Mind the Steps!
Should propositional epistemic justification
satisfy (cumulative) transitivity?
MSc Thesis (Afstudeerscriptie)
written by
Joannes Bernard Campell (born 26.09.1985 in Lavin, Switzerland)
under the supervision of Dr. Luca Incurvati and Dr. Peter Hawke, and submitted to the Examinations Board in partial fulfillment of the requirements
for the degree of
MSc in Logic
at the Universiteit van Amsterdam.
Date of the public defense: Members of the Thesis Committee:
31.08.2020 Dr. Benno van den Berg (chair)
Prof. Dr. Arianna Betti
Dr. Peter Hawke (co-supervisor) Dr. Luca Incurvati (co-supervisor) Dr. Ayb¨uke ¨Ozgun
Abstract
Arguments pertaining to the question of whether propositional epistemic justi-fication should satisfy transitivity and / or cumulative transitivity are critically examined and proposed. The goal is to explore some implications that the assump-tion of validity of (cumulative) transitivity has with respect to our understanding of propositional justification.
iii
Acknowledgements
I would like to thank the illc for allowing me to spend a couple of wonderful years in Amsterdam.
I would like to thank my supervisors Peter Hawke and Luca Incurvati for their help, invaluable comments and interesting discussions.
I would like to thank Tanja Kassenaar for her necessary help regarding admin-istrative matters.
I would like to thank the members of the Thesis Committee Benno van den Berg (chair), Arianna Betti, and Ayb¨uke ¨Ozgun for taking the time to read the thesis and discussing it at the defense.
I would like to thank Michael Andreas M¨uller, Jakob Piribauer, and Rian Zuberi for interesting discussions on the topic.
Contents
1 Introduction 1
1.1 Research question and goal . . . 1
1.2 Organization of the thesis . . . 3
2 Preliminaries 5 2.1 Doxastic and propositional justification . . . 5
2.2 A problem of abundance . . . 6
2.3 A short note on the method of cases . . . 7
3 State of the debate 9 3.1 Aikin in favour of Tr . . . 9
3.2 Smith in favour of Cut . . . 14
3.3 Black against Tr . . . 19
3.4 Nair against Cut . . . 21
4 New Arguments against the principles 27 4.1 Cut and epistemic modifiers . . . 27
4.2 Faultless slipery-slopes . . . 38
4.3 Summary . . . 45
5 Conclusion and Outlook 47 5.1 Weak and strong inferential justification . . . 47
5.2 Further work . . . 56
References 59
Chapter 1
Introduction
1.1
Research question and goal
This thesis explores the prospects of a negative answer to the question of whether inferential epistemic justification should satisfy either of the two logical principles known as transitivity and cumulative transitivity.
Roughly speaking, inferential justification is understood here as a binary re-lation that holds between two propositions when one proposition epistemically justifies the other. If ϕ inferentially justifies ψ, this is denoted by the formula ϕ ψ. The proposition on the left is interchangeably referred to as the an-tecedent, justifier or evidence and may consist of one basic proposition, or a con-junction respectively a set of propositions, while the proposition on the right will be called the consequent or target proposition.
Intuitively speaking, the inferential justification denoted by ϕ ψ is the kind of inferential justification that is sufficient for an agent to be justified simpliciter in believing that ψ, if she manages to establish ϕ (and bases her belief that ψ on ϕ). According to a virtue-epistemologist, establishing ϕ and ϕ ψ would count as fulfilling her epistemic duty, and according to an account of knowledge as justified true belief this would count as satisfying the first condition of knowledge. Consequently, the kind of justification this thesis is concerned with will also be called sufficient justification. The case where the justification is not sufficient is denoted by ϕ 6 ψ.
A few specifications about this relation are necessary and helpful. First, the antecedent ϕ is understood as the total set of evidence relevant for the consequent. This means that ϕ 6 ψ holds if ϕ on its own is justifies ψ.
This does not mean that the antecedent justifies the consequent in any circum-stances, for adding new evidence my void the justification.
Visual perception is a widely accepted example of a justifier that is defeasible. 1
If there is a perception of a tree amongst the evidence and nothing else, then this justifies the proposition that there is a tree. If however we add to the evidence that this perception takes place in a state that makes hallucinations very likely, then the proposition that there is a tree is no longer justified.
The two principles are presented in form of a rule of inference. The first principle is that of transitivity for inferential justification. It has the following form.
ϕ ψ ψ χ
Tr ϕ χ
The meaning of this principle is simply that whenever ϕ sufficiently justifies ψ and ψ sufficiently justifies χ, then ϕ sufficiently justifies χ.
The second is the principle of cumulative transitivity for inferential justifica-tion. It has the following form
ϕ ψ ϕ ∧ ψ χ
Cut ϕ χ
This principle is often regarded as expressing the following idea: If the original evidence, here ϕ, justifies ψ, then I can add ψ to my original evidence. Whatever is then justified by this cumulated evidence ϕ ∧ ψ, is also justified by the original evidence alone (see e.g. (Kraus, Lehmann, and Magidor 1990) for such a motiva-tion of Cut). An example of this is the following. Say that the original evidence, ϕ, consists out of all the propositions written in a book. John reads this book and draws a justified, but not entailed conclusion, P , from this book. He writes down this sentence at the end of the book. He then lends the book to Joseph who reads the book and mistakes the sentence written by John for a part of the book. From this ‘augmented’ book, Joseph also draws a justified but not entailed conclusion, using Johns sentence as a premise. According to Cut, Joseph would have been just as justified to draw his conclusion had John not forgotten to erase the sentence before lending him the book.
We can divide the opinions on how exactly to analyse the notion of inferential justification in a camp according to which we should give a quantitative analysis of justification in terms of probabilities and in a camp according to which a qualitative analysis is right, for example an analysis akin to ceteris paribus laws. While the principles are typically not valid according to quantitative approaches, Cut is typically regarded as valid according to the other camp. According to this camp, the logic of inferential justification is a nonmonotonic logic. The principle of Cut plays an important role for this. It is considered e.g. by (Gabbay 1985) as one of the three minimal principles next to reflexivity and cautious monotonicity that
1.2. ORGANIZATION OF THE THESIS 3 should hold for a logic to be counted as a bona fide nonmonotonic logic.1
This means that questioning the validity of Cut can be understood as ques-tioning whether the logic of inferential justification is a (standard) nonmonotonic logic. I take this to mean that the question of whether inferential justification satisfies Cut is an interesting and relevant question.
The principle of transitivity is typically rejected by both camps. It is, however, often assumed in philosophical arguments. One example are the infinite regress arguments used in the debate on how to best resolve the M¨unchhausen Trilemma for epistemic justification. According to Black (1988), infinite regress arguments proceed by denying one of the following five premises, since they form an incon-sistent set (premises (1)-(4) for example entail the negation of premise (5)).
(1) (∀x)(Ax → (∃y)(Ay ∧ xRy)). (2) (∃x)(Ax).
(3) R is irreflexive. (4) R is transitive.
(5) There is no sequence with an infinite range such that each of its elements has property A and stands in relation R to its successor.
Given that transitivity is one of them, we consider Tr as a serious contender as well.
