1. Introduction
“Epistemic safety”, which refers to an epistemic status in which the subject acquires true beliefs without involving epistemic luck, is routinely defined in terms of modality, given that the commonsensical paraphrase of “being safely true”, namely, “being not easy to be false”, has presupposed modality. I will not challenge this tradition of conceiving safety if the term “modality” is only construed in the preceding non-technical sense. However, there is a further and more technical tradition of cashing out safety-defining modality in terms of “possible world semantics” (represented by Duncan Pritchard Reference Pritchard2007, Reference Pritchard, Hendricks and Pritchard2008, Reference Pritchard2009a, Reference Pritchard2009b), and even Julien Dutant's (Reference Dutant2010) and Martin Smith's (Reference Smith2016) normalcy-based notions of safety have also taken possible world semantics as a significant component of them. However, such an approach has to largely depend on epistemologists’ ad hoc intuitions on how to individuate possible worlds and how to pick out “nearby” worlds without providing a further and unified clarification of the nature of the so-called “inter-world-distance”. My solution to this problem is to reconstruct safety-related modality in terms of the probability attribution from the ascriber's perspective without introducing any element from possible world semantics. It is noteworthy that this move would not result in the collapse of modality (which is often expressed by counterfactuals) into probability (which is not obviously related to counterfactuals), since the probability-modality boundary can still survive as the boundary between subject-based probability and the ascriber-based probability. They are viewed as two types of probability respectively, since some information accessible to the ascriber may be inaccessible to the subject, and the information gap between the two will further result in different probability values. This treatment shows minimal respect to the intuition that “counterfactual closeness goes with objective risk and chance” (Dutant Reference Dutant2010: 15), if the phrase “counterfactual closeness” can be reconstructed probabilistically and the phrase “more objective” can be consequentially cashed out in terms of ascribers’ putative omniscience of the information in the epistemic scenarios.
First of all, I need to show why the mainstream treatment of safety in terms of possible world semantics is problematic.
2. Why is the mainstream approach to safety problematic?
Pritchard's (Reference Pritchard2007, Reference Pritchard, Hendricks and Pritchard2008, Reference Pritchard2009a, Reference Pritchard2009b) definition of safety (PDS), which is the exemplary possible-world-semantics-based definition of safety, is as follows:
PDS: S's belief is safe if and only if in most nearby possible worlds in which S continues to form her belief about the target proposition in the same way as in the actual world, and in all very close nearby possible worlds in which S continues to form her belief about the target proposition in the same way as in the actual world, the belief continues to be true.
According to John Greco (Reference Greco2007), PDS can be simplified as “Weak Safety”, in contrast to “Strong Safety” as another candidate definition of safety in terms of possible world semantics:
Weak Safety: In close worlds, usually if S believes p then p is true. Alternatively, in close worlds, almost never does S believe p and p is false.
Strong Safety: In close worlds, always if S believes p then p is true. Alternatively, in close worlds never does S believe p and p is false.
But Greco, as a critic of the Pritchardian notion of safety, claims that neither conception of safety is acceptable because neither can handle both the Lottery Case and the Typing Monkey Case in a non-ad hoc manner. (The Lottery Case is the case in which a ticket holder does not know that she will lose the lottery, even if the odds of her winning are very small; The Typing Monkey Case is the case in which we do know that a monkey will not type out a copy of War and Peace by randomly banging away at the keyboard of a laptop.) Here is my reconstruction of his argument:
1. In the Lottery Case, we need to explain why the subject does not know that the bought ticket won't be a winner, whereas in the Typing Monkey Case, we need to explain why the subject knows that the monkey won't type out a perfect copy of War and Peace.
2. If we take the weak safety condition (or PDS) as the explanans, we would say that the target belief in the Typing Monkey Case satisfies this condition, since in close words, if the subject believes that the monkey won't type out a perfect copy of War and Peace, then it is a matter of fact that the monkey won't type out a perfect copy of War and Peace in these worlds. However, due to the structural similarity between this case and the Lottery Case, then the target belief in the Lottery Case should be weakly safe too.
3. But the putative weak safety of the target belief in the Lottery Case is more supportive of the subject's knowing, at least weakly, that the bought ticket won't be a winner, whereas what we need to explain is why she does not know this. Hence, step (2) conflicts (1), and hence, the weak safety condition cannot be a winner in a cross-board manner.
4. Now we turn to the strong safety condition as our backup explanans. Thereby we have to say that the target belief in the Lottery Case cannot satisfy this condition, otherwise one would absurdly assert that in close worlds, if the subject always believes that the bought ticket won't be a winner, then it is always true that the bought ticket won't be a winner. (This assertion precludes surprises, but there are surprises in lotteries.) Due to the structural similarity between the Lottery Case and the Typing Monkey Case, the target belief in the Typing Monkey Case cannot be strongly safe either.
5. But the lack of the putative strong safety of the target belief in the Typing Monkey Case entails that we do not know that the monkey won't type out a perfect copy of War and Peace, whereas what we need to explain is why we do know this. Hence, step (4) conflicts with (1), and hence, the strong safety condition cannot be a winner in a cross-board manner as well.
6. Given that neither the strong nor the weak version of safety can consistently handle the Typing Monkey Case and the Lottery Case in a non-ad hoc manner, Pritchard's safety theory cannot be smoothly applied to all epistemic vignettes without making the notion of safety “stronger” or “weaker” in an ad hoc manner.
