This article first appeared on Scientia Salon
Philosophers have traditionally defined “knowledge” as a belief that is both true and justified, a definition that sufficed until, 50 years ago, Edmund Gettier pointed out that the conditions could be fulfilled by accident, in ways that didn’t amount to what we would intuitively regard as “knowledge”.
Gettier pointed to scenarios such as:
“Smith has applied for a job, but, it is claimed, has a justified belief that “Jones will get the job”. He also has a justified belief that “Jones has 10 coins in his pocket”. Smith therefore (justifiably) concludes that “the man who will get the job has 10 coins in his pocket”. In fact, Jones does not get the job. Instead, Smith does. However, as it happens, Smith (unknowingly and by sheer chance) also had 10 coins in his pocket. So his belief that “the man who will get the job has 10 coins in his pocket” was justified and true. But it does not appear to be knowledge.”
Since Gettier’s paper many attempts have been made to patch up the “justified true belief” definition, often by adding extra conditions aimed at ruling out being right accidentally, though none of the proposed solutions has gained general acceptance and most have been shown not to work. 
Analysis of “Gettier problems” usually focuses on the word “justified”, which is often taken to be a binary condition, either fulfilled or not fulfilled. In Gettier’s paper, for instance, beliefs are taken as “justified” if there is “strong evidence” for them, but the idea of justification is not examined further . Yet, in the real world, knowledge, evidence and justifications for claims are always probabilistic.
If “justification” were taken to be infallible — justification so bullet proof that the belief cannot then be in error — then there are no Gettier problems (Smith cannot have a “justified” belief that Jones will get the job unless Jones does get the job).
But, in the absence of human infallibility, “justification” (“the action of showing something to be right or reasonable” — Oxford Dictionaries) should mean “sufficient justification”, which we can take as justification that gives a greater than 90 per cent chance of being correct (or whatever threshold we choose). Now, suppose that Smith is “justified” with 90 per cent reliability in believing that Jones will get the job, and also justified with 90 per cent reliability in believing that Jones has 10 coins in his pocket. Then he has (under these definitions) “knowlege” that Jones will get the job and “knowledge” that Jones has 10 coins in his pocket, but does not have knowledge that “the man who will get the job has 10 coins in his pocket” since the reliability of that claim is 0.9 x 0.9 = 0.8, which falls below the threshold.
That in itself does not avoid Gettier problems (since those probabilities could each have been 0.95, which does multiply to a number above the threshold), but it does suggest that the threshold nature of “justification” is central to the issue.
If we accept that “Fred is 90 per cent justified in believing truth X” counts as “knowledge of X” then we are accepting that the justification could be faulty, and thus be unrelated to the truth of the matter. But, if so, the belief can always be true entirely by accident. Thus it follows that a “justified belief” can be true entirely by accident. There is no way round that, it is inevitable for any threshold for “justification” less than 100 per cent reliability .
In response, some have simply declared that, yes, belief that is both accidentally true, and accompanied by an irrelevant and erroneous justification, really is “knowledge”. But then we would be left with a counter-intuitive interpretation of “knowledge” that amounts only to being right, even if accidentally right. No-one would accept that someone knew what the winning lottery numbers would be, just because they happened to have bought the winning ticket.
For that reason, much discussion of Gettier has attempted to add a fourth condition, designed to prevent Gettier problems, to the trinity of justification, truth and belief. These usually try to avoid being accidentally right by bolstering the concept of justification (for example, the justification must not rely on falsehoods; or it must derive from the truth). 
None of these solutions solves the issue. To see this simply consider the justification, J, together with the additional condition and call the combination J+. Now either J+ is fallible or it isn’t. If it is fallible then we still have Gettier problems; indeed all we’ve done is re-label J as J+, since we didn’t specify what was necessary for J in the first place, and at most we’ve elucidated aspects of justification. Alternatively, if J+ is infallible then, fine, we’ve avoided Gettier problems, but we might just as well have declared J infallible at the start.
