1. Safety and closure

Perhaps the biggest advantage a safety condition on knowledge has over its main rival, a sensitivity condition, is that it promises to explain, or at least not to violate, the closure of knowledge under competent deduction.Footnote ¹ Although some philosophers, most prominently Dretske (Reference Dretske1970, Reference Dretske, Steup and Sosa2005) and Nozick (Reference Nozick1981: 207f.), think that a failure of closure might help in solving the problem of skepticism, many would not be ready to give up on a closure principle for knowledge.Footnote ² From a closure-friendly perspective, safety should be favored over sensitivity because it does not conflict with closure.

A salient way of understanding closure is the following (cf. Williamson Reference Williamson2000: 117 and Hawthorne Reference Hawthorne, Steup and Sosa2005: 29):

This closure principle expresses the plausible thought that knowledge can be extended by deduction. If one knows a number of premises P ₁, …, P _n and competently deduces from them a conclusion Q and thereby comes to believe Q, then one knows Q.Footnote ³ The bracketed qualification ‘in favorable circumstances’ is meant to exclude cases where one loses one's knowledge of the premises during the reasoning process or is exposed with a defeater for the correctness of one's reasoning at the exact moment when one reaches the conclusion (see Kvanvig Reference Kvanvig2006 for discussion). I mention these possibilities only to set them aside in what follows.

With this in mind, let us turn now to safety as introduced by Pritchard (Reference Pritchard2005), Sosa (Reference Sosa1999), Williamson (Reference Williamson2000: Ch. 5) and others. A safety condition on knowledge is just this (various ways of explicating safety will be discussed below):

Here I introduce safety merely as a necessary condition on knowledge. There is work on strengthening the concept of safety, so that it might have a claim to be both necessary and sufficient for knowledge (see Grundmann Reference Grundmann2018).

What makes a safety condition seem closure-friendly? A key feature of safety is that it is about what is true at a certain range of close worlds. And if a number of premises are true across a set of worlds, then so too is anything which logically follows from these premises. This shows that a prototypical safety condition, which I'll label ‘Naïve Safety’, is closed under logical consequence:

If Naïve Safety were a plausible condition on knowledge, it would pose no threat to a closure principle. Naïve Safety is closed under logical consequence in the sense that if a number of beliefs are safe, then any belief which can be deduced from these beliefs is safe as well. So, any belief competently deduced from known premises would be safe, for competent deduction arguably requires that the conclusion follows from the premises.

The problem with Naïve Safety is that it is not a plausible condition on knowledge. A case which shows this is Nozick's (Reference Nozick1981: 179) grandmother example. As a crucial feature of Nozick's original case is not necessary to argue against Naïve Safety, the following story describes a simplified version which I relabel to mark the difference:

The grandfather knows in this case that his granddaughter is alive. However, given the circumstances, she could easily have been dead. Consequently, it is not true in all close worlds that she is alive. If so, safety as understood along the lines of Naïve Safety is not necessary for knowledge.

For this and various other reasons, current defenders of a safety condition propose more nuanced versions of safety. The issue with more nuanced versions of safety is that the easy argument in favor of safety's closure under logical consequence no longer holds. More problematically, a number of potential counterexamples have been designed to show that knowledge would not be closed under competent deduction on more refined safety conditions (Kvanvig Reference Kvanvig2004; Murphy Reference Murphy2005, Reference Murphy2006; Alspector-Kelly Reference Alspector-Kelly2011). My aim in this paper is to show that this is not correct. There are ways of articulating a safety condition which pose no threat to the closure of knowledge under competent deduction. On the upside, this means that one can coherently hold that knowledge answers to a safety condition and that knowledge is closed under competent deduction. As it turns out, the downside of this is that deduction faces the same kind of intricate issues of individuation as perception and other belief-forming methods.

The paper proceeds as follows. Section 2 reviews potential counterexamples. Section 3 introduces methods. Section 4 discusses the possibility that inferential methods are partly individuated by the premises to which they are applied. Section 5 offers a way of individuating inferential methods externally and shows that on such a view, safety poses no obstacle to knowledge being closed under competent deduction. Section 6 discusses a residual issue and section 7 concludes.

