Justified True Belief: Plato, Gettier, and Turing Rohit Parikh and Adriana Renero Abstract We examine the relationship between the justified true belief (JTB) account of knowledge and Plato’s theory about it as expounded in the Theaetetus. Considering Socrates’ remarks in the Theaetetus brings us to some concerns raised by Turing and to Wittgenstein’s famous comment explanations come to an end somewhere. We present two simple technical results which bear on the question. Finally, we look at the pragmatic aspects of knowledge attributions. In an Appendix we say a few words about Indian epistemology and Gettier problems.
It is so difficult to find the beginning. Or, better: it is difficult to begin at the beginning. And not try to go further back. On Certainty, Ludwig Wittgenstein
1 Introduction The justified true belief account of knowledge is that knowing something is no more than having a justified belief that it is true, and indeed its being true. There is a common impression that the justified true belief (JTB) definition of knowledge is due to Plato and was undermined by Gettier in his (1963) paper. R. Parikh City University of New York Brooklyn College and CUNY Graduate Center email:
[email protected] http://www.sci.brooklyn.cuny.edu/cis/parikh/ (212) 817-8197 Adriana Renero The Graduate Center, City University of New York email:
[email protected] (718) 710-5076
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Gettier himself says, “Plato seems to be considering some such definition at Theaetetus 201, and perhaps accepting one at Meno 98.” The Stanford Encyclopedia of Philosophy article on the “Analysis of Knowledge” says Socrates articulates the need for something like a justification condition in Plato’s Theaetetus, when he points out that ‘true opinion’ is in general insufficient for knowledge. For example, if a lawyer employs sophistry to induce a jury into a belief that happens to be true, this belief is insufficiently well-grounded to constitute knowledge.
Others who have attributed the JTB theory to Plato include Artemov and Nogina (2005). Kevin Meeker (2004) says, “...in the middle part of the twentieth century it appears that philosophers were remarkably unanimous in asserting that knowledge is simply justified true belief.” However, a cursory look at the Theaetetus shows that Socrates at least did not endorse the JTB theory.
It is the boy Theaetetus (who was a mere 16 years old at the time) and not Socrates who proposes the JTB account after proposing two others, knowledge as perception and knowledge as true belief. Oh, yes, Socrates, that’s just what I once heard a man say; I had forgotten but it is now coming back to me. He said that it is true judgment with an account that is knowledge; true judgment without an account falls outside of knowledge.
Socrates subjects Theatetus’ assertion to rigorous analysis and finally undermines this third, JTB account, ending with the words, …therefore, knowledge is neither perception, nor true judgment, nor an account added to true judgment.
The JTB account of knowledge, rather than being endorsed by Socrates, is explicitly rejected. But what is of interest to us in this talk is Socrates’ objection to JTB which is different from that of Gettier and arguably deeper. Gettier’s own undermining of JTB went the following route.
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Someone justifiably believes A. She deduces B from A and indeed A implies B.
And B is true. So the belief in B is both justified and true.
However, unfortunately, A is false so that the belief in B, while justified, can’t really be considered knowledge.
2 The Nature of Justification Socrates does not go this route but instead asks what a justification might be like (the Greek term here for justification is logos, which translates roughly as “account”; the corresponding Sanskrit term is pramana, see Sect. 7 below). An analogy to justification here is an analysis of the first syllable SO of his own name. SO is composed of the two letters S and O and that spelling out is rather like a justification. Both an analysis and a justification have structure and Socrates points out that the letters S and O, not having structure, cannot have an analysis. But is it possible to know the syllable SO without knowing the letters S and O? And if not, then how can we rest a knowledge of SO on a knowledge of S and a knowledge of O? It would seem that issue of the knowledge of the letter S will bring us back to perception, for we know the letter when we see it. Alas, the perception account of knowledge has already been undermined earlier by Socrates. A pig can perceive the letter S but does not know the letter S. So we are left with empty hands.
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Let us now abandon Socrates for the moment and go to another, more modern figure, namely Turing in his analysis of computation (1936/7). Turing actually faces the same problem which Socrates faced but in a slightly different form. A Turing machine reads symbols on squares on its computation tape, writing other symbols in their place, or moving right or moving left to other squares and symbols. These actions are governed by its program which is a set of quadruples (Davis 1958). Perhaps the machine, when it reads the symbol A in state q, writes a B there and switches to state q’. But how does the machine know that it is the symbol A which it is reading? If the Turing machine is a computer, then this is a design problem for the electrical engineer. But if we put ourselves in the role of the machine then we are back to Socrates’ problem. How do we know the symbol A when we see it? And if we do not know, then how can we proceed with the computation? For instance, there is indeed a Turing machine which recognizes palindromes and would recognize the string ABCBA as a palindrome while rejecting ABCAB. But its ability to recognize ABCBA is parasitical on its ability to recognize the symbol A, and that ability, while we readily grant it, requires an explanation. Turing himself does face this problem. He restricts himself to finite alphabets on the grounds that an infinite alphabet would deprive the machine of an ability to recognize the symbols on the tape. Here is what he says (1936/7, p. 249):
I assume then that the computation is carried out on one-dimensional paper,
5 i.e. on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.