The above has been discussed at length and perhaps a consensus is emerging that there is no way round the Gettier problem, at least not without declaring justification to be infallible . Yet, some are reluctant to accept that, which is perhaps strange since there is no reluctance to require the “truth” part of the definition to be absolute. Indeed, one can regard the extensive literature on Gettier as a search for a wording that stops short of adopting an infallible version of justification, while ensuring that the justification is never erroneous (because if it ever is, then in can leap Gettier).
And thus, it seems to me, that the really crucial part of the “justified true belief” definition of “knowledge” is that it is a belief that is true. This highlights a weird feature of the definition, that in order to know whether something is “knowledge” you first need to know whether it is true, and you can only know that if you already have that knowledge.
Thus the definition is not very useful: we can never use it to determine whether we have knowledge. It relies on having a “god’s eye” view where one already knows the full facts of the matter, else it cannot be applied. But, if one does have that view, then one might as well go the whole hog and declare that only infallible justification counts as justification, and thus solve Gettier’s problem.
Alternatively, if we don’t adopt the god’s-eye view, then we need to accept that, not only will our justifications be fallible, but also that what we regard as “true” might be erroneous, and thus we need reliability thresholds on both the “justified” and the “true” components of the definition.
Of course scientists have long accepted that they can never attain absolute truth about the world, but only approach it with ever-increasing, though not-total, reliability. Philosophy, however, can be considered to be a discussion about a broader “conceptual space”, in which truth could indeed be conceived of as securely known. 
So let’s declare this article to be about “knowledge” in and of the real world, and indeed discussion of Gettier is usually in terms of real-world examples (people with coins in their pocket, robotic dogs, red-painted barns, and so forth), however contrived.
On that point, and to keep this article to theme, let’s leave aside axiom-based systems such as mathematics and logic, where one might “know” something owing to it being entailed by axioms. If might be thought that within such systems one can have absolutely reliable knowledge, though even there one would be reliant on mathematicians and logicians having made correct deductions, and, without a proof of human infallibility, that is again not absolutely certain.
Is there an alternative definition of “knowledge”? Oxford Dictionaries defines the word as:
Knowledge: Facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
That is a bottom-up approach to defining knowledge, starting with information and building on it, rather than the top-down approach of starting with truth. As a scientist, the bottom-up approach seems to me in keeping with how we actually gain knowledge, by acquiring information about the world around us and interpreting it, and then building theories that best model the world. We don’t start with truth, instead we build towards truth — that’s the best we can do — and we acquire knowledge as we go.
To improve the “justified true belief” definition we should thus omit the condition “true”, since without God’s eye we cannot know what is true, and can only evaluate the probability of truth based on the justifications we have. Thus the “true” condition is redundant with the “justification” condition.
That leaves us with knowledge as “justified belief”. We can then ask what counts as “justification”, for which the best answer — if we’re talking about knowledge about the real world — is provided by the methods adopted by science. After all, the methods of science have been developed and honed over the centuries as giving our surest path towards truth about the world around us (demonstrably so, given that the engineering and technology based on science does actually work, thus showing that scientific models are a pretty good match to reality).
Is this definition vulnerable to Gettier? Yes it is, since it accepts that our evidence and justifications are fallible, deriving as they do from only a very limited sampling of the world, and yet a claim might always be true entirely by chance. Indeed, the field of statistics helps us to evaluate what role chance will be playing. The answers it gives are not certain but they are often good enough.
Thus Gettier points to an inevitable limitation on real-world knowledge, deriving from the fact that evidence will never establish anything with absolute reliability (though if one doesn’t like the messiness of the real world one can instead declare oneself omniscient, both in judging truth and in evaluating justifications, and from that pulpit bestow laurels on Gettier-proof knowledge).
 Here is the obligatory link to the Stanford Encyclopedia of Philosophy on the topic.
 The original paper by Gettier is online here.
 This has been argued by several people, for example see Ian M. Church (2013), European Journal of Philosophy, 21 (1):37–49, and citations therein.
 A bibliography of Gettier-problem papers is given here.
 See, for example, this article by Fred Dretske.
 For example Massimo Pigliucci explains that: “You can think of philosophy as an exploration of conceptual, as opposed to empirical, space …” where “conceptual space is much broader than its empirical counterpart”.