Usually, the validity or non-validity of the principles is a consequence of what is considered to be the right analysis of inferential justification. Martin Smith (2018) and Shyam Nair (2019) are however participants of a debate in which the question of which logical principles hold for inferential justification has been addressed on an abstract level, independently of concrete theories of justification. This debate can be seen as a debate about the predictions that an adequate logic of inferential justification should make. This text is intended to pertain to this debate. This means that we consider and propose philosophical arguments about what principles should or should not hold in an adequate logic of inferential justification.
1.2
Organization of the thesis
The main part of this thesis can be found in chapters 3 and 4. In chapter 3, we set the stage by critically discussing four arguments in the debate on Tr and Cut.
1Let |∼ the symbol for nonmonotonic entailment. then the two additional minimal principles
are: α |∼ β α |∼ γ Cautious Monotonicity, α ∧ β |∼ γ Reflexivity. α |∼ α
The first argument that is considered is Scott Aikin’s (2010) defence of Tr. The main point he makes in his argument is that Tr guarantees consistency in reasoning. The answer to his argument is that assuming Tr is not necessary for this, as this function can be fulfilled by other constraints and rules.
The second argument that is discussed is an argument that may be considered the main argument in favour of Cut. This argument is based on the assumption that Cut adequately encapsulates the idea that an epistemic agent can use a con-clusion drawn from a set of evidence as a premise in further reasoning. In this text, Smith’s (2018) version of his argument is discussed.
The third argument is Black’s (1988) argument against the validity of Tr. His argument is based on a probabilistic account of confirmation. His argument can easily be extended to an analogous argument against Cut. The main takeaways from his argument are the distinction between qualitative and quantitative ap-proaches to epistemic justification and the importance of not basing a general argument on a concrete analysis of justification.2 In the debate between
propo-nents of quantitative and propopropo-nents of qualitative approaches to justification, the principle of Cut plays an important role. Smith (2010; 2018) uses the validity of Cut as a premise in an argument against quantitative accounts of inferential justification. This adds to the relevance of he question of whether Cut holds for inferential justification.
The last argument is taken from Nair’s (2019) paper “Must Good Reasoning Satisfy Cumulative Transitivity?”. He argues against the claim that no plausi-ble qualitative analysis of justification invalidates Cut by providing a theory of justification where Cut might fail.
In chapter 4, two arguments are presented according to which good reasoning - i.e. inferential justification - not only must not satisfy Tr or Cut, but according to which they should not satisfy them. The first argument is directed against both principles and considers the role of consequents augmented by epistemic intensifiers. The second argument, again directed against both principles, is based on the Sorites paradox and the use made of it in (van Rooij 2012; Cobreros et al. 2012a; Cobreros et al. 2012b).
But before we get to these arguments, the important distinction between propo-sitional and doxastic justification and some methodological considerations are in-troduced in chapter 2.
2See section 2.2 for my argumentative strategy employed in order to deal with the great
Chapter 2
Preliminaries
2.1
Doxastic and propositional justification
Work on epistemic justification has made use of distinction between two kinds of epistemic justification, namely between doxastic and propositional justification (see (Firth 1978) for an early introduction of this distinction). Doxastic justifica-tion is concerned with the justificajustifica-tion of beliefs, while proposijustifica-tional justificajustifica-tion is concerned with propositions alone.
While the adequacy of this distinction can be seen as common ground, there is a debate going on about how the two types of justification are related to each other. The position that comes closest to being mainstream in this debate holds that propositional justification grounds doxastic justification. Roughly speaking, and given that propositions are the contents of beliefs, this means that a belief A is doxastically justified if it is based on a belief B whose content propositionally justifies the content of belief A (see e.g. Kvanvig 2003; Cruz and Pollock 2004; Swain 1979). Jonathan Kvanvig (2003), for example writes the following about the relationship between doxastic and propositional justification.
Doxastic justification is what you get when you believe something for which you have propositional justification, and you base your belief on that which propositionally justifies it.(Kvanvig 2003)
There are views that oppose this mainstream position. One is the view that we should reverse the relation and explain propositional justification in terms of doxastic justification (see e.g. Turri 2010). Another position holds that neither is more fundamental but that they are somehow intertwined (see Melis 2018).
We agree with all participants to the debate that there are this two different kinds of justification. This thesis is concerned with the structural properties of
propositional justification and the argument will therefore be that propositional justification is not (cumulatively) transitive.
Another point about which (almost) all participants seem to agree is that these two relations come together in harmony in cases of agents with beliefs that are justified by other beliefs. What I mean by this is that there is a robust way in which doxastic justification suggests or implies the propositional justification of the respective propositions and in which propositional justification creates - in certain circumstances - affordances for doxastic justification. The debate about how propositional and doxastic justification are related to each other is about how this harmony comes about, not about whether this harmony exists. We try to remain neutral on the question of how this harmony comes about, but do assume this harmony in order to use considerations about doxastic justification in the arguments that follow.
There is one last general assumption to mention. In at least one of the argu-ments, we will assume that justification is a normative concept. This assumption is shared by the majority of epistemologists, however not by all (e.g. Fumerton 2001).
2.2
A problem of abundance
The idea of contemplating whether inferential justification is transitive or cumu-latively transitive faces a serious challenge. The challenge derives from the fact that there is an ever growing cornucopia of different relations being proposed as an analysis of propositional justification (The theories discussed in the text to follow should illustrate this point).
This makes a bottom-up approach to the questions of transitivity unpromising. Instead of arguing that a specific notion of justification is correct and then pro-ceeding to show whether this notion of justification is transitive or not, a different strategy is pursued here. The strategy here is to move the discussion up to a more general level.
The idea is to work with the intuitive notion of inferential justification that is the target of any philosophical analysis. Assuming that there might be multi-ple adequate theories of justification, we then work not with properties assumed in a concrete analysis of the concept, but with properties that might be called plausible candidates for necessary conditions of an adequate analysis of inferential justification.
While arguing that propositional justification is sometimes based on a relation R and then showing that R is not transitive would count as an argument that propositional justification is not transitive, the dialectic force of such an argument is questionable. It allows for the response that if this is so, then ϕRψ is not
2.3. A SHORT NOTE ON THE METHOD OF CASES 7 sufficient for ϕ ψ (see section 3.3 for such an argument and response). An argument that is based on general assumptions that delimit acceptable analyses of propositional justification does seem more promising.
This strategy is successful if all assumptions made about the relation of infer-ential justification in a counterexample to or an argument against a transitivity are indeed necessary conditions for an adequate account of inferential justification. Even if the properties used in an argument remain disputed with regard to their status as necessary conditions for an adequate analysis, the arguments and counterexamples deliver - if valid - conditions on a nice level of generality under which inferential justification is not (cumulatively) transitive. This strategy is accordingly also robust.