I am quite sympathetic to Greco's diagnosis, and my own position is even more radical than his. My criticism is: PDS is simply circular. This is my analysis:
First of all, any attempt to satisfy PDS (or even a strong safety condition) cannot avoid involving a three-step procedure as follows:
(1) To individuate possible worlds as discrete units.
(2) To order them according to their “distances” to the actual world.
(3) To discriminate nearby/very nearby possible worlds in which p is still true from those in which p is not true.
However, to execute (1) is not only appealing to some ad hoc-ness related to specific vignettes but also circularly presupposing (2). But why? Just to take a simplified version of the Fake Barn Case (Goldman Reference Goldman1976) as an example: if there are 6 barn-alike objects evenly dispersed around the subject, then the subject would see the second object when she turns her head from the firstly seen barn-alike object to her left/right side at an angle of 60°. Hence, it may be expected that a different possible world will be created whenever she turns her head to her left/right side at an angle of 60° from then on. As a result, there should be 5 possible worlds presented in the vignette (excerpt the actual world in which she luckily perceives the real barn). But the question is: why the scenario-builder cannot create another possible world simply by supposing that the subject turns her head to her left/right side at an angle of, say, 20°, or even 10°? Surely there is no metaphysical reason for preventing one from creating more and more possible worlds in this way. And the only reason to pick out only 5 possible worlds among all these candidates seems to be that only these 5 worlds permit the subject to form the same barn-relevant belief in the same way as she does in the actual world. Or alternately, only these 5 worlds are epistemologically relevant to the purpose of creating the whole Fake Barn Case. However, since “to be epistemologically relevant to the actual world” seems to be synonymous to “to be (epistemologically) close enough to the actual world”, to execute step (1) has assumed that step (2) has been executed, and this patently results in circularity.
In addition, Timothy Williamson's (Reference Williamson2000) conception of safety, in which metaphorical terms like “close” also play a role, is not less troublesome than Pritchard's theory. This is his conception of the relationship between “knowledge” and “safety from error”:
The Connection between Knowledge and Safety: For all cases α and β, if β is close to α and in α one knows that C obtains (wherein “C” refers to a condition which itself can obtain without obtaining safety), then in β one does not falsely believe that C obtains, or in other words, C is safe from error in β (Williamson Reference Williamson2000: 128, my paraphrase).
But how to further define the term like “close” in Williamson's theory? One of his suggestions is to cash it out in terms of “similarity”. However, the assessment of the similarity between two items cannot be executed without a pre-fixation of a parameter (e.g., the v-parameter) for doing the comparison. Hence, if “|v(α)v(β)|< c” means that “the gap/dissimilarity between the v-value in case α and the same value in case β is smaller than the threshold-value c”, then Williamson's notion of epistemic safety will be unpacked as follows (Williamson Reference Williamson2000: 129, wherein the italicized part is my own supplementary account):
The Condition of Epistemic Safety: For all cases α and β, if |v(α)v(β)|< c and in α one is in a position to know that C obtains, then C will obtain in β, or in other words, epistemologically speaking, C will be with epistemic safety in β in addition to in α.
Williamson's own illustration can be provided here (cf. Williamson Reference Williamson2000: 127). Suppose that the v-parameter is about the height of the tree. Accordingly, if the gap between the height of the tree in case α and the height of tree in case β is smaller than the threshold-value c, then one believing that the tree is at most 50 feet high (which is a numerical determination insofar as height is concerned) in α will also believe that the tree is at most 50 feet high in β. Hence, according to the preceding condition, a subject's belief that the tree is at most 50 feet high is safe in both α and β.
Nonetheless, quantitatively measurable parameters like the height of trees are not typical to see in most epistemic vignettes. For instance, I cannot see how to find such a parameter in the Fake Barn Case. (Is it the size of the barn-alike objects or the physical distance between the subject and the objects? Nothing seems to be helpful here.) Moreover, the discreteness between cases α and β has assumed the possibility of individuating two possible worlds containing α and β respectively, but Williamson, similar to Pritchard, still owes us a standard for individuating them in different vignettes.
In my view, even Martin Smith's (Reference Smith2016) normalcy-based notion of safety appears to be assuming possible world semantics. It is noteworthy that his primary goal is not to account for safety but to account for justification in the light of “normic support”, according to which “a body of evidence E normically supports a proposition P just in case the circumstance in which E is true and P is false requires more explanation than the circumstance in which E and P are both true” (Smith Reference Smith2016: 40). Though this definition does not appear to be patently possible-world-semantics-involving, Smith's reworking of it is definitely in terms of “variably restricted quantification over possible worlds”:
The Condition of Normic Support (S1): A body of evidence E normically supports P just in case there is a world at which P is true in all the most normal worlds in which E is true (Smith Reference Smith2016: 42).
But the question is: how to distinguish normal worlds from abnormal ones in this theory? This is his construct:
The Consequence of Normic Support (S2): If evidence E normically supports a proposition P, then the most normal E-worlds are P-worlds (Smith Reference Smith2016: 44).
Nonetheless, if we combine the connotations of S1 and S2 into one sentence, then we will have the following one:
The Integration of S1& S2 (S3): A body of evidence E normically supports P just in case there is a world at which P is true in all the P&E-permitting worlds in which E is also true.
