Efectos Secundarios y Colaterales de la Quimioterapia

Jim and Max are having a conversation about their plans to attend the theatre tonight:

Jim: I doubt Sarah will be at the theatre tonight.

Max: But Paula won’t be there unless Sarah is.

Max: So you must doubt that Paula will be there too.

The inference made by Max seemsprima facie valid. Indeed it looks like an instance ofmodus tollenswhere ‘It is doubted that. . . ’ replaces negation, ‘It is not the case that. . . ’ The reason the argument looks valid is that if Jim doubts B and believes A implies B then he certainly cannot believe A (supposing

belief is closed under believed implications and Jim is rational). (One might recognize this principle as a thesis of Richard Kilvington.) But then he knows he must hold thatAis false or dubitable, and if he holds that A is false—i.e. hedisbelieves A—then, on certain accounts, it is dubitable. So in either case Jim ought to doubt A (or at least he may, though in this case the stronger obligation seems more appropriate). It looks then as if doubt, just like denial, is closed under (believed)converse implication: ifBis dubitable andAimplies B thenAtoo is dubitable.

An account of doubt according to which disbelief implies dubitability is reasonable on some ordinary uses of the term, but not on all. There is a no- tion of doubt asagnosticism according to which disbeliefrules out doubt, for if one disbelieves a statement she cannot also be agnostic with respect to it. (By being agnostic with respect toA I mean that neither A nor its negation is believed, i.e. that neither Ais believed nor disbelieved.) I shall fix on this latter notion of doubt since it seems to be one that arises naturally in con- structive mathematics or more generally theories in which assertion and denial are epistemically constrained.10 _{A natural interpretation of the conditions un-} der which a statement is assertible, deniable and dubitable—in the context of mathematics—is that the statement be proved, refuted or neither, respec- tively. That is, the following conditions plausibly constrain assertion, denial and doubt:

(Assert) Ais assertible iffAis proved;

(Deny) Ais deniable iffAis refuted;

(Doubt) A is dubitable iffAis undecided.

10_{In [Sal95], Salmon defines doubt inAdisjunctively as either disbelief in or agnosticism}

with respect toA. This notion of doubt is weaker than doubt as agnosticism and, on the assumption that agents are consistent in their beliefs, equivalent to (modulo the usual normal modal logics of belief, e.g.K4) doubt asnon-belief. For the most part I shall, but needn’t, make this consistency assumption (typically encoded by the schema “D”,$A→♦A), and so Salmon’s disjunctive account turns out not very plausible.

We need not think these constraints apply only to constructivists. Pre- sumably they ought to apply to classicists as well. It would seem irrational for a classicist to assert (deny) Goldbach’s conjecture without having actually obtained a proof (refutation). And without a proof that the conjecture is ei- ther provable or refutable one would not be warranted in asserting that that is assertible either. Of course that does not amount to denying an instance of excluded middle so the classicist is not in any trouble, and she may even still hold that the conjecture is either assertible or deniable if she holds that a sentence is assertible (deniable) iff it is true (false). But being assertible hardly seems equivalent to being true for truth seems at most a necessary condition on assertibility and not a sufficient one, and likewise for deniability and falsity. Doubt as agnosticism is not closed under (believed) converse implication. Looking back at the example conversation we can see why. Jim might doubt that Sarah will be at the theatre while disbelieving that Paula will be. Since disbelief is incompatible with doubt in the present agnostic sense, Jim cannot doubt that Paula will be at the theatre. (Sometimes in contexts like the one Max and Jim are in, ‘I doubtA’ means ‘I disbelieveA’. e.g. consider ‘I doubt you can hit the bullseye!’ uttered during a game of darts. In these cases doubt as disbelief is closed under converse implication.) So if the agnostic sense of doubt is neither constrained by implication nor converse implication, by which notion of consequence, if any, is it constrained?

In answering that question let us consider again the properties of being proved, refuted and undecided. Clearly they are pairwise mutually exclusive and jointly exhaustive over the set of all statements, i.e. every statement has precisely one of the properties. Suppose we take these properties as semantic values in a theory of assertion, denial and doubt. Perhaps surprisingly we find that they behave just like the values in a strong three-valued Kleene matrix. We have:

• A_∧B is undecided iff both A,B are, or if just one is then the other is not refuted;

• A_∨B is undecided iff both A,B are, or if just one is then the other is not proved.

(Implication would be the only tricky case under a constructive notion of proof, but under a classical notion of proof we can simply define A _→B in one of the usual ways, e.g. by_¬(A_{∧ ¬}B), whereuponA_→B would be undecided iff both A and _¬B are, or if just one is then the other is not refuted. If one is inclined to think constructively about proof, then implication will be left out of the analogy between undefinedness in Kleene semantics and being undecided.)

Clearly the following constraints on assertibility and deniability hold.

(Assert-Cl) IfA is assertible and it has been proved thatA impliesB then

B is assertible.11

(Deny-Cl) IfB is deniable and it has been proved thatA impliesB thenA

is deniable.

That is, assertion and denial are closed under proved implications and the converse of proved implications, respectively. Now what sort of analogous con- straint could we place on dubitability? When the relevant notion of implica- tion is the same for each assertibility, deniability and dubitability, then no such analogous constraint exists. We need a notion of implication as preservation of undecidedness.12

11_{It is important to note that ‘assertibility’ means ‘may be asserted’ and not ‘has been}

asserted’ or ‘must be asserted’. Obviously one may assertAwithout having any sort of obli- gation, even rational, to assertBwhenAimpliesBand even when he knows the implication holds, for in the former case maybe he has never even considered A, or in the latter case maybe it would take more than a lifetime to assert it.

12_{Suppose we had decided to opt for provability rather than provedness and likewise for}

refutedness and undecidedness. Then even for intuitionistic logic, the general case of con- dition (Deny-Cl) requires a consequence relation preserving refutability (in a multi-premise setting) that differs from the converse of the provability-preserving relation, since provability- preserving consequence does not anti-preserve refutability.