"Knowing that P is at least a matter of having a belief that P which is both true and justified."
Is this an adequate definition of knowledge?
If not, how should it be improved?

No, this is not an adequate definition of knowledge.  Firstly, there are some well respected theories of knowledge that would deny the question's premise that "knowing that P"  involves a "justified true belief".  Secondly, even accepting the question's assumption of the "Justified True Belief"  theory of knowledge, this single sentence does not provide an adequate understanding of just what is meant by either "true" or "justified".  Different philosophers have offered different interpretations of both of these concepts, and the simple sentence above offers no guidance as to which variation of "truth"  or "justification"  is being referred to.  Thirdly, however "truth"  and "justification"  are to be understood, there is an additional difficulty that is not addressed by this simple statement.  And that difficulty is commonly referred to as the "Gettier"  problem (after Edmund Gettier who first explored this issue in 1963).

Because of the limitations in time and space, I will not deal here with alternative theories of knowledge, the various theories of truth, or the various theories of justification.  I will focus instead on the Gettier problem.  And I will do so from the perspective of the question's presumption of a "justified true belief" theory of knowledge.

In preparing a definition for a term like "knowledge", philosophers generally employ a process that combines a suggestion for the conditions that are individually necessary and jointly sufficient to warrant the claim to knowledge, with a search for counter-examples where the definition is at odds with the common use of the concept.  It turns out that it is relatively easy to compile scenarios where the definition of knowledge as a "justified true belief" runs counter to our intuitive understanding of how we employ the concept of "knowledge".  Obviously, the definition of knowledge as simply a "justified true belief" is missing some key factor.  The study of the Gettier Problem is the study of how the various suggested definitions of knowledge fall short of our common use of the word.

Regardless of the particular theory of justification that is added to expand the definition provided above, there is always the possibility that the defined conditions for a "justified true belief" are satisfied, but satisfied in a way that is not generally accepted as knowledge.  It is possible that P could be true, that S believe that P, and that S is considered properly justified (however that is to be understood) in believing that P.  Yet the satisfaction of these three conditions together might be totally accidental.  All the evidence (or other beliefs) that are considered by S to support the truth of P might not be supportive of P at all. 

Edmund Gettier, along with many philosophers who have studied the problem, have framed numerous ingenious examples.  Here is one of them.  Consider the case of Fred's Ford.  John believes that Fred owns a Ford.  In fact, Fred does indeed own a Ford (a Ford Explorer, actually).  Yet all of the evidence that John has that suggests to him that Fred owns a Ford, is derived from John's observations of Fred and a particular Ford Mustang.  John concludes from the evidence that Fred owns a Ford.  But actually the evidence equally supports the alternative that Fred has borrowed his sister's Ford Mustang.  It is generally acknowledged that John does not actually know that Fred owns a Ford because John's belief that Fred owns a Ford is quite unjustified.  It is just "epistemic luck" that Fred does in fact own a Ford.  Fred could actually not own a car at all, or might own any other kind of car besides a Ford, without changing any of the justification that John has for his belief.

Gettier cases in general share a few key characteristics.  In each case, a subject holds a belief that P that the proposed scenario stipulates is well justified, yet is supposedly generally acknowledged as not knowledge.  The justification that is described in the example is fallible in that while it strongly suggests that P is true, it is not conclusive proof that it is.  Each example contains the key element of luck (called, in this context "epistemic luck" ).  Each case is constructed so that it is pure chance that P is true, because the evidence that justifies the belief that P is also consistent with not-P.  The challenge that the Gettier problem presents to the justified-true-belief model of knowledge is to identify the additional factor that must be added to a "justified true belief" to render the proper judgement of whether the belief in question is (or is not -- depending on the scenario) knowledge.