But the italicized part of S3 only encodes a tautology, and no meaningful definition of anything can take a tautology as its precedent. Hence, there is no way to convert such a definition of normic support into a workable model to predict the behaviors of justification-ascription.
Given that Smith's notion of safety is based on the preceding problematic notion of normic support, his following notion of safety has to be problematic as well:
Smith's Safety: For E to be safe evidence for P, E and P must dovetail throughout a certain set of significant possible worlds (Smith Reference Smith2016: 105).
And the expression “a certain set of significant possible worlds” here is further defined in terms of their similarity to the actual world. However, how to fix the similarity-issue in any world, in Smith's theory, is up to whether E and P dovetail in the same world. Hence, his definition of safety is as uninformative as his definition of normic support. Therefore, Smith's normalcy-based approach is incapable of eluding problems which have already been posed to Pritchard's approach.
It is noteworthy that Julien Dutant's (Reference Dutant2010) normalcy-based approach is also intended to refine the Pritchardian counterfactual closeness-based narrative rather than provide a systematic replacement of it. Hence, problems left for Pritchard would be mostly left for him as well. But it is also noteworthy that when he is addressing the problem of ordering possible worlds, he introduces the notion of “normalized safety”, according to which the modal distance between the actual world and a counterfactual world is dependent on how normalized the counterfactual situation is when we view it from the perspective of this world. This move has already assumed an ascriber's perspective (especially her background knowledge on what is normal), and hence somehow predicts some aspects of my theory. Nonetheless, with contrast to Dutant's approach, mine will not assume possible world semantics from the bottom to the top for the sake of integrity.
3. A new definition of safety
This is my alternative definition of safety (DS):
DS: For any ascriber S0 who ascribes safety/knowledge to any other subject S in a context C (composed also of the vignette information on S, which is accessible to S0), S holds a belief p safely iff according to S0's evaluation, which is based on her C-triggered background knowledge (hereafter “Kso”), the following conditions could be satisfied:
(i) p is taken as true by S0, or Pr(p|Kso) = 1;
(ii) S0 takes it to be true that the subject in the vignette holds p, or the value of Pr(S-holds-p|Kso) = 1;
(iii) the probability of the occurrence of the truth-maker (hereafter “Tm”) of p is taken to be high enough by S0, or in other words, the value of Pr(Tm-of-p|Kso) is high enough. (Here ‘to be high enough’ means to be above a ‘safety-threshold’, which is pre-fixed in S0's mental architecture and cross-contextually invariant.)
Now I will immediately clarify DS, and then explain why it is epistemologically interesting.
Firstly, it is noteworthy that when the epistemic vignette is presented to the ascriber in the form of a thought experiment, the satisfaction of both (i) and (ii) will be granted, given that most vignette-creators will directly tell the ascribers that the subject has the target belief, which is assumed to be true (especially in Gettierized cases). Due to this reason, from now on I will simply ignore the first two conditions when DS is being applied.
Secondly, I want to explain why the representation of S0 is irreducible in DS. The main reason is: “to know” contains elements that the subjects within the epistemic vignettes may be inaccessible to, such as elements encoding information on whether her belief is true of reality or luck-precluding. Since these elements should be all put on the table for knowledge ascription, it looks fairly natural to ask a question like “On whose table are they expected to be presented?”, and the only sensible answer to this question is: On the table of knowledge/safety ascribers working on a higher level. Hence, to analyze how the story on S is presented is tantamount to analyzing how it is represented from the eyes of S0.
Thirdly, I want to explain how a systematic replacement of the possible-world-based narrative by virtue of my probability-based narrative can be warranted. Prima facie, as some critics may contend, it is entirely misleading to confuse closeness of worlds with probabilities, since probability looks like an internal element of justifying beliefs, while safety, which is based on the inter-world distances, is a matter of stability under perturbations of conditions external to the belief-processing procedure. But this accusation can work only when the “external” factors cannot be internalized on a higher level, whereas it simply can be internalized in my narrative. To take the Fake Barn Case again: although the fact that most barn-alike objects surrounding the subject are fake is external to S, it is fairly internal to S0, who is on a higher level, and S0 can accordingly attribute a probability-value to the Pr(Tm-of-p|Kso)-value from her perspective. This maneuver will not result in a confusion of an ascriber's ascribing process with the justifying process of p on the subject's level, as on this lower level, the justifying process will actually involve another proposition, say, Pr(Tm-of-p|Ks).
Fourthly, I want to explain why I need to mention “truth-maker” in DS. Throughout this paper, I will be satisfied with generally defining “the truth-maker of p” as “anything other than p which makes p true”. This term is introduced for showing my minimal sympathy for epistemic externalism in the sense that I concede that the issue on safety cannot be neatly set merely by virtue of elements accessible to the subjects within the epistemic vignette. Accordingly, safety has to be viewed as some conceptual device measuring the “tightness” of the connection between at least two variables encoding external entities (although they are still both internal to the ascribers): the first is the truth of the target belief (since it is meaningless to attribute safety to a false belief), and the second is the truth-maker of that belief (since a true belief cannot be “safe” if its truth-maker is not “in its vicinity”). More technically, we may formalize the “tightness” between the two variables in terms of the joint probability of the belief & truth-maker pair. Thus, we may measure safety in terms of Pr(p&Tm-of-p). Since Pr(p&Tm-of-p) = Pr(p|Tm-of-p)×Pr(Tm-of-p), and Pr(p|Tm-of-p) =1 (otherwise Tm-of-p could not be the truth-maker of p), we can assess the degree of safety simply in terms of Pr(Tm-of-p). And given that a body of background knowledge is supposed to be accompanying all forms of probability attribution, to assess the value of Pr(Tm-of-p) is tantamount to assessing the Pr(Tm-of-p|Kso)-value, as DS proposes.