And later (1936/7 p. 250):
The behaviour of the computer at any moment is determined by the symbols which he is observing,1 and his “state of mind” at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite. The reasons for this are of the same character as those which restrict the number of symbols. If we admitted an infinity of states of mind, some of them will be “arbitrarily close” and will be confused.
Is the problem which Turing discusses the same as the problem discussed by Socrates? Not necessarily, because Turing assumes that the problem is purely geometrical, but the difficulty of recognizing symbols adequately may arise for non-geometric reasons. But Turing does not address the question how the machine recognizes symbols even from a finite alphabet. The issue is that when we reduce complex problems to simpler ones, we do still have the problem of addressing the simpler problems, per-
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Note that Turing is neutral between thinking about the computer as a person or as a machine and his mentalistic terminology is defensible.
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haps the simplest ones that our analysis will come up with. And now we still have to deal with these.
3 Wittgenstein In a slightly different context Wittgenstein does (sort of) face this question in his five red apples example in the Philosophical Investigations (2009, §1): “But how does he know where and how he is to look up the word ‘red’ and what he is to do with the word ‘five’?” ---Well, I assume that he acts as I have described. Explanations come to an end somewhere. “But what is the meaning of is
the word ‘five’ ?” No such thing was in question here, only how the word ‘five’ used.
We are left with what looks like a gap. It almost seems like Wittgenstein’s answer is an evasion. But we will see that there isn’t really a gap. Let us proceed by recounting a joke. Someone calls up on the telephone and a boy of five answers.
Can I speak to your father?
He is not here.
Can I speak to your mother?
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She is not here either.
All right, will you write them a note that Socrates called?
OK, how do you spell Socrates?
S..O..C..R..A..T..E..S.
There is a long pause and then the child says, How do you make an S? The child’s problem is the same problem which both Socrates and Turing faced. An explanation or an account or a justification is of no use to someone who does not know the basic elements. The basic elements are the individual symbols of the alphabet in this context. In other contexts they might be something else. But in any case, an explanation is made up of parts, made up of further parts. And when we reach bottom, we reach things for which there is no further decomposition. But these elements are furnished for us because we belong to the same species and have similar training. The fiveyear-old boy does belong to our species but lacks the corresponding training.
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4 Agent Based Justification So let us replace the notion of “justification for p” to “justification for p for agent A”, and allow that different agents will need different justifications even for the same conclusion. Suppose someone asks why people who were brought illegally to the US as children are eligible for some program. Then we can say, “Because Obama ordered it.” Someone else may need “Because Obama ordered it and he is President” and someone else may even require a lesson in the US constitution. There is no one size fits all justification. There are justifications which suit the occasion. “How do you know it is an S?” can be answered by, “Because I can see it is an S.” But “How do you know Obama is the president?” cannot be answered by “Because I can see he is the president.” The first explanation (knowledge as perception) which Theaetetus gave was not wholly wrong. Knowledge can coincide with perception on occasion. But having that perception and interpreting it does not require an ‘account’ or a ‘justification’. It requires membership in a community for which explanations have come to an end because they are not asked for – normally. Or consider another, similar case.2 Samuel Johnson having argued for some time with a pertinacious gentleman; his opponent, who had talked in a very puzzling manner, happened to say, “I don’t understand you, Sir;” upon which Johnson observed, “Sir, I have found you an argument; but I am not obliged to find you an understanding.” 2
Thanks to Melvin Fitting for telling us about this example.
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We will now present two simple examples of the way in which justification may differ from person to person. Suppose that the law
(∀x)(Px ∧ Qx ∧ Rx → Sx)
is known to three agents A, B and C. Moreover there is some person D whom they all know who has properties P, Q and R and thus also S. However A only knows PD ∧ QD , B only knows QD ∧ RD and C only knows RD ∧ PD. Then the justifications of SD will consist of RD for agent A, PD for agent B and QD for agent C. They will all learn the same fact, but the justifications will differ. In the example we just gave, there is a single justification PD ∧ QD ∧ RD which would work for all three but we can easily imagine cases where this will not work. Suppose that we have instead the law, for a total number of properties N,
(∀𝑥)([⋀ 𝑃𝑖 (𝑥): 𝑖 ∈ 𝑁] → 𝑆(𝑥))
And agents Ai already know some (proper) subset of properties holding of an x:
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(⋀ 𝑃𝑗 (𝑥): 𝑗 > 𝑖).