2. Problem cases

In response to problems with Naïve Safety, the notion of safety has been refined in various respects. Perhaps the most important change is to focus not merely on whether the proposition is true in close worlds but rather on whether the proposition is true if the subject believes it to be (Sosa Reference Sosa1999; Williamson Reference Williamson2000: Ch. 5; Pritchard Reference Pritchard2005):

This gets around the problem of known propositions which could easily have been false. To wit, recall the grandfather case. Although what the grandfather believes, namely that his granddaughter is alive, could easily have been false, his belief is modally robust. He would not have believed that his granddaughter is alive in circumstances in which she had been dead, for he would then not have seen her in her room.

The most prominent counterexamples to safety's closure are directed at the belief version of safety. Consider this case presented by Murphy (Reference Murphy2005):Footnote ⁴

Given that, as a matter of fact, everything is normal – there is a red barn in the field, lighting conditions are good and there are no barn facades nearby – Murphy (Reference Murphy2005) is right, I think, in assuming that Harriet knows

1. (1) There is a red barn in the field.

If knowledge is closed under competent deduction, it should be possible for Harriet to gain deductive knowledge of (2) on this basis:

1. (2) There is a barn in the field.

Note that ascribing deductive knowledge of (2) is very plausible: we would credit Harriet with knowledge that there is a barn in the field if she deduced this from her knowledge that there is a red barn in the field.

But now observe that Harriet's belief in (2) does not appear to be safe. In some close worlds, the mayor would have put up a green barn facade. Harriet would have believed that there is a green barn in front of her from which she would then have inferred (2) by way of the same inference rule as in the actual world. But (2) would have been false, for there would only be a barn facade in the field. For this reason, Murphy (Reference Murphy2005) concludes that Harriet's belief in (2) is not safe.

It is not fully clear whether one should follow Murphy in his diagnosis of this case. One way to resist his argument would be to deny that worlds in which barn facades are put up are close. Despite the high antecedent probability of a barn facade having been put up, such a maneuver would not necessarily be ad hoc, for safety is a modal rather than a probabilistic notion (Pritchard Reference Pritchard2014).Footnote ⁵

But even if one finds fault with the details of Murphy's case, the structure behind it reveals a recipe for constructing interesting problem cases. Take any safe belief P ₁ from which Q is competently deduced. Given that logical consequences are often weaker than the premise from which they are inferred, there are plenty of cases where Q could also have been inferred from a different premise P ₂ which is logically independent of P ₁. Now see to it that there are close worlds in which P ₁, P ₂ and Q are false but the subject believes P ₂ and infers Q from it in the same way as in the actual world. Then Q is not true in all close worlds in which it is believed and so safety would fail.

Alspector-Kelly (Reference Alspector-Kelly2011) presents a similar case which instantiates this recipe. It goes like this (slightly adapted):

It is evident why this is another problem for closure. Luke's belief that his car is in B1 may very well be knowledge. Seeing his car on a well-functioning app is almost as good as direct perception. Although there are close worlds in which his car is not in B1 (either because it was stolen or because it was moved to another lot), these are worlds in which he does not believe that it is in B1 because he would not see it being in B1 on his app. However, in some of these worlds, Luke still believes his car has not been stolen despite the fact that it has. In such worlds, he concludes that it has not been stolen from it being in another lot which is something he believes in those worlds in which the car is not in B1. Again, despite knowing that his car is in B1, the conclusion that it has not been stolen does not appear to be safe.

I find it hard to object to Alspector-Kelly's example by deeming the problematic worlds not to be close. After all, some cars are moved every night and one car even gets stolen. If Luke's car could not be said to have ‘easily been moved’ and ‘easily been stolen’, it would be hard to keep a firm grip on the intended sense of ‘close world’.

If we grant the counterexamples (or at least the possibility of instantiating the recipe for constructing counterexamples mentioned above), what are our options? The most radical option would be to conclude that knowledge is not closed under competent deduction. But recall that this conclusion is not supported by the problematic cases. To the extent that it is plausible that the relevant premise is known in these cases, it is equally plausible that the conclusion is known. Harriet knows that there is a barn in the field when inferring this from her knowledge of a red barn being in the field. And Luke knows that his car has not been stolen when inferring this from it being in B1. Therefore, the examples are best seen as challenging whether safety is necessary for knowledge. If the conclusion of the inference in the two cases is known but not safe, then safety is not necessary for knowledge.