2.3
A short note on the method of cases
Before we present the arguments, we first make a note on the argumentative strat-egy of using counterexamples in order to disprove the validity of the principles. This means that we work with intuitions and thought experiments (for an overview on the topic of intuitions, see (Pust 2019), for a critique of this type of ‘armchair philosophy’, see (Machery 2017)). Sometimes the method of cases and counterex-amples is deemed unsuccessful on the basis that the counterexcounterex-amples to a natural and plausible concept or principle are unappealingly elaborate and theoretical. In defending the natural and plausible concept or principle against the counterex-amples it is argued that having concepts that carves nature at its joints and is robust enough to correctly predict a vast majority of the cases is worth biting the bullet. As an example, Weatherson (2003) proposes this response to the attacks on the analysis of knowledge as justified true belief driven by Gettier style exam-ples. Having this analysis of knowledge that carves nature at its joints is worth accepting that Smith did in fact know that the man who will get the job has ten coins in his pockets, according to Weatherson.
Do we have a similar situation with respect to the principles where we can base a plausible defense of the principles on the claim that the counterexamples are too elaborate and specific to carry the theoretical weight necessary to invalidate the principles? In order to answer this we have to distinguish two ways of biting the bullet with respect to the counterexamples. One way of doing so is to accept that the concept or principle does in fact make the wrong prediction in such a hypothetical case. The second way is to accept that the concept or principle does in fact make a surprising and unintuitive prediction in these cases, but then biting the bullet by accepting the prediction of the concept or principle. Accepting that Smith did in fact know that the man who will get the job has ten coins in his pockets is an example of the second way of biting the bullet. Accepting that
knowledge as justified true belief makes the wrong prediction in this case would be an example of the first way of biting a bullet.
With respect to the first way of biting the bullet, I claim that this option is not open for a defendant of Tr and/or Cut as valid logical principles for patent reasons. To claim the validity of something as a logical principle just comes with the claim of absolute absence of counterexamples, no matter how elaborate they are. That is just part of what it means for something to be a logical principle. To accept some counterexamples and at the same time defend Tr and Cut on the basis of the principles being robust enough to correctly predict the cases that are most likely to happen in real life is to change position by reducing the claim and their status from a valid logical principle to a good rule of thumb. While I endorse both Tr and Cut as good rules of thumb, it is their status as valid logical principles that is at stake here, and in that case, any possible counterexample counts - no matter how elaborate and remote it is.
The second response is open to a defendant of Tr and Cut. Here we accept the predictions of the principles, even though our initial judgment may be that those predictions are wrong. Whether this response is plausible and we should accordingly discard these initial judgments in order to uphold a theory containing Tr or Cut depends on the respective pull of counterexamples and theory. The aim of this thesis is not to claim that one way to balance this intuitions is the correct one, but it aims at providing relevant considerations for this balancing act.
I argue in sections 3.1 and 3.2 that the intuitive pull to include Cut or Tr in the theory is not as strong as some people have suggested. In chapter 4, I argue that some counterexamples carry enough theoretical weight to at least balance the scale. The assumptions used in the counterexamples accordingly play a crucial role in the argumentation. In addition to the brute plausibility of these assumptions, a further argument can be made for the option of giving up on the principles. Given that the principles are argued to be incompatible with different other intuitive ideas in each counterexample, there is an economic rationale against accepting the principles. Giving up on the principles solves the conflict in all cases, while maintaining that the principles are valid forces one to give up on different ideas in each of the three counterexamples (and in potentially many more that could be presented).
Chapter 3
State of the debate
In this chapter, four arguments are discussed. For each principle, there is an argument that has been made in favour of it, and one that is critical of it. The goal is to blunt the arguments in favour of the principles, to see what room is left for critical arguments, and in general to give the context of my own arguments that are presented in the next chapter. We start with the arguments in favour of the principles.
3.1
Aikin in favour of Tr
Aikin (2010) argues that justification is transitive. Before looking at the argument it is useful to describe the concepts that Aikin is working with and the relations between them.
Aikin’s interest in the transitivity of justification stems from it being a premise in the regress arguments of justification that he uses to argue for an infinitist posi-tion on inferential justificaposi-tion. Another premise in this argument is the principle of inferential justification. He works with the following formulation of it:
If S is justified in holding P , then there is some Q that S is justified
in holding, and Q supports P . (Aikin 2010, p. 24).
The interpretation of this quote is that Aikin calls ‘justification’ what we have been calling ‘doxastic justification’ and he calls ‘support’ what we have been calling ‘propositional justification’. His use of words when he defines a chain of reasons suggest that while this interpretation seems to be on the right track, there is a little twist to it:
Let C be a chain of reasons that supports P for S if C is a series of reasons wherein:
(C1) each member of C is justified
(C2) each member of C is supported, and
(C3) each member supports the next member, except the ultimate
member, which supports P . (Aikin 2010, p. 24).
The twist consists in the fact that support is presented as a ternary relation, with the agent in a certain epistemic state being one of the relata next to the two propositions. If we consider the agent as fixed in a chain of reasons and cancel it out, this reduces to a binary relation between the members of the chain.1 It is this relation that Aikin is concerned with. Concerning this relation, he claims that transitivity fulfils a dual role:
The first role is a formal way of stating that failure of justification earlier in the chain yields justification failure down the chain.
The second role is that what is earlier in the chain plays a limiting role on what may be added later down the chain. There may be am-pliative judgments that go beyond what’s provided in the chain, but the further judgments may not contradict what’s come earlier. (Aikin 2010, p. 27f., his emphasis).
First a short note on ampliative judgments. This simply means that justifi-cation is non-monotonic and that it allows for things to be justified that are not entailed by the evidence. They go beyond what is given in the evidence in the sense that if we add the justified proposition to our beliefs, then there is more in-formation in our beliefs now than the beliefs deductively guaranteed before adding it.
When it comes to the first role of transitivity that Aikin mentions, there is an ambiguity at play in the sense of it not being entirely clear what he means by a failure of justification, be it earlier in or down the chain. In trying to get rid of this ambiguity, two interpretations are considered. Take the following chain of reasons:
ϕ0. ϕ1 ϕ2 ϕ3 ϕ4 · · ·
A failure of justification can either mean that a proposition fails to justify the very next proposition in the chain, or it can mean that a proposition fails to justify a proposition further down the chain. Applied to a failure of justification earlier in the chain, this means that we have one of the following exemplary cases: For the first case, say that ϕ4 fails to justify the very next link ϕ3. Then we have the
following:
1There is however a consequence of this that becomes relevant in the next chapter, namely the
3.1. AIKIN IN FAVOUR OF TR 11 ϕ0. ϕ1 ϕ2 ϕ3 ϕ4 · · · 6
For the second case, let the direct links be intact, but this time ϕ4 fails to justify
something deeper down the chain, say ϕ1. Then we have the following.
ϕ0. ϕ1 ϕ2 ϕ3 ϕ4 · · · 6
If we look at the first case, i.e. at a chain with a broken direct link, then I do not see what transitivity accomplishes here, for we can take the transitive closure of this chain and nothing happens to the broken link or with the justification down the line, the reason being that a broken direct link only ever figures as a premise but not a conclusion of Tr.
The failure of justification in the second case, ϕ4 6 ϕ1, can however figure as
a conclusion of Tr, and accordingly we have the case that if this chain where to be transitive, then one of the direct links in between would have to fail as well.2
This gives us the following understanding of the first role of transitivity. The first role of transitivity is to guarantee that if a proposition fails to justify a proposition down the line, then there must be a failure of justification between two adjecent propositions in between.
