that sets are closed under logical In essence, this means that if we ‘belief entailment’. know that and implies then we also know that p, p q, q. Closure of Belief Sets Under Logical Entailment This claim, however, seems implausible and generates a requirement to consider sceptical hypotheses [h], such as the possibility that genium Descartes’ is introducing falsehoods into our minds without our knowledge. malignum ‘Knowing that has the logical implication that we know that skeptical hypothesis is not the p’ h case (ie the is not deceiving us about genium malignum p): Ks Ks [~h] p → Since we do not know for sure that is not the case, then neither do we have h knowledge of p. Ks Ks [~h] p → ~Ks [~h] ………………………….. ~Ks p As well as being a disaster for knowledge-claims in general, the demands of closure lead to particular problems for requirement, by which we set Nozick’s ‘sensitivity’ such great store earlier. Nozick (3), we recall, requires that were not to be the case, p we would not believe it. But under the influence of the we might genium malignum, believe it anyway, by being in a state insensitive to the contrafacticity of The p. sensitivity requirement stipulates that in an alternative world in which ~p obtains, we would believe that ~p. But if this world is one in which demon has free Descartes’ reign, we might (wrongly) believe that being subject to his undetectable deceptions. p, Sosa claims that the failure of to allow closure of knowledge under ‘sensitivity’ implication is a reason to support his rival notion of but Jonathan Kvanvig has ‘safety’, shown convincingly that too suffers from this shortcoming. ‘safety’ There are two ways of dealing with this problem: we can ignore the sceptical challenge or we can deny that the lack of closure of knowledge under implication matters. The first move is the educational version of banishment of the sceptic: if we David Hume’s
that ‘belief sets are closed under logical entailment’. In essence, this means that if we know that p, and p implies q, then we also know that q. Closure of Belief Sets Under Logical Entailment This claim, however, seems implausible and generates a requirement to consider sceptical hypotheses [h], such as the possibility that Descartes’ genium malignum is introducing falsehoods into our minds without our knowledge. ‘Knowing that p’ has the logical implication that we know that skeptical hypothesis h is not the case (ie the genium malignum is not deceiving us about p): Ks p → Ks [~h] Since we do not know for sure that h is not the case, then neither do we have knowledge of p. Ks p → Ks [~h] ~Ks [~h] ………………………….. ~Ks p As well as being a disaster for knowledge-claims in general, the demands of closure lead to particular problems for Nozick’s ‘sensitivity’ requirement, by which we set such great store earlier. Nozick (3), we recall, requires that were p not to be the case, we would not believe it. But under the influence of the genium malignum, we might believe it anyway, by being in a state insensitive to the contrafacticity of p. The sensitivity requirement stipulates that in an alternative world in which ~p obtains, we would believe that ~p. But if this world is one in which Descartes’ demon has free reign, we might (wrongly) believe that p, being subject to his undetectable deceptions. Sosa claims that the failure of ‘sensitivity’ to allow closure of knowledge under implication is a reason to support his rival notion of ‘safety’, but Jonathan Kvanvig has shown convincingly that ‘safety’ too suffers from this shortcoming. There are two ways of dealing with this problem: we can ignore the sceptical challenge or we can deny that the lack of closure of knowledge under implication matters. The first move is the educational version of David Hume’s banishment of the sceptic: if we