I think this might be the right conclusion to draw as long as safety is understood along the lines of Safety (belief version). But independently of issues with closure, the debate on safety indicates that Safety (belief version) might not be the best way of explicating a plausible safety condition anyway. If this is so, then one should first look at safety conditions independently offered as improvements of Safety (belief version) before concluding that safety is at odds with knowledge's closure. This is what I'll try to do next.

4. Methods and premises

One way of understanding an inferential method essentially involves the premises which it takes as input. On this thought, one does not use the same method if the premises of one's reasoning differ. This would be so even if there is no difference in the inference rules applied. Thus, when evaluating whether a given belief is safe, one would look only at close worlds in which this belief is based on the same premises. This gives rise to the following constraint on the individuation of methods:

This constraint specifies a necessary condition on the individuation of methods. It allows one to distinguish two deductive methods which start out from different premises. The constraint is not meant to fully individuate deductive methods.

It is easy to see that Safety (method version) can circumvent the counterexamples against closure when inferential methods are construed as involving the premises essentially. Start with Murphy's barn case. When Harriet falsely believes in a close world that there is a barn in the field, she infers this from the belief that there is a green barn in the field. In the actual world, however, she has a different ground for her belief. There she believes this because she believes that there is a red barn in the field.

A similar consideration applies to Alspector-Kelly's parking lot case. In the actual world, the belief that the car is not stolen is inferred from it being in B1, while in the counterfactual scenario, the belief is inferred from a different premise, namely the belief that the attendant has moved it to a different lot.

Alspector-Kelly (Reference Alspector-Kelly2011: 136f.) objects to a strategy of this kind with a modified counterexample. The idea behind the modified counterexample is simple: just as one may believe the conclusion of an argument for different reasons in close worlds, so one could also believe the premises for different reasons. If this makes the premises false in some close worlds, then the conclusion may be false as well despite being believed for the same reason as in the actual world. Here is the modified counterexample:

Take one of the worlds in which Luke's car is stolen. In such a world, Luke believes by testimony that his car is (still) in B1 and infers from the latter that it has not been stolen (which is false). So, there are close worlds in which Luke's inferred belief is false despite being based on the same premise as in the actual world.

Given that in the actual world, Luke sees his car in B1 via a well-functioning app, it is hard to deny that he knows that his car is in B1. Although he would falsely believe that his car is in B1 in a close world, this belief would be based not on what the app shows him but rather on a different method, namely on testimony (what the attendant tells him). This aspect of the case is structurally similar to Nozick's grandmother case. Again, it also seems that the inferred belief – that his car has not been stolen – constitutes knowledge. As far as what the case prima facie suggests, we are not looking at a problem for closure but rather at a problem for safety.

From the perspective of a defender of Safety (method version) who individuates deductive methods partly by their premises, a fairly natural response would be to draw a distinction between direct premises and indirect premises. A direct premise is the penultimate step in a chain of reasoning. Indirect premises are all beliefs which occur at an earlier stage of the reasoning as intermediate conclusions or as ultimate, non-inferential grounds the reasoning is based on. A premise (or ground) of a belief could then be construed either in a narrow sense meaning direct premise or in a broader sense meaning direct or indirect premise.

In a similar vein, an inferential method can be construed in a narrow sense where the methods just cover the process of inferring the conclusion from the direct premises it is based on. Or, alternatively, methods can be construed broadly to cover the whole belief-forming process with all its intermediate steps. On this approach, the method a subject uses would even include the way the most fundamental beliefs are formed. For this reason, there are broad methods which are hybrid in the sense of being neither purely deductive nor purely non-deductive. For instance, a base belief may be formed non-deductively, by perception say, and then a further belief is formed by deduction. The broad method for the latter is then a chain consisting of a perceptual and an inferential link.