Then agent Ai can learn S(D) by learning the remaining properties hold:
⋀ 𝑃𝑗 (𝐷): 𝑗 ≤ 𝑖 and
⋀ 𝑃𝑗 (𝐷): 𝑗 ≤ 𝑖
will be agent Ai’s justification for S(D).
Thus different agents will have different justifications. But if their conscious language is finite and they learn about properties in piecemeal ways, then there will be no universal justification that ever can be used.
We suggest that
a) an explanation will differ from person to person, depending on what that person already knows (their background).
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b) When we get down to the ‘atoms’ of explanation, then further explanation could be asked for, but typically isn’t because membership in a community implies that we do not ask further questions.
There are no complete explanations because every explanation presupposes something. But nevertheless knowledge and justification still work, and can be systematically represented in a logic.
Finally, while we have questioned Gettier’s account of Plato, we have no quarrel with Gettier’s own result. Gettier and Socrates offer different challenges to the JTB account and both challenges have their merits.
5 Pragmatic Encroachment Pragmatic encroachment can be motivated by intuitions about cases. Jason Stanley’s book Knowledge and Practical Interests (2005) argues that it is the best explanation of our intuitions about bank cases like these:
Low Stakes Hannah and her wife Sarah are driving home on a Friday afternoon. They plan to stop at the bank on the way home to deposit their paychecks. It is not important that they do so, as they have no impending bills. But as they drive past the bank, they notice that the lines inside are very long, as they often are on Friday afternoons. Realizing that it wasn’t very important that their paychecks are deposited right away, Hannah says, I know the bank will be open tomorrow, since I was there just two weeks ago on Saturday morning. So we can deposit our paychecks tomorrow morning. High Stakes Hannah and her wife Sarah are driving home on a Friday afternoon. They plan to stop at the bank on the way home to deposit their paychecks. Since they have an impending bill coming due, and very little in their account, it is very important that they deposit their paychecks by Saturday. Hannah notes that she was at the bank two weeks before on a Saturday morning, and it was open. But, as Sarah
12 points out, banks do change their hours. Hannah says, I guess you’re right. I don’t know that the bank will be open tomorrow. (Stanley 2005, 34)
The quandary of the couple can be easily understood if we consider a little decision theory. Consider that we are thinking of performing a procedure If B then do W else do S and we think that B is true so we do W. What are the risks? In the following, Table 1, B is “Bank is open tomorrow”, W is wait, and S is “stand in line.” We assume that the preferences are 1) go on Saturday 2) wait in line and 3) go on Monday (least preferred but not considered a disaster)
13 Table 3.1 B W
5
S
10
¬B
15 10
If p(B) is more than one half then “wait” dominates “stand in line.”
Now let us change the payoffs. Table 3.2 B
¬B
W
5
100
S
10
0
Now “stand in line dominates” unless p(B) > .95 So while “Bank is open” was “known” before, it isn’t known any more. Note that we are using cardinal utilities and subjective probabilities in our explanation. This maneuver is not really justified since people do not have such numbers in mind when they speak. However, there is at present no convenient way to use only ordinal utilities to make a distinction between two cases where
1. B is worse than A
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2. B is much worse than A
And this implies, again, that we must distinguish between points of view when we individuate justifications.
Low Attributor-High Subject Stakes. Hannah and her wife Sarah are driving home on a Friday afternoon. They plan to stop at the bank on the way home to deposit their paychecks. Since they have an impending bill coming due, and very little in their account, it is very important that they deposit their paychecks by Saturday. Two weeks earlier, on a Saturday, Hannah went to the bank, where Jill saw her. Sarah points out to Hannah that banks do change their hours. Hannah utters, “That’s a good point. I guess I don’t really know that the bank will be open on Saturday”. Coincidentally Jill is thinking of going to the bank on Saturday, just for fun, to see if she meets Hannah there. Nothing is at stake for Jill, and she knows nothing of Hannah’s situation. Wondering whether Hannah will be there, Jill utters to a friend, “Well, Hannah was at the bank two weeks ago on a Saturday. So she knows the bank will be open on Saturday”. It seems that Jill is making a mistake but a natural one since she does not know about Hannah’s high stakes.
This is an issue involving the Premack-Woodruff Theory of Mind (1978). Jill does have a theory of mind about Hannah, but it is the wrong one, as is shown by the fact that she attributes to Hannah a false belief. That very possibility turns, however, on her being in a different situation than Hannah is.