But some may still ask: how to make the formulation of a truth-maker of a belief substantially distinguishable from that of the belief itself, given the widely accepted idea that the truth-maker of the belief that “The cat is on the mat” is nothing but the fact that this cat is actually on this mat?
My response: given that most representations bear less information than represented entities (as for instance, if the word “sexy” is used to represent the sexy appearance of somebody, the word itself definitely carries much less information than the vivid appearance of the person in question), we may immediately conclude that a truth-maker of a belief routinely carries more information than the corresponding belief. And this asymmetry between the belief and its truth-maker could be fairly useful for analyzing Gettier's (Reference Gettier1963) case on the “Ford owner”, as we will see in section 5.
But now there is still one DS-related problem remaining: what sort of “probability” do I mean, given that there are many interpretations of “probability” in the literature of philosophy of science? To answer this question, I need to firstly introduce a bipartite taxonomy of background knowledge of ascribers.
4. The bipartite taxonomy of ascribers’ background knowledge
Since empirical truth and formal truth (including mathematical and logical truth) are mutually exclusive and jointly exhaustive, we may accordingly divide background knowledge into “causal background knowledge” (Kso-c) and “formal background knowledge” (Kso-f): the former corresponds to empirical truth, while the latter corresponds to formal truth. To be more specific, Kso-c refers to the background knowledge in which beliefs are arranged in accordance with an existing map of causal laws relevant to the vignette being assessed by the ascriber. Hence, when Kso-c is the salient background knowledge that S0 is appealing to, S holds p safely iff the value of Pr(Tm-of-p|Kso-c) is fairly high. By contrast, “formal background knowledge” (Kso-f) refers to the ascriber-perceivable formal features of the epistemic vignette, hence, when Kso-f is the salient background knowledge that S0 is appealing to, according to DS, S holds p safely iff the Pr(Tm-of-p|Kso-f)-value is fairly high.
A case of throwing a die in the gambling game may help in clarifying the foregoing bipartite taxonomy. Such a game is either fairly played or unfairly played. If it is really fairly played, then the hexahedron-shaped die itself has to be homogeneous, and the gambler's hand performing the throw also has to move arbitrarily each time, etc. Hence, there would be no biased causal link between the material quality of the die and the result of the gambling, and hence, there would be no informative causal story available to the ascriber to inform her about which facet of the die would be selected. So the ascriber has no choice but to turn to her Kso-f, namely, her background knowledge about how many possible results of a throw are formally involved, and this in turn hinges on how many facets a die has.
However, when such a game is unfairly played in the sense that, for instance, the material quality of the die is heterogeneous so that the density of the material near one specific facet is much higher than that near all other facets, then the ascriber's Kso-c, especially his storied knowledge about gravity, will induce her to ascribe safety to beliefs reporting the results favoring certain facets. Though the formal features of the game are still there now, their role in the process of predicting the results of gambling is now overridden by that of the information about how cheating is empirically involved.
But how could an ascriber know which type of background knowledge has to be triggered in a specific vignette? This is the formulation of a proposed two-step heuristic:
(1) The ascriber receives the information from the vignette (which constitutes the main part of the “context” for epistemic evaluations), and thereby searches for contextual factors which may be the illustrations of her stored causal laws in Kso-c, and if these factors are found, the ascriber will ultimately send out a Pr(Tm-of-p|Kso-c)-value. Otherwise she will go to execute step (2). And if the attributed value is over a pre-fixed “safety-threshold”, she will ascribe safety to the target belief. Otherwise the belief is still unsafe.
(2) The ascriber searches for contextual factors which may be triggering the formal features of the target belief in Kso-f, and if these factors are found, the ascriber will ultimately send out a Pr(Tm-of-p|Kso-f)-value. And if the attributed value is over a pre-fixed “safety-threshold”, she will ascribe safety to the target belief. Otherwise either the belief will be judged as unsafe or the degree of its safety will be judged as undecidable.
As to why the sequence for the ascriber to pick up candidate perspectives is “Kso-c→ Kso-f”, the reason is: when an epistemic vignette is presented, it is natural for the ascriber to firstly search for the most informative background to evaluate the target belief (since the more information is at hand, the more efficiently the job of safety-evaluation could be done), and the sequence from Kso-c to Kso-f is exactly the sequence from the more informative background to the less informative one. This is somehow analogical to the contrast between a smart-heuristic-driven artificial chess-player and an artificial player merely with the capacity of exhaustively enumerating and analyzing all possible further games following the preceding one: other things being equal, the first player can definitely work more efficiently than the second, since experience encapsulated in the heuristics already provides shortcuts to the desired solutions. Hence, unless such heuristics are not at hand, there is no reason not to try to apply them first.Footnote 1
Now is the right time to explicate which type of “probability” is involved in either the evaluation of “Pr(Tm-of-p|Kso-c)” or “Pr(Tm-of-p|Kso-f)”.