Formulated in this way, in essence as the contrapositive of Tr, it sounds admittedly rather plausible. If I reason myself to a proposition that is not supported by my starting point, so goes the claim, then I must have made a mistake along the way. While it may be very sound advice to trace back the steps once one loses the justificatory connection with the starting point, it nevertheless poses a limitation that is questionable. It has to do with the possibility of ampliative judgments in the single steps. What Tr guarantees is that no matter how many ampliative steps there are in a chain, each moving the proposition just ever so slightly from the preceding proposition in the chain, the sum of all these cannot result in a proposition that is not justified by the starting point. I will argue in section 4.2 there are in fact chains where the steps are acceptable, but where the starting point is at best irrelevant for the truth of the ending point. If these arguments are successful, then we should elevate the status of Tr from a good rule of thumb to a valid principle.
The second role that Aikin ascribes to transitivity is is a restrictive role. Given a chain of justification that runs from left to right, such as
2Suppose that all direct links are intact. Then, by transitivity, we have ϕ
3 ϕ1. Using
ϕ4 6 ϕ1 and contraposition of Tr, we then either have ϕ4 6 ϕ3 or ϕ3 6 ϕ1. In the first case
we are finished, in the second case we have, again by contraposition of Tr, ϕ36 ϕ2 or ϕ26 ϕ1
ϕ0, ϕ1 ϕ2 ϕ3 ϕ4 · · ·
the role of transitivity is according to Aikin to exclude the possibility that any proposition contradicts a proposition that is on the left of it. This means that Tr should exclude a situation like the following.
ϕ0. ϕ1 ϕ2 ϕ3 ϕ4 · · · contradictory
How is transitivity supposed to exclude a situation as depicted above? I take Aikin to have the following in mind. By transitivity, we have ϕ4 ϕ1. We now
have a situation where a proposition justifies an other proposition that contradicts it. This seems less than ideal, but in order to exclude it we need a principle in addition to transitivity, something like a consistency requirement (CR).
Local CR: For any propositions ϕ and ψ we have that ϕ ψ =⇒ ϕ and ψ are not contradictory.
Given this, Aikin’s argument in favour of transitivity with respect to the second role would be that we should assume justification to be transitive, because tran-sitivity in combination with the local consistency requirement guarantees that no chain of justification contains two contradictory propositions.
I am not convinced by this argument, for it seems that the local consistency requirement does the heavy lifting and if we give a slight reformulation of this consistency requirement, then transitivity is no longer needed to exclude the jus-tification chains containing a contradiction. The following requirement should do the job.
Global CR: For any propositions ϕ, ψ0, . . . , ψn, χ such that ψi ψi+1
for 0 ≤ i < n, we have that
ϕ ψ0 and ψn χ =⇒ ϕ and χ are not contradictory.
This principle is admittedly a bit more cumbersome, but it achieves exactly the same without having to assume transitivity.3 Given this, the conclusion is that
even if one wishes to exclude the chains that contain contradictory propositions, transitivity is not needed for that.
3There might be a worry that they do not achieve the same thing because the local consistency
requirement can be written down in form of one rule, say ϕ ψ
Local CR ϕ, ψ 6|= ⊥ ,
3.1. AIKIN IN FAVOUR OF TR 13 An analoguous argument can be made with respect to another interesting re-quirement of consistency, according to which, given two chains that start from the same proposition, each proposition of one chain has to be pairwise consistent with all propositions of the other chain. This requirement excludes situations such as the following. ϕ χ1 χ2 χ3 χ4 ψ1 ψ2 ψ3 contradictory
Given transitivity, we can exclude such situations with a principle that, in form of a rule, claims the following.
ϕ ψ ϕ χ
ψ, χ 6|= ⊥
In a similar fashion as above and using the notation introduced in footnote 3, we
while it is not so clear that this can be done with the global consistency requirement. If this where the case, then this might be seen as an argument in favour of transitivity. However, by introducing a notion of justificatory ancestry, denoted by ϕ ψ and with the meaning that there is a natural number n such that I can get from ϕ to ψ in n justificatory steps, we can express the global consistency requirement as the following rule.
ϕ ψ
Global CR ϕ, ψ 6|= ⊥ .
The relationship between and is goverened by the following three rules: ϕ ψ -I ϕ ψ ϕ ψ ψ χ -EL ϕ χ ϕ ψ ψ χ -ER ϕ χ
The rule -I allows for the introduction of , while the other two rules allow for an Expansion to the Left, respectively to the Right of the relation.
Enriching the language in this fashion has the benefit that we can represent a more fine grained understanding of propositional justification by being able to specify which conclusions demand premises with direct justification and for which conclusions it is enough that propositions belong to some chain of justification.
do not need transitivity and can exclude such situations by the this rule:
ϕ ψ ϕ χ
ψ, χ 6|= ⊥
3.2
Smith in favour of Cut
In this section an argument in favour of Cut is critically examined. It is Smith’s (2018) version of what seems the most popular argument in favour of Cut, namely that Cut reflects the idea that we are allowed to use our conclusions in order to derive further justified conclusions.
Martin Smith’s argument in favour of Cut proceeds in three steps. First we are invited to agree that a certain idea about reasoning and inferential justification is very plausible. In a second step, we are presented with a principle that is intended to encapsulate this idea. Smith then argues in a third step that given two assumptions, we can reformulate this principle in a formal way as Cut. The four ingredients of his Argument are the following.
Intuitive Idea: “[. . . ] it’s very plausible to think that, if I have justification for believing a proposition P , then I also have justification for using that proposition as a premise in theoretical reasoning.”(Smith 2018, p. 3864).
Principle: “If one has justification for believing P and one’s ev-idence, along with P , provides justification for believing Q, then one has justification for believing Q.” (Smith 2018, p. 3861). Evidentialist Assumption: “[...] if one has justification for be-lieving a proposition then this justification is provided by the total
evidence that one possesses” (Smith 2018, p. 3862).
Propositionalist Assumption: “[...] one’s evidence consists of a stock of propositions, or a conjunction of propositions.” (Smith 2018, p. 3862).
3.2. SMITH IN FAVOUR OF CUT 15 (P1) The Intuitive Idea is true.
(P2) The Evidentialist Assumption is true. (P3) The Propositionalist Assumption is true.
(P4) If the Intuitive Idea is true, then the Principle is true.
(P5) If both the Evidentialist and Propositionalist Assumption and the Principle are true, then Cut is valid.
(C) Cut is valid.
It is clear that in trying to establish that Cut is not valid, we should have some-thing to say about this argument. We will focus on the intuitive idea mentioned in the first premise. We will argue that there are two readings of the idea (and accordingly also of the principle). One reading is plausible but not strong enough to derive Cut from it. The second reading is strong enough to derive Cut from it, but not plausible. If this counterargument is successful, then we manage to blunt the argument in favour of Cut, while simultaneously explaining the intuitive appeal of the principle and idea.
The intuitive idea in the first premise states that I am justified in using some-thing that is justified by my evidence as a premise in theoretical reasoning. But what exactly does that mean? Smith gives the following partial answer to this question:
[. . . ] Part of what this means is that, by using the proposition to justify further beliefs, I won’t be led to believe any propositions for
which I lack justification. (Smith 2018, p. 3864).
