matter ("P, Q, therefore, P"), so an argument with truth functionally equivalent premises only looks more persuasive: Q [contains] P, Q, therefore, P. The apparent persuasiveness can be increased by further tactics of obfuscation such as rewording. But rearrangements of the same information don't increase knowledge. Repackaging a circular argument leaves it circular. However, a combination of these camouflage techniques can take us from "P therefore P" to any valid argument form. Therefore, all deductive reasoning is impotent. Or so runs this self-defeatingly interesting deduction.

Notice now that these implications can be expanded by ascent to higher order appearances, People cannot commit fallacies and cannot even appear to. People cannot be informed by deduction and cannot even appear to. This extra layer of commitment topples extremists into an abyss of absurdity.

IX. Closure Conflicts with Universal Consistency

In 1968, Leonard Linsky pointed out that Jaakko Hintikka's (1962) modeltheoretic account of belief condemns as indefensible the belief that someone else is inconsistent. Roughly, a proposition is doxastically indefensible if one cannot consistently believe it even though it might be true. For example, Moore's sentence "It is raining but I do not believe it" is logically consistent but believing it commits the speaker to conflicting beliefs. Thus Hintikka's system generates a weak analogue of impossibilism. Instead of dismissing "Someone believes an impossibility" as a contradiction, it condemns belief in that proposition as indefensible.

This implication might be viewed as awkward side-effect of using model theory A model set cannot contain an inconsistency because it represents a (partial) description of a possible set of affairs. So a believer in "Someone else believes an inconsistency" cannot embed the object of belief in a model set. This inability of the belief to come out true is not the same thing as logical inconsistency. There is no way to make belief in "No one has a belief" come out true but that does not show that "Someone has a belief" is a necessary truth.

Not all view the result as a technical embarrassment. Some make a virtue out of necessity and defend the result as a bold discovery: although "Someone else believes an inconsistency" does not sound odd, it is a subtle Moorean sentence. Bolder still are the impossibilists who think that Hintikka's result is an understatement; instead of being merely indefensible, "Someone else believes an inconsistency" is a contradiction (Purtill 1970).

However, the infallibility of belief in "Some impossibilities are believable" suggests that it is Linsky's result that is understated. Linsky's complaint is that Hintikka's system cannot permit belief that someone else is inconsistent. The real problem is that a model theoretic approach cannot require belief in the believability of impossibilities. Hintikka's believers are supposed to have deductively closed beliefs. Hence, every necessary truth should appear within every belief system. But given the necessary truth, "It is possible to believe an impossibility," each believer must then think that a belief system can be inconsistent. However, a model set is inherently consistent. Thus all of Hintikka's ideal believers reject Hintikka's theory of belief.

The ingratitude generalizes. Any theory that correlates belief and necessity will create Frankensteins who repudiate the theorems of their creators. In Brian Ellis' epistemic semantics, necessity is defined in terms of belief. In particular, a proposition is possible if and only if it is believed in at least one rational belief system. A proposition is necessary only when believed in every rational belief system. So the necessary truth of "Some impossibilities can be believed" demands its presence in every system. Consequently, each system is committed to the possibility of a rational belief system containing an impossibility. But this is possible only if there is at least one rational belief system containing an impossibility. Such a rational belief system is inconsistent. Thus the attempt to reflect all necessities by means of ideal belief comes into conflict with the requirement that all ideal belief systems be consistent.

The same conflict between closure and universal consistency arises with the converse attempt to analyze belief in terms of necessity. For example, Robert Stalnaker (1984) takes the object of belief to be a subset of possible worlds. When I believe my cat or my dog tipped a vase, I believe I am in a possible world in which my cat or my dog tipped a vase. After learning that the dog was outdoors when the vase broke, I narrow down the range of possible worlds to those in which the cat is the culprit. The possible worlds account neatly models contingent beliefs by providing a clear contrast between the worlds I may occupy and those I exclude. However, this contrast is not to be found for necessary truths. Necessary truths hold in all possible worlds, so they all have the same content, namely, the set of all possible worlds. Since necessary truths appear in every possible world, everybody believes all of them. Stalnaker deftly defends deductive closure and the second consequence of consistency. However, the details of this defense do not protect his account from the objection used against epistemic semantics. As a necessary truth, "Some impossibilities can be believed" must appear in every possible world and hence every belief state. But for an impossibility to be believed is for it to appear in a belief state, that is, a possible world. But no possible world contains an impossibility. Hence closure conflicts with universal consistency.

Stalnaker does have a resource for accommodating the appearance of inconsistent beliefs. He allows a single agent to be in more than one belief state. That is, in one complete belief state, p holds and in another complete belief state ~p holds. Each belief state is "a separate center of rationality." Deduction is "the integration of the separate belief states of a single agent" (Stalnaker 1984, 87). However, this maneuver does not apply when the object of belief is a single necessary truth. For "Some impossibilities are believable" must be reflected within each and every belief state. There is no variation in belief to integrate, so belief in the believability of impossibilities does not stimulate the search for deductive relief of internal dissonance.

Neighborly conflict between closure and consistency should be distinguished from the self-referential sort associated with Kurt Godel's incompleteness theorem. Roughly, Godel showed that one can express "This statement cannot be proved in this system" in a language strong enough to do arithmetic. This creates an internal conflict between consistency and completeness. The conflict precipitated by "Some impossibilities are believable" is between systems. For a belief system need not attribute belief in an impossibility to itself. It can say that some other belief systems contain impossibilities. The attribution pattern for inconsistency could resemble one that some anthropologists allege for cannibalism; all groups abominate some groups as cannibals but no group admits to being cannibals. This self-serving finger-pointing carries a price. An anti-cannibal group cannot say that its forbearance is guaranteed by human nature. Likewise, a belief system that abominates inconsistency cannot claim that its own consistency is guaranteed by the nature of belief systems.

Belief in cannibalism could be a cultural universal without there being any cannibals. Perhaps each group accuses other groups to increase local solidarity. However, those who link belief and necessity cannot hope that the believability of impossibilities is a universal myth. For they take the presence of a belief in every system to be sufficient for its truth, indeed, it must count as a necessary truth.


I have purposely reviewed the impossibilists' arguments in broad historical strokes. The modern refinements surpass Plato's first efforts. Nevertheless, they inherit a rare disorder. Philosophers can typically help themselves to an appearance/reality distinction. A Francis Herbert Bradley or a George Berkeley or a Zeno can admit that their thesis appears false and then triumphantly argue that it is, against all odds, really true. Indeed, much of the value of speculative reasoning lies in the prospect of dispelling an illusion. The preconditions for illusion are so modest that they are commonly ignored. However, impossibilism is an unprecedented case in which this minimal condition of speculative debate is violated. Consequently, students of belief are in a unique position to lay down an adequacy condition with instructive connections to puzzles about reasoning, representation and analysis: All discriminative accounts of belief must permit belief in impossibilities.*


(*) I salute some troopers on the bloopers: Jonathan Adler, Curtis Brown, John Carroll, Daniel Goldstick, and Carsten Hansen. Audiences at the CUNY Philosophy of Logic Seminar and the University of Rochester corrected meta-bloopers. All meta-meta-bloopers are mine.


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Date:Jul 1, 1996
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