Insofar as the Pr(Tm-of-p|Kso-f)-value is concerned, it is obvious that the “probability” on this level has to be interpreted in a classical manner, according to which when there is no evidence with enough weight to favor any outcome of a preceding event, the probability of any type of outcome is simply the fraction of the total number of possible outcomes in which the target outcome occurs. It is noteworthy that the assessment of “the total number of possible outcomes” has to be based on ascribers’ own mental capacities and background knowledge, and hence bears the mark of “subjectivity”.
In contrast, the explication of the “probability” involved in the Pr(Tm-of-p|Kso-c)-value looks a bit more complicated. I tend to view it in terms of “subjective frequency probability”. But what is it?
First of all, it is based on the frequency interpretation of probability, according to which the probability of an attribute A in a finite reference class B is the relative frequency of actual occurrences of A within B. Hence, if there is an empirical law asserting that As are Bs with a certain frequency value, then she can easily derive a frequency value concerning whether this A is B.
However, the preceding notion of “frequency” simpliciter is still unsatisfactory for my purpose, unless it is further modified by the adjective “subjective”, which implies that the frequency-value-attribution is up to the ascriber's own background knowledge (rather than the priori history of the objective environment external to any ascriber). But why is this modification necessary?
The first reason: traditional frequentism, which is objectivity-oriented, cannot handle cases in which only one unrepeatable event (like “Brutus murdered Caesar”) occurs, while the target belief involved in safety-ascriptions may be encoding an unrepeatable event. My remedy is based on the ascriber's subjective intellectual capacities for assessing frequency in these cases. To be more specific, by doing analogical reasoning, the ascriber may find that this unrepeatable event, say, C, is analogical to A, which is repeatable, hence, the frequency concerning whether C is B can still be derived from the frequency of whether As are Bs. However, I concede that if the ascriber fails to mentally associate the target belief with any stored empirical law, she could hardly ascribe any safety-value to the belief. But this is not a defect of my theory yet, since in some occasions, we do intuitively feel it difficult to ascribe a certain safety-value, whether high or low, to some beliefs.
The second reason: as Alvin Goldman (Reference Goldman1988–1989) points out, the task of explaining “knowing” is basically an issue of how to provide a “plausible, parsimonious explanation of people's (linguistic) intuitions”, which are largely within the explanatory scope of psychology rather than a theory about the a priori history of the world external to the ascribers (Goldman Reference Goldman1988–1989: 199). His psychology-oriented notion of “knowing” can be easily extrapolated to that of “epistemic safety” due to the following observation: in some cases, the objective frequency is obviously irrelevant to safety-ascriptions, if their results are expected to be in accordance with most epistemologists’ intuitions. As for instance, even though real winners of the lottery game are extremely rare in the objective sense, an ascriber may underestimate this rareness by mentally retrieving some typical exemplars of the expression “winners of the lottery game” due to the impacts of certain psychological heuristics. Hence, a subjective-frequency-based treatment could be very suitable to handle the asymmetry between the Typing Monkey Case and the Lottery Case, as I will show in section 5.
5. Typical epistemic vignettes revisited
Now I will apply my theory to some typical epistemic vignettes to test its explanatory power.
Case 1: The Lottery Case
Suppose that Smith holds two tickets A&B in a fair lottery involving n participants (wherein “n” is a very large number), and comes to believe that (a) ticket A will be a winner; (b) ticket B will be a loser (and both A and B are randomly selected). Are these beliefs safe or not?
Belief (a) is intuitively unsafe and hence non-knowledge, and my theory can quickly explain why. A safety-ascriber, according to my theory, will firstly try to consider the truth-maker of the target belief in her Kso-c, a truth-maker like the event that “The ticket with the number of 388888 is selected”.Footnote 2 However, she finds no clue in the scenario to present a biased causal link between betting on a random ticket and the occurrence of this particular truth-maker. Hence, she has no choice but to consider the truth-maker in her Kso-f by shifting her attention to the formal features of the whole lottery game, and thereby evaluates the safety of the target belief in terms of Pr(Tm-of-p|Kso-f)-value, which is obviously as low as 1/n.Footnote 3
The preceding unsafety-revealing analysis can also quickly explain why belief (b) is safe, since (b) is nothing but the negative form of (a)-like beliefs. However, many may still feel reluctant to take (b), being safe notwithstanding, as knowledge before the result of the lottery is actually presented to the public. And reluctance of this type is related to a further fact that probabilistic reasoning here is now interfered by the so-called “availability heuristic” (Tversky and Kahneman Reference Tversky and Kahneman1973), according to which the evaluation of a belief psychologically hinges on mentally emerging instantiations related to this belief. Specifically, the process of attributing truth to a belief like “Ticket B will be a loser” is interfered by availability-heuristic-driven considerations like “My uncle Tom randomly bought a ticket last month, but it was a winner”. Hence, the strength of the assertion that “S knows (b)” is now undermined by Kso-c-based considerations of the preceding type, although it is also simultaneously strengthened by Kso-f-based considerations, which merely highlight the formal features of the game. Conceivably, the seesaw struggle between the two types of considerations will finally make the ascription of a high-safety value to the target belief difficult. However, the preceding form of hedging mechanism could be largely ignored when the non-safety/knowledge status is being attributed to belief (a), given that the stability of the safety/knowledge-status of a belief is much more sensitive to undermining factors of the same belief than its non-safety/knowledge counterpart, just like the purity of white is more sensitive to the “invasion” of motley colors than that of black.Footnote 4
Case 2: The Case on the Lottery-Results Reported by Newspaper
This case is taken from Pritchard's (Reference Pritchard2009a: 28–29) criticism of the probabilistic treatment of safety.Footnote 5 Suppose that Smith gets the information that his ticket is not a winner from the local newspaper. Now the question is: Is this belief safe?