Even with this answer, there still remains an ambiguity in both the intuitive idea and the principle: the evidence that I have for Q is not specified in either of them or in the quote above. The intuitive idea speaks of ‘not lacking’ evidence for Q, while the principle only claims that ‘one has justification’ for Q, without specifying what this justification exactly consists in. This allows for different readings, depending on what one takes this justification to consist in. As long as we leave that unspecified and only make an existential claim, then the idea and principle are very plausible and hard to argue with, for there will always be the original evidence E in conjunction with P that is justification for Q. The idea and principle read in this way are however clearly not strong enough to motivate Cut. Another plausible reading includes that justification comes in degrees and that I am left with at least some rather than no degree of justification. This also seems plausible, but again is not strong enough to entail Cut, as the degree of justification we are left with may not be sufficient justification.
It seems clear that Smith must have in mind the reading according to which the original evidence E is sufficient on its own. Only this interpretation of the idea gives rise to a reading of the principle that is strong enough to derive Cut from it:
If one has sufficient justification E for believing P and one’s evi-dence E, along with P , provides sufficient justification for believing Q, then E is sufficient justification for believing Q.
The plausibility of the idea and the principle understood in this way are however not obvious. Let’s look at exactly what kind of epistemic behaviour gets condoned by it.4
1. I draw a conclusion P from my evidence E.
2. I treat P “[...] as another piece of evidence from which to draw further conclusions” (Smith 2018, p. 3861).
3. From this new evidence, E ∧ P , I draw a conclusion Q.
4. I can ignore that I used P in order to get to Q and consider Q sufficiently justified by my original evidence E alone.
Note first how this epistemic behaviour reflects the assumption that Cut is valid. Step one is equivalent to establishing the left premise
E P.
Step three accordingly reflects establishing the right premise of Cut, according to which
E ∧ P Q.
Step four reflects the idea that Cut is valid, i.e. that from the two premises it follows that
E Q.
In the following few lines I aim to give some reasons as to why it is not obvious, why the epistemic behaviour
The fallibility of propositional justification means that E P allows for ex-ceptions, i.e. for the possibility that E is true, while P is not. Using the apparatus of possible worlds to talk about modalities and using [[ϕ]] to denote the truth set of ϕ, i.e. the set containing all possible worlds at which ϕ is true, this means that for some E and P we have both
E P and [[E ∧ ¬P ]] 6= ∅.
4Note that (Nair 2019, p.129) and others presents a very similar understanding of what it
3.2. SMITH IN FAVOUR OF CUT 17 The fact that the existence of worlds in [[E ∧ ¬P ]] does not undermine this justifi-cation means that these worlds are in some way not relevant enough to undermine the justification of P by E.
Moreover, this worlds are inconsistent with the evidence of the right premise and for this reason are not relevant when it comes to the justification of Q by E ∧ P . This assumption is made plausible by the fact that even for the strictest of implicative relations (e.g. (ϕ → ψ)), worlds inconsistent with the evidence respectively the antecedent are not considered to be relevant. Another way of arguing that inconsistency with the evidence result in non-relevance is to point out that such worlds do not constitute an exception to the justification.
With respect to the conclusion, it is necessary that two types of worlds are deemed irrelevant. These are worlds belonging to either of the two following sets, whose non-emptiness is consistent with both the premises and the conclusion.
[[E ∧ P ∧ ¬Q]] and [[E ∧ ¬P ∧ ¬Q]].
Since worlds belonging to either set constitute cases where the antecedent of the conclusion is true while the consequent is not, these worlds must be ruled irrelevant in the sense of being exceptions that do not break the justification for the conclu-sion to go true. For ease of reference, worlds belonging to the left set will be called P -exceptions and the others ¬P -exceptions. Note further that the claim that Cut is valid is assumed here to imply that the irrelevance of both types of exceptions to the conclusion is guaranteed by the truth of the premises, for true premises with relevant exceptions to the conclusion would result in a false conclusion and there-fore in a counterexample to Cut. The aim at this point is however not to show that Cut is not valid, but only to show that this guarantee of irrelevance is based on assumptions that constitute answers to substantial theoretical and philosophical questions and are therefore debatable. The argument is accordingly not directed against Cut but speaks in favour of not being too sure about it.
Let’s begin with ¬P -exceptions. One story about epistemic justification that allows for these worlds to be irrelevant for the premises while remaining potentially relevant for the conclusion is that the relevance of an exception is not always solely dependent on the evidence alone, but is at least sometimes a function of both the antecedent and the consequent of the justification. If we assume this, then ¬P -exceptions would be irrelevant exceptions to to the left premise, because the justification goes from E to P . For the right premise they would still be irrelevant due to not being consistent to the evidence. When it comes to the conclusion however, their relevance is not ruled out, since these worlds are consistent with the evidence, and the consequent of the justification in the conclusion is different form the consequent in the left premise. Excluding these types of possibilities boils down to the claim that the set of relevant scenarios is determined alone by the
evidence. While the claim of evidence based uniformity of relevant scenarios is not implausible, it does also not seem obvious and undebatable.
Even if we accept the latter claim of uniformity, there is still a possibility for relevant exceptions to the conclusion in form of P -exceptions. Since both E and P are true in such worlds, they do not constitute exceptions to the left premise and hence might be part of the scenarios fixed as relevant by the shared evidence E of left premise and conclusion. Given this, it seems that if we want a guarantee of irrelevance, we must get it from the right premise. Since P -exceptions are also exceptions to the right premise, this means that they are irrelevant for the justification claimed in this premise. In this case we can use the fact that the evidence in the right premise and conclusion is not the same. One possible way to get P -exceptions that are relevant for the conclusion is to say that it makes a difference whether something, in this case P , is part of the evidence or not. This would result in the following story explaining the discrepancy of relevance of certain P -exceptions. Some are relevant to the conclusion, but are deemed irrelevant for the right premise due to the fact that P is part of the evidence in the right premise. A proponent of Cut seems to be committed to negate this story. And again, negating it is not necessarily implausible, but it is the result of a theoretical choice.
Coming back to the epistemic practice depicted in the four steps above, this means that we should never be worried about any exceptions becoming relevant for the conclusion. No matter how delicate the subject matter or how many times we repeat the steps. For this is what the validity of Cut means, namely that this is not only good or acceptable epistemic practice, but indeed perfect epistemic practice. Given the possibilities mentioned above, I do not think that the ideas and argument presented by Smith above are conclusive enough to consider the validity of Cut as something “[. . . ] that precedes substantial theorising about justification as something that helps to delimit the very notion that we’re theorising about.” (Smith 2018, p. 3861).
Hoping to have given some reasons to consider the possibility of Cut being dependent on some other, less than necessary choices, I will leave the argument at that. The two considerations offering a possibility of the invalidity of Cut will however be further discussed in the text to follow. Nair (2019) offers an argument towards the conclusion that it is not the case that propositional justification must satisfy Cut. He argues so by offering a theory of justification where it does make a difference whether a proposition is part of the antecedent or not. This argument will be discussed in section 3.4. In section 4.1 I will present a conception of propositional justification according to which the relevance of exceptions can be dependent on both the antecedent and the consequent of and according to which there are counterexamples to Cut.