The distinction offers an alternative understanding of safety:

One would pair this understanding of safety with a revised version of how to individuate inferential methods:

Alspector-Kelly's adjusted counterexample would work on a narrow sense of ‘method’, for the direct reason – that the car is in B1 – is kept constant over the relevant class of close worlds. Interestingly, however, the counterexample would not challenge safety on the revised account. In those close worlds in which Luke falsely believes that his car has not been stolen based on his false belief that the car is in B1, Luke believes the latter on different grounds than in the actual world. In the actual world he believes that his car has not been stolen because he sees his car in B1 on his app, while in the relevant counterfactual world he believes that his car is in B1 because the attendant tells him so. The problematic worlds are therefore worlds in which the broad method differs.

Although this protects the present approach against Alspector-Kelly's modified counterexample, is there any guarantee that a different counterexample could not easily be constructed? In other words: Can one devise a positive argument that Safety (broad method version) is consistent with closure under competent deduction when methods are partly individuated according to Inferential Methods (broad grounds version)?

This question has a positive answer. The principle one can show to follow is this:

The principle states that beliefs competently deduced from known premises are safe.Footnote ⁷ Counterexamples of the kind proposed by Murphy and Alspector-Kelly could then not be found. For any time there is a case of an inferential belief based on known premises, Safe Deduction would imply that the inferred belief is safe.

With the present constraints on safety in place, Safe Deduction can be established in the following way. Take the premises P ₁, …, P _n to be known and assume that Q is competently deduced from them. Is the subject's belief in Q safe? Let W be the set of close worlds in which the subject uses the same broad method. By Inferential Methods (broad grounds version), one can conclude that W includes only worlds in which Q is inferred from P ₁, …, P _n. Moreover, the broad method the subject uses for inferring Q has as parts the broad methods the subject has used to arrive at P ₁, …, P _n. Therefore, the set W of worlds in which the subject believes Q by way of the same broad method is a subset of the set W ^′ of worlds in which P ₁, …, P _n are believed by way of the same broad method as in the actual world. Given that P ₁, …, P _n are known in the actual world, they are, by assumption, safe (here one makes use of Safety Condition). For this reason, they are true in all worlds in W ^′ and therefore, given W⊆W ^′, in all worlds in W. Now add to this that a deduction is only competently made if the conclusion follows from the premises (see the upcoming section for more on this) and one sees that Q must be true in W. This shows that Q is safe in the sense of Safety (broad method version) as desired.

These findings comprise the good news. If safety is constrained by the method one uses and if methods are understood broadly to pertain to all steps of the way leading up to the target belief and if they are, moreover, constrained by the content of the premises involved, then safety poses no threat to knowledge's closure.

The envisaged account of methods individuates methods very finely. Inferential methods A and B can only be the same if they represent exactly the same “inferential tree”: all direct and indirect premises must be the same. This makes it very hard for the same method to be applied by two persons or by the same person more than once.Footnote ⁸ Moreover, this way of individuating methods is at odds with a plausible pre-theoretical constraint on methods. A method should be applicable to a variety of input states.Footnote ⁹ But on the present account, a given method can only be applied to the same premises or grounds. The only variety a method would allow for are different tokens of the same premises. This may not be a knock-down objection, for in a theory of knowledge ‘method’ is partly a technical term coined for representing ways in which a belief is formed.Footnote ¹⁰ Nevertheless, even if construed in a technical sense, extremely finely individuated methods reduce the logical strength of a safety condition. As the individuation of methods gets finer and finer, there are fewer and fewer (close) worlds in which beliefs are formed by the same method. Hence, there are fewer possibilities for a subject's belief to violate safety. In other words, individuating methods more finely makes it easier for a belief to be safe. If this is pushed to the extreme, safety becomes a fairly weak condition on knowledge in danger of losing much of its explanatory power. For this reason, it is worthwhile to see whether one cannot find a way of defending closure which provides methods with a broader realm of possible application.

5. External individuation of methods

A different defense strategy can be used for making safety compatible with knowledge's closure. It exploits the fact that the debate on safety indicates that methods must be individuated externally for reasons independent of issues with closure (Broncano-Berrocal Reference Broncano-Berrocal2014; Grundmann Reference Grundmann2018). In light of this, it seems reasonable to see what an external individuation could mean for deduction.