This belief is intuitively safe, and my formulation will tell why. For the sake of illustration, we may formulate the truth-maker of the target belief as the event that “Ticket number 488888 was actually selected as a winner”, and additionally, the ticket that Smith bought is 38888, the number of which is not presented in today's newspaper. In this situation, a safety-ascriber will firstly try to consider the truth-maker of the target belief in her Kso-c. Since the belief is supposed to be acquired by the subject via testimony, the attribution of the Pr(Tm-of-p|Kso-c)-value is basically an issue for estimating the reliability of the relevant source of testimony. Suppose that the probability for the local newspaper to be immune to misprints is 0.999, then the ascriber will accordingly assign a high value to Pr(Tm-of-p|Kso-c) as well. Surely there is still a probability for the newspaper to misprint a winner as a loser and vice versa. However, this consideration will undermine the safety of the target belief only in a fairly small degree, and in most ordinary cases of practical reasoning, such undermining factors will be ignored. Thereby the ascriber will still feel that the target belief is safe. Surely when we calculate the probability of the occurrence of the relevant truth-maker by suspending the causal links related to the testimonial source, a resulted Pr(Tm-of-p|Kso-f)-value may turn out to be different. However, as my formulation predicts, an ascriber would not like to suspend these causal links since they do exist in the given scenario and they do trigger the ascriber's Kso-c-based considerations, which overrides her Kso-f-based considerations. Hence, it would be pointless to follow Pritchard (Reference Pritchard2009a: 28–29) by criticizing the probabilistic treatment of safety if my formulation has been assumed in advance.
Case 3: The Ford-Owner Case
This case, originally formulated by Gettier (Reference Gettier1963), should be divided into two phases:
Phase One: When the subject, say, Smith, has a justified true belief that he has only one Ford, that belief is made true by the truth-maker that he actually has a Ford with the manufacturing code of, say, FD007.
Phase Two: Suppose that the original Ford owned by Smith with the manufacturing code FD007 is now destroyed in an earthquake in California, and Smith, who is now giving a lecture in Kyoto, has no idea about what has happened to his Ford. However, at the same time, and still unbeknownst to Smith, his lottery ticket becomes a winner, and his prize is nothing but a new Ford with the manufacturing code of FD008. Hence, surprisingly, the original belief held by Smith, namely, “I have only one Ford”, is still true. In this phase, the truth-maker of this belief is the joint form of the two sub-events: a FD007-destroying event and a FD008-ownership-forming event, without any of which the target belief will be rendered untrue. Here comes the question: Is his belief that “I have only one Ford” safe now?
Intuitively speaking, it is unsafe, and my formulation will tell why. As I have mentioned, in the second phase of this scenario, the truth-maker of the target belief is the joint form of the two sub-events: a FD007-destroying event and a FD008-ownership-forming event. Prima facie, it looks natural to consider both truth-makers in Kso-c, since both the earthquake and the possession of properties are instantiating causal laws. But a more careful scrutiny of the case will lead the ascribers to pay more attention to the joint part of the two sub-events, since it is this joint part that substantially contributes to the truth of the target belief. And such observation will lead ascribers to calculate the probability for the temporal overlapping of the two sub-events, and such calculation may involve considerations of the following factors:
(i) The quantity of Fords of the same model available in the market (and FD007 and 008 are just two of them);
(ii) The odds for winning something in the game that Smith participated in;
(iii) The odds for winning a Ford rather than, say, a Toyota;
(iv) The odds for a California earthquake to destroy FD007;
(v) The odds for a FD007-destroying event and a FD008-ownership-forming event to occur simultaneously, etc.
Hence, it seems that Smith has been unconsciously involved in a series of lotteries, and the final odds for a winner surviving all games has to be estimated in terms of the conjunctive probability of being a winner in all single lotteries. Accordingly, the resulted Pr(Tm-of-p|Kso-f)-value should be very low, which will induce a fairly low safety-value.
Case 4: The Zebra Case
This case was originally formulated by Dretske (Reference Dretske1970). Suppose that Smith is watching a putative zebra, which may be a real zebra, or a cleverly painted mule (and this is not decidable by either Smith or the ascriber). And Smith forms a belief that “What I see is a zebra”. Is this belief safe?
In this case, there are two competing explananda for explaining the occurrence of the relevant truth-maker of the target belief, i.e.:
EA: This belief is induced by a real zebra;
EB: This belief is induced by a cleverly painted mule.