3.3. BLACK AGAINST TR 19
3.3
Black against Tr
The third argument that we consider is Oliver Black’s (1988) argument that jus-tification is not transitive. The conclusion of Black’s argument is that doxastic justification is not transitive. The first assumption of his argument is that doxas-tic justification is grounded by confirmation, a relation between two propositions. Then he provides a counterexample to the transitivity of probabilistic confirma-tion.
In the following, we will present his argument, show how it can be extended to an argument against cumulative transitivity and then show why this argument is not suitable for the purpose of showing that propositional justification in general should not be considered to be transitive or cumulatively transitive.
3.3.1
Black’s argument
The first premise of Black’s argument is that doxastic justification can be analysed in terms of propositional justification. Black works with the following definition of the binary relation of epistemic justification R (Black 1988, p. 428):
Definition 1 (Doxastic justification according to Black). A belief B1 is a reason
for belief B2 if and only if there exist a person N and propositions ϕ1 and ϕ2 such
that
(i) B1 is N ’s belief that-ϕ1,
(ii) B2 is N ’s belief that-ϕ2,
(iii) ϕ1 confirms ϕ2 and
(iv) B2 is based on B1.
Note first that ‘confirmation’ and ‘is a reason for’ are but terminological vari-ants for ‘propositional justification’ and ‘doxastic justification’. In fact this way of characterising doxastic justification is very much in line with the standard view on the relation between doxastic and propositional justification mentioned in section 2.1. This means that his counterexample for what he calls probabilistic confirma-tion can be read as a counterexample to the transitivity of probabilistic proposi-tional justification.
Black defines probabilistic confirmation as holding whenever the conditional probability of the target proposition is higher than 0.5 given the justifying propo-sition. With this in place, he presents a simple counterexample by adequately populating three predicates with individuals (see figure 3.1)
Figure 3.1 Black’s counterexample to transitivity of probabilistic confirmation (Black 1988, p. 432).
1. The probability of an individual being G, given that it is F is 0.8, and hence F (x) probabilistically confirms G(x) for any x.
2. The probability of an individual being H, given that it is G is 0.66, and hence G(x) probabilistically confirms H(x) for any x.
3. The probability of an individual being H, given that it is F is 0.0, and hence F (x) does not probabilistically confirm H(x).
Note first that this sort of counterexample is not dependent on the threshold being 0.5. We can build an analogous counterexample for any threshold r with 0 < r < 1 by populating the predicates F, G, H in the following way (|F | denotes the cardinality of F for any set F ):
|F ∩ G| ≥ r · |F |; |G ∩ H| ≥ r · |G|; |F ∩ H| < r · |F |.
Note also that this type of counterexample can be turned into a counterexample for Cut. For simplicity, let r = 0.99. Figure 3.2 then depicts a counterexample `a la Black to Cut.
3.3.2
Impact of the argument
Black shows at least one thing: A simple analysis of propositional justification in terms of conditional probability is incompatible with the claim that propositional justification is transitive or cumulatively transitive. This is not enough to establish that the intuitive concept of inferential justification is not transitive, as he leaves two options for a theorist who wants to maintain that propositional justification does satisfy Tr and/or Cut. The first option is to abandon probabilistic justification
3.4. NAIR AGAINST CUT 21 10 0 10 z 1000 x y F G H
Figure 3.2 A counterexample to cumulative transitivity for probabilistic confir-mation with threshold r = 0.99:
1. F (x) G(x), because the probability of G given F is 0.99.
2. F (x) ∧ G(x) H(x), because the probability of H given F ∧ G is 0.99. 3. F (x) 6 H(x), because the probability of H given F is 0.98.
altogether. The second option is to give a more complex analysis of justification in terms of probability, perhaps using auxiliary conditions in such a way that the principles remain valid.
For an example of the second approach with respect to transitivity, see (Shogenji 2003; Roche 2012; Roche 2017). Martin Smith (2010; 2018) opts for the first op-tion. He argues on the one hand that even very high probabilities do not grant justification to believe a proposition. On the other hand, he argues in favour of Cut and uses this conclusion to argue in favour of a quantitative approach to jus-tification. Convincing counterexamples should accordingly fit especially well with quantitative approaches to justification.
3.4
Nair against Cut
Nair (2019) uses, like Black, a bottom up approach in his argumentation. He presents a qualitative theory of justification that should be consider at least to be plausible and wherein Cut may fail. A further difference that makes Nair’s argument worthwhile to consider is that he advocates for a qualitative approach to justification instead of a probabilistic account.
3.4.1
The argument
Nair sketches what he calls a minimal foundationalist theory of justification before arguing that Cut might fail therein. The theory has three components, two of them are epistemological and one is psychological in nature (cf. Nair 2019, p. 137):
1. Epistemic component 1 : The epistemological permissibility of non-foundational beliefs depends on the foundational beliefs of an agent: A non-foundational belief ϕ is permissible, iff
(i) It is good reasoning to conclude ϕ from A, (ii) it is permissible to believe A, and
(iii) A consists of all and only the foundationalist beliefs of the agent. 2. Epistemic component 2 : The permissibility of foundationalist beliefs is not
determined in this way.
3. Psychological component : An agent’s beliefs are structured in foundational beliefs and non-foundational beliefs. Non-foundational beliefs are those which are based on foundational beliefs.
The epistemic components express a typical foundationalist way of dealing with a potential infinite regress of reasons. They express that all belief are in need of a justifying set of beliefs, except for the foundational beliefs.The psychological com-ponent is intended to allow for the distinction between premises and conclusions. The premises of an agent S at a given time t are all her foundational beliefs.
According to his theory, good reasoning to a non-foundationalist belief always has all and only all foundational beliefs as premises.5 Nair uses the following notation for this kind of reasoning:
A ⇒St β1, . . . , βn.
In the formula above, A is the set of all and only all foundational beliefs of an agent S at time t, and β1, . . . , βnare all the non-foundational beliefs that the agent
can permissibly reason to at that time.
The next step in his Argument is to supplement the theory with a theory of learning. The idea here is that updating your foundational beliefs by adding a new foundational belief might impact the permissibility of some non-foundational
5This is expressed in condition (iii) of the firs epistemic component and excludes situations
where reasoning to a non-foundational belief from a subset B of A is defeated by a foundational belief in A \ B.
3.4. NAIR AGAINST CUT 23 beliefs. Let A be set of foundational beliefs of an agent S. Suppose further that α and β are the only permissible non-foundational beliefs for S, i.e. that
A ⇒St α, β.
If the agent updates her foundational beliefs to A0 = A ∪ {δ} (notation: A, δ refers to A ∪ {δ}), for example by learning δ through perception, this may have, amongst other, one of the two following consequences: The agent might be allowed to reason to a further non-foundational belief χ, i.e.
A, δ ⇒St0 α, β, χ.
or the agent may have to retract some non-foundational belief that was permissibly hold before the update, i.e.
A, δ ⇒St0 α and A, δ 6⇒St0 β.
These cases of learning relate in the following way to the question of whether good reasoning satisfies Cut according to Nair. The principle of Cut tells us that updating the set of foundational beliefs with a permissible non-foundational belief cannot increase the set of permissible non-foundational beliefs. Put in other words, updating the foundational beliefs with an already permissible non-foundational belief cannot result in a more powerful set of foundational beliefs according to Cut.