For starters, let us explicitly distinguish deduction as mere reasoning from deduction as a belief-forming mechanism. Deduction may take as its starting point any kind of assumption. Perhaps an assumption is merely made for the sake of argument. Perhaps an assumption is considered possible and one is interested in what would follow from it should it turn out to be true. Whatever conclusion follows in such cases, the reasoning does not usually produce a belief in the conclusion, for one does not believe the premises one started with. Still, the reasoning employed may look the same as in a case in which one does believe the premises and therefore comes to believe the conclusion. The same inference rules may be applied to the same propositions in the same order. So, there is a sense of deduction where it refers to deductive reasoning which does not by itself constrain which epistemic attitude one forms towards the conclusion.

In addition to deduction as mere reasoning, there is also a sense in which it refers to the way a belief is formed. In this sense, using deduction as a method implies that one believes the conclusion of the deductive reasoning. Deduction in this sense also implies that one believes the premises. One believes the conclusion (at least in part) because one believes the premises.Footnote ¹¹ Roughly speaking, then, deduction as a belief-forming process has a tripartite structure: a condition on the input epistemic state, a condition on the reasoning based on the input epistemic states and a condition on the output epistemic state.

It is clear that extending our knowledge by competent deduction saliently targets the sense of deduction as a belief-forming mechanism, for we are interested in a belief being formed by way of inferring it from a number of premises. With this in mind, we can now ask how deductive methods should be individuated.

What I'd like to draw attention to is that the input states a deductive method accepts provides interesting options. It is clear that a deductive method which delivers beliefs as outputs should only accept beliefs as input. But as far as individuation is concerned, there is the option of going further: forming a belief based on known premises may not be the same method as forming a belief based on believed premises, where a believed premise will occasionally not amount to knowledge. Deductive inference from knowledge may not constitute the same method as deductive inference from belief. The point of forming a belief based on knowledge is that one transfers the good epistemic standing of the premises to the inferred conclusion. On this line of thought, inferring a conclusion from an instance of knowledge would not be the same method as inferring the same conclusion by way of the same reasoning from a belief with the same content if this belief does not constitute knowledge.

An approach along these lines interestingly differs from the approach set out in the previous section. On the present idea, the identity of the content of the input states – the premises – would not have to be held fixed. Rather, it would be the epistemic quality of the input states – knowledge versus mere belief – which would contribute to the individuation of methods.Footnote ¹² Let us make this explicit:

Applied to Safety (method version), this means that if one knows the premises of one's deductive inference in the actual world, then a belief in the conclusion is safe only if the conclusion is true in all close worlds in which it is believed when inferred from known premises.Footnote ¹³ If in close worlds the premises are not known, then an inferential belief would not count as being formed by the same method if the premises are known in the actual world. It is clear that this approach blocks the counterexamples by Murphy and Alspector-Kelly, for they require one of the premises to be false in a close world in which the same method is applied.

Before we turn to the question of whether counterexamples can be blocked in general, a word on ‘competent deduction’ as used in the characterization of Closure. It seems clear that in order to gain knowledge of a given conclusion, more is required than just getting it right. Someone who is not competent in making a certain inference and is therefore prone to get it wrong in very similar situations may not know what she inferred from something she knows. This is so even if, in fact, the conclusion follows from the premises. For this reason, one may require from competent deduction not only (i) that the conclusion in fact follows from the premises but also (ii) that in close worlds in which the subject starts a similar reasoning process from potentially different premises, the conclusion she reaches follows from the premises. Although even more may be required, for present purposes it suffices to assume that competent deduction satisfies these two conditions (see section 6 for further discussion):

Note that, strictly speaking, clause (i) is redundant. As the actual world is a close world, clause (ii) implies (i). I include it only for reasons of emphasis.