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6 Conclusion We have surveyed some of the old and more recent observations about the nature of knowledge, and more specifically, the nature of justification, and argued that the notion is agent relative. We also point out that this issue has been discussed by Socrates, Turing, and Wittgenstein and that the responsibility for the much maligned JTB theory of knowledge cannot fairly be laid at the door of Socrates. He was far more subtle than people seem to have noticed
7 Addendum After we wrote this paper and after it was proof-set, we discovered that we had said nothing about the Indian contributions to epistemology (IE from now on). Work in IE begins with Gautama (not the Buddha) around the 3rd century CE and proceeds through various developments to the 17th century. Two major figures in IE are Gautama, the 3rd century author of Nyaya-sutra and founder of the Nyaya school, and Gangesa, the 14th century author of Tattva-chinta-mani and the founder of Navya Nyaya (new Nyaya). (Gautama’s Nyaya school is sometimes called Pracina or ancient Nyaya to distinguish it from Gangesa’s later school). Both the schools mentioned, Nyaya and Navya Nyaya are Vedic, but there are other Vedic schools as well as some atheist (or non-Vedic) schools, as well as some Buddhists who showed keen interest in knowledge (prama). Some Buddhists are presentists, believing that nothing except the present is real. For a fuller list, please consult (Phillips 2015). However, two persons who seem most aware of Gettier-like problems are Prasastapada in the 6th century and Sri Harsa in the 11th. (All dates are rough). A common example used is: Suppose smoke is seen on a mountain, smoke is associated with fire and so one infers fire on the mountain. Actually, what was seen was steam or dust, mistaken for smoke; but as a matter of fact there is a fire on the mountain.
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Does one know then that there is fire on the mountain?
Both Prasastapada and Sri Harsa say No, but oddly enough, Gangesa, who followed them both seems to regard at least some Gettier cases as cases of actual knowledge. Gangesa, who is very much a physicalist, only allows certain kinds of errors and some particular ones do not fall in his taxonomy of errors. Sources of knowledge (pramana) are often classified into a) perception b) inference c) testimony and d) similarity. But some Indian philosophers have rejected all of these except for perception. Testimony is suspect because one does not know that the speaker is honest and knowledgeable. If one infers Gx from Fx and (∀x)(Fx→Gx) then the inference is vulnerable to one’s being wrong about (∀x)(Fx→Gx). And even perception might be veridical only when it is nonconceptual. A pre-lingual child who sees a cow “knows” what she sees but cannot tell us what she saw. But an adult saying, “I saw a cow” is not safe from concepts and the ills they are heir to. Not being experts in this area we will not say more, but will refer the reader to Saha (2003), Ganeri (2007) and Phillips (2015). All three have very kindly helped us with our investigations, but of course any errors are our own.
Acknowledgments Thanks to Aranzazu San Gines for some of the references. Arthur Collins also made a useful suggestion.
References Burnyeat, M. (1990). The Theaetetus of Plato, M. J. Levett, translator, Indianapolis, IN: Hackett Publishing. Davis, M. (1958). Computability and Unsolvability, Inc., New York: McGraw-Hill Book Co.
17 Ganeri, Jonardon. ”Epistemology in Pracina and Navya Nyaya (review).” Philosophy East and West 57.1 (2007): 120-123. Gettier, E. L. (1963). "Is Justified True Belief Knowledge?" Analysis, 23(6), 121-123. Ichikawa, J. J., and Steup, M. (2013). "The Analysis of Knowledge", E. N. Zalta, (ed.) The Stanford Encyclopedia of Philosophy (Fall 2013 Edition). URL = http://plato.stanford.edu/archives/fall2013/entries/knowledge- analysis/. Meeker, K. (2004). "Justification and the Social Nature of Knowledge." Philosophy and Phenomenological Research, LXIX, 156-72. Phillips, Stephen, ”Epistemology in Classical Indian Philosophy”, The Stanford Encyclopedia of Philosophy (Spring 2015 Edition), Edward N. Zalta (ed.). Plato. (1997). Plato, Complete Works, Indianapolis, IN: Hackett Publishing. Premack, D., and Woodruff, G. (1978). "Does the Chimpanzee Have a Theory of Mind?" Behavioral and Brain Sciences 1(4), 515-526. Saha, Sukharanjan, Epistemology in Pracina and Navya Nyaya, Jadavpur University (2003). Stanley, J. (2005). Knowledge and Practical Interests, New York: Oxford University Press. Turing, A. M. (1936/7). "On Computable Numbers, with an Application to the Decision Problem." Proceedings of the London Mathematical Society, 2(42 (1936-7)), 230-265. Wittgenstein, L., Anscombe, G. E. M., Hacker, P. M. S., and Schulte, J. (2009). Philosophische Untersuchungen = Philosophical investigations, Malden, MA: Wiley-Blackwell.