The next step in the argument is an argument for the claim that defeasi-bility conditions might refer to the organisation of beliefs in foundational and non-foundational beliefs and to the way of how a belief became a foundational belief.
Based on this Assumptions, Nair presents the following abstract counterexam-ple to Cut:
“Here’s the example: Suppose
(1) the beliefs corresponding to A defeasibly permit the belief that β and
(2) the beliefs corresponding to A together with the belief that β defeasibly permit the belief that χ.
(3) [. . . ] if we unpack [A ∧ β χ]’s ‘unless’-clause it says the beliefs corresponding to A together with the belief that β defeasibly per-mit the belief that χ unless the belief that β is solely based on the beliefs corresponding to A.”
According to this example, the following situation is possible: based on (1), we have
A ⇒t β.
If the agent adds any belief δ 6∈ A to her foundational beliefs, we have that A, δ ⇒t0 β, χ.
The Agent can reason to χ at t0 because β is no longer based solely on A. In other words, the update with δ defeats the unless-clause mentioned in (3). However, so Nair, adding δ is not the only way to defeat this unless-clause. By adding β itself to the foundational beliefs, β is no longer based solely on A, and hence we have a situation, where
A ⇒t β and A, β ⇒t0 β, χ.
This latter situation is according to Nair a situation where Cut fails: “Returning to the less rich notation that we used in the critical portion of this paper, the picture of what happens after adding β tells us that A, β χ. This is because if your foundational beliefs cor-responding to A are permissible and your foundational belief that β is permissible, then you would be permitted to believe that χ. The picture before adding β shows us that A β. And it shows us that A 6 χ because the claims about defeasible permissibility that we are considering say that the beliefs corresponding to A and the belief that β permit you to believe that χ unless the belief that β is based solely on the beliefs corresponding to A.
(Nair 2019, p. 142).
3.4.2
Impact of the argument
There are three things to say about Nair’s argument. The first is that his interpre-tation of Cut as claiming that a set of premises enriched by propositions justified by this set can never grow in inferential power is a subtle and important realisa-tion. In the response to the main argument for Cut, discussed in section 3.2, this idea is used to argue that the fallibility of propositional justification suggests that a set of premises enriched by propositions justified by it actually and normally does grow in justificatory power. This puts some pressure on a proponent of Cut. The second point has to do with the fact that the minimal foundationalist the-ory is part of his argument. Basing his argument on what he calls a “. . . credible albeit controversial theory in epistemology” (Nair 2019, p. 137) is sufficient for
3.4. NAIR AGAINST CUT 25 his purposes. For the aim of this text, it is not. When presenting our own argu-ments, we will therefore try to avoid assumptions about how concrete theories of justification could or should look like.
Thirdly, his counterexample is rather abstract. In which sense is it abstract? In the sense that neither the propositions being justified or the ones providing the justification have a specified first order content. The unless-clause moreover makes reference to the the status of the propositions as either foundational or not and to the overall justificatory structure. If we allow for this level of abstraction and for these types of references in the clauses and propositions, then some arguably simpler counterexamples are possible: Let β 6∈ A be any first-order belief that is justified by A. Let ϕ be the belief that β is amongst the foundational beliefs. In this case, we have A β and A ∧ β ϕ while A alone does not support ϕ. Lacking any kind of concrete content, examples such as the latter do look a bit like flat out denials of Cut. The aim for the next chapter is therefore to find more concrete candidates for counterexamples.
There is another possible issue with the counterexample by Nair that stems from the abstractness of it: While the content of the propositions involved is left open, the unless clause is rather specific. It seems however plausible to think that the unless clauses are not something that we can stipulate next to the propositions, but that these clauses are a consequence of the concrete propositions under consid-eration. If this is right, and the unless clauses are accordingly indeed consequences of the concrete propositions, then Nair presents the structure that a counterex-ample could have, while letting it open whether there actually are propositions that work in this manner when slotted in the relations of justification. Given the modality in his claim, this does not mean that his argument does not work, but it does again point to the usefulness of working with concrete first order propositions. In the next chapter, the goal is accordingly to do the more or less opposite of Nair. Instead of submitting an abstract counterexample for a concrete theory, the aim is to present concrete counterexamples based on an abstract notion of propositional justification.
Chapter 4
New Arguments against the
prin-ciples
In this chapter, two arguments against the validity of Tr and Cut are presented. The first argument in section 4.1 takes Smith’s (2018) argument against Tr as a starting point. I then argue that the assumptions that he uses can be used in an argument that questions Cut’s status as clear and obvious.
In the section 4.2 an argument is presented that Cut and Tr are not valid for any notions of justification that satisfy the principle of tolerance. It is then explored whether similar arguments can be made against the principles without relying on the principle of tolerance.
4.1
Cut and epistemic modifiers
In this section, Smith’s (2018) counterexample to Tr will be discussed. We will then explore whether the assumptions he makes can be used to produce counterexamples to Cut.
4.1.1
Smith against Tr
Smith’s counterexample to Tr is the following.1
Let ϕ be the proposition that the wall is white and illuminated by tricky red light, ψ be the proposition that the wall appears to be red and χ be the proposition that the wall is red. Plausibly, the proposition that the wall is white and illuminated by tricky red light provides
1This sort of examples are often encountered in the literature on bootstrapping and easy
knowledge. See e.g. (Cohen 2002).
justification for believing that the wall appears to be red, and the proposition that the wall appears to be red provides justification for believing that the wall is red. But the proposition that the wall is white and illuminated by tricky red light does not provide justification for believing that the wall is red. (Smith 2018, p. 3864). It is built up with the following propositions.
ϕ: The wall is white and bathed in red light emanating from an invisible source. ψ: The wall visually appears red.
χ: The wall is red.
Note that the counterexample has the same structure as the counterexamples generated by the recipe presented in the previous section. The conclusion ϕ χ is wrong, as before, because ϕ and χ contradict each. But how do we get from one sentence to a contradictory sentence in just two steps, each of which is claimed to be epistemically justified? A possible answer is that the trick is to change the domain about which the claims are made. ϕ and χ both speak about what is the case. They make claims about the realm of being so to speak. The intermediary proposition ψ however is situated in the realm of appearances. This gives us the following picture (see figure 4.1).
Realm of appearances
Realm of being
ϕ: White wall and red light. ψ: Red appearance.
χ: Red Wall.
Figure 4.1
While this picture might explain some features of the counterexample, there is another way of describing the mechanism in this counterexample. It is the following. If we consider the proposition ϕ then we see that ϕ really is a nothing but a thinly disguised version of the conjunction ψ ∧ ¬χ. As such it contains a
4.1. CUT AND EPISTEMIC MODIFIERS 29 source of justification (ψ) and a condition that defeats the justification of what the justifier normally justifies (¬χ). The same mechanism drives the counterexamples generated by the recipe mentioned above.