There have been many different attempts to "de-Gettierize" the Justified-True-Belief theory of knowledge by adding what is termed "the fourth condition".  Unfortunately, the Gettier problem remains an unresolved issue.  As each new idea has been presented to the philosophical community, someone has come up with an alleged counter example.  Here are some of the attempted solutions:

Infalliblism -- This approach limits the concept of "knowledge" to those beliefs for which we have sufficient evidence to rule out any possible alternative.  Hence, our claim to knowledge must be infallibly justified.  Since neither induction nor abduction can give rise to inferences with sufficient certainty for this approach, infalliblism limits valid claims to knowledge to beliefs arrived at by deductive inference.  However, most of our beliefs are formed on the basis of induction or abduction (or perhaps directly from our sensory experiences).  Most of our justifying reasons for claiming knowledge are far from sufficient to rule out any logically possible alternative.  As a matter of acceptable common usage, most of the time our justifying reasons are just barely sufficient to rule out the more likely alternatives.  The problem with this approach, therefore, is that it rules out most of what we normally refer to as knowledge.  While it should be noted in passing that one response to an offered Gettier example is to deny the premise that the belief in question is (or is not) in fact properly to be called knowledge, infallibilism is so at odds with the general usage of the term that it is no longer a seriously proposed solution.

Defeasible Evidence -- This approach demands that the evidence the subject draws upon to justify his belief that P not be opposed by any evidence to the contrary.  There are two degrees of this approach.  The weaker theory is that the subject himself cannot be aware of any defeating evidence.  However, it is unclear that this approach will resolve many of the proposed Gettier examples.  In the example cited above, John is not aware of any evidence that might defeat his belief that Fred owns a Ford.  Yet it is generally accepted that in this case John does not know that Fred owns a Ford.  The stronger theory is that there must not exist any defeating evidence, whether or not the subject is aware of it.  This would resolve the case of Fred's Ford.  But there have been counter examples designed for this suggestion.  In what is referred to as the "assassination example" there is no evidence that does in fact defeat the belief, but there is a lot of false evidence that, if believed, would be considered defeating.  So even though there is in fact no defeating evidence, the presence of the false evidence is supposedly sufficient to withhold the warrant of knowledge.  Such examples demonstrate that this approach, while promising, is not a complete solution.

No-False-Inference -- This approach demands that the subject include no false inferences when justifying their belief that P.  In the example of Fred's Ford, John employs the false inference that the Ford Mustang that all his evidence is about actually belongs to Fred.  But like all other proposed Gettier solutions, counter-examples have been offered for this one as well.

However, there is a variation of the No-False-Inference approach that I believe is proof against these counter-examples.  Examine the belief that P and focus on the concepts being employed rather than the words being used.  If one then decomposes P into its "atomic beliefs" (which is easily doable from the context provided by each example), then it seems to me that the false belief involved becomes obvious.  Since the JTB definition of knowledge demands that the belief in question be true, after conceptual decomposition all of the Gettier examples result in an acceptable granting or denying of knowledge in the specified example.  In the case cited above, P is initially framed as "Fred owns a Ford".  But this belief is based on a lot of evidence having to do with a Ford Mustang.  To decompose this belief into its conceptual context is to render it as "Fred owns that particular Ford Mustang for which I have all this evidence".  In the famous case of "Barn County",  the P that is offered is "I see a barn".  But conceptual decomposition renders this as "What I see over there is a barn".  If one is employing an externalist concept of justification, then the decomposed belief is obviously either true or false accordingly.  On the other hand, if one is employing an internalist concept of justification, the contextual decomposition makes it obvious that the evidence does not in fact justify what the subject thinks it does.  Since, by design, each proposed Gettier example involves a non-particular belief that is extrapolated from some particular beliefs drawn from the evidence, this approach should be generally applicable.  (Especially for those examples where P is a conjunctive or disjunctive proposition.)

The no-false-belief approach without the contextual decomposition, combined with an internalist conception of justification, can easily suffer from "creeping scepticism".  The scepticism arises because it is never clear from the inside when some particular supporting belief we are employing in the justification of our belief that P is actually or even likely false.  We are none of us infallible.  At any given time, it is always possible that some of our beliefs are in fact false.  If that is the case, then it is possible that we don't know what we think we do.  Concepttual decomposition prevents this creep to scepticism by making it clear, even to the subject, which belief is the unjustified one.  The no-false-belief approach without the concepttual decomposition, combined with an externalist conception of justification, quickly devolves into the next offered solution.