Given these considerations, it is now straightforward to see that Safe Deduction, as explicated in the previous section, holds.Footnote ¹⁴ To this end, suppose the conditions (i)–(iii) are satisfied. What we want to show is that the subject's belief in Q is safe in the sense of Safety (method version). So, consider a close world w in which the subject believes Q by using the same method, which is, by (iii) in Safe Deduction, a competently executed deduction. By (i) in Safe Deduction, the subject knows the premises of the inference in the actual world. Therefore, by Inferential Methods (externalist version), the subject knows the premises (which may differ from the actual premises) in w as well. Now, by Competent Deduction, competently using a deductive method requires that the conclusion follows from the premises in all close worlds. By assumption, that is (ii) in Safe Deduction, the subject competently inferred the conclusion from the premises. Therefore, the conclusion follows from the premises in w. Given that these premises are known, they are true and so it follows that Q is true in w as desired.

This demonstrates that individuating methods in terms of the epistemic standing of the input states is strong enough to show that beliefs competently inferred from knowledge are safe. Methods, on this picture, are thus individuated partly externally. Given that a subject can sometimes not know whether or not an input belief constitutes knowledge, they could sometimes not know which method they are actually using.

Independently of the issues surrounding deductive knowledge, it has been noticed (Broncano-Berrocal Reference Broncano-Berrocal2014; Grundmann Reference Grundmann2018) that Safety (method version) is most plausible on an external individuation of methods. In this debate, where non-inferential knowledge is at issue, the external components in method individuation usually concern instruments, informants or environmental conditions. The externalist holds that looking at a well-functioning clock may be a different method than looking at a clock (which may or may not be well-functioning). She also holds that forming a belief based on a reliable informant may be a different method than forming a belief based on an informant (which may or may not be reliable). Finally, the externalist holds that perceiving an object in epistemically friendly conditions may be a different method than perceiving the object under conditions which may or may not be epistemically friendly. All this is meant to hold even if the methods are indistinguishable from the subject's internal point of view.

To illustrate the parallel between non-inferential and inferential knowledge, consider a case of a counterfactually unreliable informant:Footnote ¹⁵

It is plausible and in line with parallel cases (see Grundmann Reference Grundmann2018: Sec. 4 for discussion) that Jane knows in the actual world that John is not coming to the party. After all, John is a reliable informant under the actual circumstances (he is not scared by Jane's actual dress). However, an externalist may disagree with Comesaña (Reference Comesaña2005) and Kelp (Reference Kelp2009), who claim for similar cases that Jane's belief is not safe and therefore that safety is not necessary for knowledge. One might think that Jane could have easily acquired a false belief because she could have easily dressed up as a killer in which case she would have been misinformed in those close worlds in which John decides to go to the party but tells her “No” because he is scared. An externalist about method individuation like Grundmann (Reference Grundmann2018) can respond that Jane uses a different method in the problematic worlds. The method she actually uses is to be externally individuated and corresponds to something like trusting a reliable informant.

If one accepts the externalist's response to challenges of the present kind, then one is in a position to draw a close parallel between an externalist individuation of non-inferential methods and an externalist individuation of inferential methods. Inferring a conclusion from knowledge would be like trusting a reliable informant, while inferring a conclusion from a belief which may or may not be knowledge would be like trusting an informant who may or may not be reliable. Given this parallel, the present approach to deduction fits well with general safety-based theories of knowledge.

6. A residual issue

The present account is based on two main elements. External individuation of methods ensures that whether a belief is safe or not depends only on good cases, namely on what the subject believes in situations in which the premises are known (provided they are known in the actual world). The second element is competent deduction. This ensures that the subject is not prone to make a mistake in reasoning from the premises to the conclusion.

With this story in place, it can be shown that beliefs competently deduced from known premises are safe. One may wonder, however, whether this argument does not show too much. Suppose one inferred the conclusion from known premises in the actual world by an application of modus ponens. If the method itself requires the premises to be known in every world in which it is used, then it may seem that the method is absolutely fail-proof: as modus ponens never leads one from true premises to a false conclusion, beliefs formed by this method will be automatically true in all possible worlds. This seems too strong.Footnote ¹⁶

The present issue rests on the assumption that the validity of a deductive method is an essential property of the method in question: if the conclusion is validly inferred in the actual world, only a valid method in some possible world can be identical to the method which was actually used. Admittedly, it is very tempting, from a pre-theoretical standpoint, to individuate deductive methods in this way. But I think this move should be resisted, for it does not square well with the purpose that a safety condition is supposed to play in a theory of knowledge.