Hence, the ascriber has no choice but to attempt to attribute the value of Pr(Tm-of-p|Kso-c&EA) and Pr(Tm-of-p|Kso-c&EB), respectively. And by relying on Bayes’ theorem, the ascriber may calculate them in the following way:Footnote 6
(i) Pr(Tm-of-p|Kso-c& EA) = Pr(Tm-of-p)×Pr(Kso-c&EA|Tm-of-p) /Pr(Kso-c& EA)
(ii) Pr(Tm-of-p|Kso-c& EB) = Pr(Tm-of-p)×Pr(Kso-c& EB|Tm-of-p) /Pr(Kso-c& EB)
Insofar as (i) is concerned, it is fairly acceptable to attribute “0.5” to the Pr(Tm-of-p)-value, since it is indeterminate whether there is a real zebra in the vignette. As to the Pr(Kso-c&EA|Tm-of-p)-value, it seems to be 1, since when a truth-maker of p is given, it is natural to suppose that a normally conceived explanandum of p, in which the presence of this truth-maker is predicted, is confirmed by its presence. As to the Pr(Kso-c&EA)-value, it is nothing but the Pr(EA|Kso-c)-value, given that Pr(EA&Kso-c) = Pr(EA|Kso-c)×Pr(Kso-c) and that the Pr(Kso-c)-value = 1. I tend to estimate the Pr(EA|Kso-c)-value as 0.9, since the existence of a zebra-belief-inducing zebra in the physical world looks fairly permitted by the existing Kso-c of the ascriber. Hence, Pr(Tm-of-p|Kso-c& EA) = 0.5×1/0.9 ≈ 0.56.
Insofar as (ii) is concerned, I think it is fairly acceptable to attribute “0.5” to the Pr(Tm-of-p)-value, since it is indeterminate whether there is a zebra-alike mule in the vignette. As to the Pr(Kso-c&EB|Tm-of-p)-value, I tend to estimate it as 0.1 (which is already an overestimation), since when there is a real zebra in the vignette, to claim that the target belief is induced by a cleverly painted mule would be fairly unreasonable. As to the Pr(Kso-c&EB)-value, it is nothing but the Pr(EB|Kso-c)-value, given that Pr(EB&Kso-c) = Pr(EB|Kso-c)×Pr(Kso-c) and that the Pr(Kso-c)-value = 1. I tend to estimate the Pr(EB|Kso-c)-value as 0.3 (which is already an overestimation), since the existence of a zebra-alike-non-zebra in the physical world looks a bit abnormal according to the Kso-c of most ascribers. Hence, Pr(Tm-of-p|Kso-c& EB) = 0.5×0.1/0.3 ≈ 0.16.Footnote 7
Therefore, given that Pr(Tm-of-p|Kso-c & EA) > Pr(Tm-of-p|Kso-c & EB) (since 0.56 > 0.16), EA, compared with EB, can thereby make the target belief safer insofar as we construe “safety” in terms of DS. And if we, for the sake of simplicity, further suppose that EA and EB have exhausted all explananda for the presence of a real zebra, then we may calculate the Pr(Tm-of-p|Kso-c)-value by appealing to the “total probability rule”:
$$\eqalign{Pr( {Tm-of-p\vert Ks_{o-c}} ) &= Pr( Tm-of-p\vert Ks_{o-c} \& EA) \times Pr( {EA\vert Ks_{o-c}} ) \cr &\quad + Pr( Tm-of-p\vert Ks_{o-c} \& EB) \times Pr( {EB\vert Ks_{o-c}} ) \cr & = 0.56 \times 0.9 + 0.16 \times 0.1 \cr & = 0.52} $$Now a new question comes: is the target belief safe, given this Pr(Tm-of-p|Kso-c)-value? To recall, according to the two-step heuristic for safety ascription (which is presented in the last section), a Pr(Tm-of-p|Kso)-value will not immediately lead to the delivery of the label of “safety” or “non-safety” unless this value is compared with a pre-fixed threshold-value. Conceivably, the higher the threshold-value is, the higher the Pr(Tm-of-p|Kso-c)-value will be required for delivering the safety-label.Footnote 8 Hence, whether a Pr(Tm-of-p|Kso-c)-value as high as 0.52 will result in the delivery of the safety-label will in turn depend on whether the threshold-value itself is above 0.52 or not.
But how to determine the threshold-value?
My response to this query is quite simple: I have no philosophical theory on this, since the determination of the threshold-value is simply a psychological/statistical matter. Specifically, skeptical people's threshold-values tend to be statistically high, whereas normal people's threshold-values tend to be statistically low. Hence, whether vignettes like Case 4 will evoke skeptical considerations mainly depend on whether the invited ascribers tend to be suspicious about what they see and hear.
I believe that the analysis of Case 4 can be easily applied to some more typical skeptical cases, say, the brain-in-a-vat Case (or BIV-Case). Surely the BIV-Case is different from Case 4 in the sense that the BIV-speculation will undermine the truth of all beliefs, most of which may not be about zebras or mules. But the BIV-Case is still similar to Case 4 in the sense that the BIV-conjecture, like the cleverly-painted-mule-conjecture, is still one possible explanans of the given explanandum, and still like Case 4, in the BIV-Case, the “normal explanandum” which is not BIV-involving (hereafter NE) is still left on the table. Hence, the estimation of the Pr(Tm-of-p|Kso-c)-value in the BIV-Case will still appeal to the “total probability rule”, according to which the result of “Pr(Tm-of-p|Kso-c&NE)×Pr(NE|Kso-c)” has to be put into the sum of the Pr(Tm-of-p|Kso-c)-value. Given that the result of “Pr(Tm-of-p|Kso-c&NE)×Pr(NE|Kso-c)” cannot be low, it is quite probable that the resultant Pr(Tm-of-p|Kso-c)-value is not very low even in the BIV-Case. Hence, even in the BIV-Case, we can hardly decide whether this value will lead the ascriber to deliver the safety-label or non-safety-label, unless a pre-fixed threshold-value of the ascriber is known.