Global Justification Externalism -- Like the strong defeasible evidence theory mentioned above, this approach demands that there must not exist any defeating evidence, and considers it irrelevant whether or not the subject is aware of it.  But it goes further than the defeasible evidence approach by examining all the potential evidence (practically available or not) and considers how that evidence, if it were known to the subject, would impact the process of justification.  This approach nicely deals with all of the Gettier examples because it considers, from a global omniscience perspective, all of the evidence provided by those examples.  The problem that this approach faces is that it makes the concept of knowledge something that is determinable only externally by an omniscient intellect, or at least by an outside observer.  It provides no guidance to us fallible and limited individual knowers on whether any claim to knowledge is properly justified.  Like with the strong version of defeasible evidence, global justification externalism is at odds with many philosophers'preference for the alternative of justification internalism.

Causal-Chain -- This approach demands that the subject's belief that P be caused in "an appropriate way" by the fact that P.  Furthermore, the subject's justification for the belief must include the logic by which the truth of P caused the belief that P.  This nicely rules out all of those Gettier examples where the truth of P is not the cause of the subject's belief that P.  In the example cited of Fred's Ford, John's belief that Fred owns a Ford is not caused by the fact that Fred owns a Ford Explorer.  The difficulty faced by this theory, however, is that it is difficult to specify exactly how the truth of P can cause a belief that P without recourse to the very concept of knowledge that is being defined.  The other shortcoming of this approach is, of course, that the literature abounds with counter-examples.  So this approach also is demonstrated to be incomplete.

Ridiculous Complexities -- I group in to this category all those fancifully complex theories of justification that have been generated as the number of Gettier examples has multiplied.  Every time some innovator comes up with a new suggested Gettier example, the theories of justification become more and more complex, with additional logical conditions designed to rule out the newly proposed scenarios.  Shades of Ptolemaic epicycles.  The whole approach to the challenge of identifying necessary and sufficient conditions that cover all cases is based on the premise that our concept of knowledge is fundamentally an exercise in intellectual reasoning.  As such, however, the approach it runs counter to the common practice of attributing knowledge to animals and pre-intellectual children, and the common intuition that knowledge can sometimes be gotten quickly without intellectual reasoning.  The growing complexity of the anti-Gettier conditions is one of the factors driving theorists to alternative conceptions of knowledge.

Reliablism -- This is perhaps the most popular of the alternatives to the standard conception of knowledge as an intellectual exercise.  Reliablism is the thesis that propositional knowledge consists of a true belief that is arrived at through some reliable (but fallible) process.  It is an externalist theory because the subject is not in a position to determine whether the processes involved are reliable in any particular case.  Conveniently, it is always possible to understand the word "justified" in broad enough terms that we could consider a belief formed by some reliable process as suitably "justified.  The advantage of reliablism as an approach to the Gettier problems, is that a reliable process need be only more reliable than not.  A reliablist conception of knowledge permits subjects to err in their claims to knowledge.  John may properly claim to know that Fred owns a Ford, and yet be mistaken in that claim.  We, with more information, can reach a different reliable conclusion.  Likewise, our common employment of the concept of "knowledge" need not be totally precise.  We need only be more correct than not.  So instances of an incorrect granting or withholding of the warrant of "knowledge" can be quite acceptable as long as we do not err too often.  A definition consisting of necessary and sufficient conditions to properly cover all conceivable scenarios is not required.  Some slack is tolerable.

Regardless of which approach is chosen to address the challenge of the Gettier problem, it is clear that even within the Justified-True-Belief definition of knowledge, knowledge is more than simply a justified true belief.  While knowledge may be conceived of as at least a justified true belief, any definition that purports to mimic our common use of the concept must be a whole lot more.

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