4.1.2
Variations in perception and confidence
In this section, I will argue that the assumptions that Smith uses in his argument against Tr lead to an incompatibility of Cut with arguably natural and plausible assumptions about epistemic justification. These ideas have to do with justified propositions that contain epistemic modifiers such as ‘might’, ‘likely’, or ‘certainly’. In order to show the incompatibility of these ideas with Cut, I will use Smith’s assumption that appearances justify the corresponding factual claims in the first version of the argument and his assumption that evidence should be considered as evidently given in the second version of the argument. The former assumption can be formulated as the following principle:
APP: Sensory appearances justify the corresponding factual propo-sition,
The general idea of this section is to explore alterations of the principle APP. The variations of this principle will be the result of slightly alter the forms of perception in the antecedent and of adding different epistemic modifiers to the consequent.
A simple example of an altered form of perception in the antecedent is the following. Sometimes we take the perception of a reliable but defeasible sign as sufficiently justifying the belief that something is actually the case. Suppose you walk in an office and see the case of a computer on the table. From this perception you reason that there is a CPU inside of it. Although there was no direct perception of the CPU, I take your belief to be justified. The reason is that the case of a computer in such a context is a very reliable sign that there is a CPU inside. The sign is however fallible, because it is possible that the CPU was removed in order to replace it with a new one. This type of reasoning is widespread and arguably acceptable from an epistemic point of view. This is expressed in the following variation of APP:
APPsign: Sensory appearances/perceptions of reliable but fallible
signs of p justify the corresponding factual proposition that p.
The perception part can not only differ according to what we perceive, but also according to the circumstances and conditions in which we perceive something. The conditions might be very close to ideal, as in a laboratory, or very suspicious, for example in a thick fog or under heavy rain. By altering the conditions in which we perceive something, we get plenty of variants of APP. Some will be valid, others
not. Suppose for simplicity that there are five kinds of conditions: ideal, good, normal, slightly below par and very suspicious. Then we have the following five variants of APP:
APP++: Perceptions of p in ideal conditions justify the
correspond-ing factual proposition that p.
APP+: Perceptions of p in good conditions justify the corresponding
factual proposition that p.
APP: Perceptions of p in normal conditions justify the correspond-ing factual proposition that p.
APP−: Perceptions of p in slightly below par conditions justify the
corresponding factual proposition that p.
APP−−: Perceptions of p in very suspicious conditions justify the
corresponding factual proposition that p.
First a short comment on the structure of the evidence. The idea here is that the conditions under which one perceives something are transparent to the person doing the perception. Accordingly, the antecedent can be thought of as having the form of a conjunction, where one conjunct contains the perception and the other the conditions under which this perception has taken place. The transparency of the conditions should exclude skeptical and similar scenarios in which the conditions might seem good, but are in fact completely misleading. Understood in this way, the acceptability of such principles depends on how robust we take the link between perception and reality to be. Although nothing depends on it, I take the principles of somewhere around APP and APP−and upwards to be
valid. The principle APP−− is not valid. Note however that we get a much more
plausible principle if we add an appropriate epistemic modifier to the proposition p in the consequent:
APP−−−−: Perceptions of p in very suspicious conditions justify the
corresponding proposition ‘It might be the case that p’.
If the evidence is not up to par to justify the proposition, we can (almost) always use an epistemic modifier to get a justified proposition. It is however important to point out that perceptions can be trumped by perceptions under better conditions. If my evidence contains a suspicious perception of p and a perception under near ideal conditions of ¬p then I am not justified to believe ‘It might be the case that p’. The assumption that the antecedent contains all available information relevant
4.1. CUT AND EPISTEMIC MODIFIERS 31 to the question of whether p is, is not, or might be the case should get rid of these worries. In other words, to claim that such a principle is valid means to claim that if all information relevant to the consequent is at least compatible with the mentioned perception, then the consequent is justified.
To have the skills to know which is the strongest epistemic modifier to use for a proposition to still be justified by the evidence on ha sis an important aspect of a proficient epistemic agent.
Adding an epistemic modifier can also make a good principle invalid. Suppose the principle APP− is valid. By adding an epistemic intensifier to the consequent
of that principle, we get the following principle:
APP++− : Perceptions of p in slightly below par conditions justify the corresponding proposition ‘It is absolutely certain that p’.
This principle seems more questionable than APP−, if not just false. If
per-ception can justify the target proposition of this new principle at all, it must be perception under better conditions than the ones given in the principle.
Consider the target proposition to be the unmodified proposition p, as in the original principle APP proposed by Smith. Consider now all the valid variations of APP with this unmodified proposition as its target proposition. Let perceptionmin
be the weakest kind of perception mentioned by these principles. It might be some form of indirect perception as in APPsign or perception under certain just sufficient
conditions. The corresponding principle is APPmin:
APPmin: Perceptionsminof p justify the corresponding factual
propo-sition that p.
Having this in place, two arguments are presented, both aiming at raising doubt about whether Cut should be considered a valid principle. The general structure of the arguments is similar in that both suggest that Cut is incompatible with a general idea about justification that is at least debatable, if not plausible. The first argument is based on a correspondence between propositional justification and justifiably updating a system of beliefs. The second argument is based on a theory according to which both the antecedent and the consequent play a role in determining which exceptions to the justificatory relation are relevant and which aren’t.
4.1.3
Argument 1
The first argument is based on the assumption that there is the following cor-respondence between propositional justification and justified updates of a belief
system.2 Given a belief system such that ϕ
0, . . . , ϕn are the contents of all beliefs
in the system relevant for ψ, the following holds.
ϕ0 ∧ · · · ∧ ϕn ψ corresponds to ‘given all beliefs in the system
that are relevant for ψ , the agent would be epistemically justified to add the belief that ψ to the system of beliefs’.
While trying to remain neutral on what exactly puts this correspondence in place - whether doxastic justification is for example grounded by propositional justification or vice versa - this assumption is made in order to be able to switch perspectives in the text that follows.
As an illustration, take the principle APPsign. According to this principle, a
sensory perception of a reliable sign of p justifies the corresponding factual propo-sition that p. Using the notation for propopropo-sitional justification, this is expressed as
Perceptionsign(p) p.
Transferring this to the belief system, the belief that one has perceived a reliable sign of p (notation: B(Perceptionsign(p) p)), will be the belief in the system relevant for p, and the belief that p (B(p)) will be the belief with which the belief system is updated. Typically this proposition p will be some proposition stating something about the world. Sometimes however this proposition will be about our beliefs, say B(q). Updating a system with the corresponding belief would result in adding a higher-order belief, namely B(B(q)). This higher-order belief makes a claim about the system, namely that B(q) is in there. Accordingly, whether one is epistemically justified to add this very much depends on whether B(q) is part of the system or not. If B(q) is already in the system then one is justified to add this. Moreover, because B(B(q)) makes a claim about B(q) and B(q) alone, it does not matter which other beliefs are in the system in order for the update to be justified. If on the other hand B(q) is not part of the system, then things are different. It is hard to justify the addition of a belief that one has a belief that q while actually not believing q. Translating this to propositional justification, the proposition is that ϕ B(ϕ) normally holds.
If the above is right, then the following should also hold. Perceptionsign(p) ∧ p B(p).
The proposition p is amongst the evidence for B(p) and, as argued above, the other piece of evidence does not act as a defeater.
2For the present purpose it suffices to say that a belief system can be understood as a set of
beliefs, typically containing all beliefs that an agent has at some time, and updating this system with a belief simply means adding this belief to the set.