So far, so good. But there is still a further question remaining unanswered: given the varieties of different ascribers’ threshold-values and their background knowledge, how can my theory handle the cases in which one safety-ascriber does not agree with another? Or to raise a relevant question: Ascribers are not only fallible, but may also be unreliable to the point that they may fail to understand some relevant aspects of the description of the given vignette. Hence, how could my theory preclude these disqualified (or even irrational) ascribers from my narrative?
These questions could be answered by resorting to the notion of “normalcy”. However, unlike Julian Dutant's (Reference Dutant2010) and Martin Smith's (Reference Smith2016) treatment of “normalcy”, this notion is introduced here not for specifying the needed features of epistemic safety, but for specifying the needed features of ascribers’ intellectual virtues.
6. Some clarifications of a “normal ascriber” as concluding remarks
For the sake of simplicity, I assume that the ascriber introduced into my theory is a “normal ascriber”. By “normal ascriber”, I mean an ascriber whose background knowledge and threshold for epistemic safety is “depersonalized” so that she will behave as most normal ascribers would do when the vignettes are presented to them. Accordingly, my putative ascriber only needs to have an average level of intellectual capacities (reading abilities included) as well as minimal commonsense. Hence, “irrational” ascribers could be easily precluded from my story if the term “irrationality” is going to be construed in terms of “lacking average intellectual capacities and minimal commonsense”.
This treatment of the “normalcy” of ascribers seems to be fairly vague, but I believe that such vagueness is something already assumed in post-Gettier epistemology. Please reconsider the “Smith” in the Gettier Case: “Smith” is actually a place-holder of any normal person with average intellectual capacities and minimal commonsense so that he can be qualified as a potential possessor of some form of knowledge. Hence, there is no need to talk about the diversities among subjects within the Gettierized vignettes. In my theory, I just take a further move to additionally assume that such notion of normalcy could also be applied to ascribers who are assessing the epistemic statuses of subjects on a higher level. This treatment goes hand in hand with the intuition-based methodology of mainstream epistemologists, since by “intuition”, what they actually mean is nothing but “cognitive deliveries of normal ascribers”. Accordingly, in my treatment of the Zebra Case, the reason why I estimate the Pr(EA|Kso-c)-value as 0.9 and the Pr(EB|Kso-c)-value as 0.1 is simply that I share the following commonsense with most ascribers: cleverly painted mules are rare to see, although their existence is conceivable.
But what if there is a cleverly-painted-mule-prevailing community so that ascribers living there will tend to attribute a higher value to Pr(EB|Kso-c)? In my view, in order to handle the difference between our community and theirs, we need to resort to statistics. For instance, we can statistically reconstruct an average portrait of a normal safety-ascriber from empirical data taken both from our society and theirs, hence, we can get an “intermediate value” representing the putative disposition of safety-ascription of such a reconstructed ascriber. Or we may also select one society as a more reliable provider of normal ascribers due to some arbitrary reason (for instance, we may arbitrarily assume that a cleverly-painted-mule-prevailing society itself is simply abnormal), and thereby marginalize the data taken from their society. The similar treatment could be easily extrapolated to the difference among safety-thresholds.
However, it is noteworthy that a “normal ascriber” does not mean an “idealized ascriber”, if to be “idealized” means to be immune to any inferential fallacies and cognitive biases. On the contrary, as I have mentioned, there are many studies showing that normal people are not immune to psychological biases such as the availability heuristic (Tversky and Kahneman Reference Tversky and Kahneman1973), hence, a statistical summary of their behaviors would also statistically summarize how these “biases” influence their behaviors. But I don't think this is a trouble for my theory; rather, the employment of these heuristics is quite useful for explaining Greco's (Reference Greco2007) following puzzle: given that there is a structural similarity between the Typing Monkey Case and the Lottery Case, why do most people feel that they know that a monkey will not type out War and Peace, while they feel that they do not know that the subject will lose the lottery game? According to my theory, this asymmetry is simply based on the asymmetry between most knowledge-ascribers’ monkey-related experience and their lottery-based experience: the instantiation of a novel-typing monkey is nearly unavailable in the former whereas the instantiation of a lottery-winner is fairly available in the latter. Accordingly, to believe that a monkey will not type out a novel would be much safer and hence nearer to the bar of knowledge (cf. note 4).
Surely here some may still contend that the availability of instantiations of a type, say, A, in an ascriber's mental architecture is not the same thing as the frequency of A's instantiations in the objective world, but the question is: by appealing to objective frequency alone, it looks hopeless to solve Greco's (Reference Greco2007) puzzle, since novel-typing monkeys and lottery winners seem to be equally rare in the objective world. Or one may bite the bullet by simply claiming that ascribers treating the Typing Monkey Case and the Lottery Case differently are themselves irrational. But in my view, this move lacks minimal respect of normal people's intuition and thereby makes the intuition-based methodology in mainstream epistemology quite problematic as well. In contrast, my theory is much more friendly to mainstream epistemology by showing its minimal respect to normal ascribers’ intuition. And it can also be converted to be more friendly to experimental epistemology (which often highlights cultural diversities and their relevance to knowledge attribution), especially when statistical surveys of the behaviors of ascribers from different communities do prove that the behavioral difference among these communities are really statistically salient.Footnote 9