Safety articulates the idea that a belief formed in a certain way could not easily have been false. Applied to a piece of deductive reasoning, this can be fleshed out by saying that the subject's way of reasoning could not easily have led her from true premises to a false conclusion. This leaves open whether the subject would have arrived at a false conclusion in a (non-close) world in which, say, her reasoning skills are seriously impaired. If this is right, a theoretically more fruitful way of thinking about deductive methods in the context of a safety-based theory of knowledge could be stated in terms of the modal properties of the subject's actual reasoning. Most prominently, this would involve the reasoning dispositions the subject actually has. Could the subject easily have arrived at a different, possibly false conclusion when reasoning from the actual premises? Could she easily have arrived at a false conclusion when reasoning from a similar but different premise set? On the envisaged way of individuating deductive methods, instantiating a valid inference pattern in the actual world does not by itself guarantee that one cannot go wrong in other worlds by using the same method.Footnote ¹⁷

There is a limiting case which may still prove problematic. Consider logical theorems which can be validly inferred from an empty premise set. On the present notion of safety, a belief in a logical theorem (or any necessary truth for that matter) is automatically safe, for there exists no possible world in which the belief is false.

Given that safety has only been construed as a necessary condition on knowledge, one need not worry that automatic safety of logical truths would result in automatic knowledge of logical truths. However, one route for explaining why a particular belief in a logical truth does not constitute knowledge would be blocked. One could not explain the absence of logical knowledge in terms of a lack of safety.

Yet, on the picture elaborated so far, one can explain the absence of knowledge through an absence of competent deduction. If a belief in a logical truth was not competently deduced, that is if one's reasoning could easily have led one astray, then one may cite this as a reason why the subject does not know the logical truth in question.

From a slightly different approach, one could strengthen safety by requiring not only that the actual belief is true in close worlds if arrived at by the same method, but also that similar (Williamson Reference Williamson2009a) or all (Grundmann Reference Grundmann2018) beliefs arrived at by the same method must be true. Then a belief in a logical theorem may not be safe because the subject could easily have arrived at a different and false belief. As far as I can see, this approach could be integrated into the present picture without jeopardizing the overall conclusion that beliefs competently deduced from knowledge are safe.

7. Concluding remarks

There is the plausible assumption that knowledge is closed under competent deduction in favorable circumstances. One prominent approach to knowledge takes knowledge to answer to a safety condition. Yet a number of potential counterexamples challenge this pair of assumptions. On closer inspection, it turns out that the problematic cases are best seen as challenging a safety condition on knowledge, for in these cases the subjects always seem to know the conclusion of the relevant inference.

The line of response developed in this paper is based on the observation that a plausible safety condition on knowledge must be relativized to methods for independent reasons. This provides room for maneuver: one can plausibly hold that in the problematic cases, the belief-forming methods are not held constant across possible worlds.

The difficulty with this strategy is to say something on when inferential methods count as the same. One option is to hold that the premises are essential for the way a derivative belief is formed. When understood broadly, so that even the grounds of one's direct premises must be kept constant, this way of individuating methods can be used to defend closure under competent deduction. However, one faces the objection that this proposal individuates methods too finely by restricting the range of input states that a method accepts to only one kind.

A better way of individuating inferential methods is to partly individuate them externally. It would be partly constitutive of the way a derivative belief is formed that it was deduced from something the subject knows. Deduction from knowledge is not the same method as deduction from belief. On this account, the premises of an inference are seen in close analogy to the environment in which perception takes place. Just as one does not always know whether the environment one is in is epistemically friendly or not, so one will not always be able to know whether what one treats as premises is of the good (knowledge) or the bad (mere belief) kind.

This account of deduction rebuts the counterexamples and provides the means for a general argument that beliefs competently deduced from knowledge are always safe. It also reveals that deduction faces the same kind of intricate issues of individuation as non-inferential belief-forming methods such as perception. Thus, deduction does not come easy: explaining deductive knowledge is (almost) as hard as explaining perceptual knowledge.Footnote ¹⁸