Condition 4b is a 'no defeaters' condition or a 'no undermining evidence' condition. With this condition, it is assumed that in order for S to have knowledge, there cannot exist additional evidence (or theory) not possessed by S, such that if this additional evidence were possessed by S, this evidence would undermine (i.e., undercut) or defeat (i.e., override, rebut), S's previous evidential premises in condition 4a for believing p. In formal terms, condition 4b maintains that there cannot be evidence q, such that if S comes to believe q, this evidence would lead S to doubt p. If there is evidence q that suggests ~p and if S becomes aware of this evidence, then S must have (or acquire) adequate evidence to dismiss (or resolve) the counter-evidence, or admit to not knowing p.

A no-defeaters condition was first introduced by Keith Lehrer and Thomas Paxson, Jr. in their "Knowledge: Undefeated Justified True Belief" (1969). Since the publication of this article, it has been argued that there are unresolvable problems with any specification of a no-defeaters condition. First and foremost, it is objected that there are instances where irrelevant and misleading evidence, not accessible to S, can undermine S's true belief p held for relevant reasons, and a no-defeaters condition would wrongly deny these cases of knowledge. Second, it is claimed that there almost always exists a true proposition that S is unaware of, that would undermine almost any of S's justifications for any p, and as a result condition 4b mistakenly allows very little human knowledge. Third, there is the possibility of a regress of both undermining and vindicating evidence for any belief; thus making 4b an untenable condition. These objections are addressed in the course of chapter one.

Below, I respond to the initial objection that condition 4b allows the existence of inaccessible misleading evidence to unfairly defeat a person's strong justification and putative knowledge. I argue that misleading evidence, even if S is unaware of it, can prevent S from knowing, even when S possesses strong and relevant reasons for believing a true p. This is contary to some philosophers intuitions. To illustrate my position, I discuss hypothetical cases from the literature to defend my intuitions. The first case is from Gilbert Harman (1973).

Example #1: Jill and the Newspaper: A Political Leader is Assassinated
A political leader N is (in fact) assassinated in a foreign country. N is dead. His political associates fearing a violent coup, quickly decide to plant a false news story saying that the bullet hit someone else, and that N is safely in protective seclusion. On the cameras of CNN global television network, the associates declare "the assassination plot has failed, but that an associate of N was killed." But, shortly before this false and misleading live television announcement is made, a newspaper reporter in full-view of the assassination emails his story to the New York Times newspaper, stating that "N has been killed." The newspaper prints the (true) story of the successful assassination in its early edition.

Jill buys the early-edition New York Times on the way to work. Without access to CNN television, she reads the newspaper story of the assassination. From the newspaper information, Jill believes p: 'the leader N has been assassinated.' Here, p is true, and Jill's belief is based upon a single relevant premise for why p should be believed: A usually-reliable newspaper states that leader N has been assassinated. Jill believes a true p, based on relevant information for why p should be believed, and thus knowledge conditions 1, 2, 3 and 4a are satisfied.

But because CNN television network is simultaneously stating a contradictory headline that "N is safe" (filming the hoax announcement) and CNN is similarly a reliable source of news, it is clear that Jill does not know that N has been assassinated. The usually-reliable (but false) television reports not considered by Jill would weaken Jill's belief that N was assassinated, if Jill had access to this evidence. Jill possesses a justified belief p, and is personally justified (given her evidence) in believing p, but she doesn't know p. Condition 4b is violated. If Jill was aware that CNN television is reporting failure of the assassination attempt, it is unlikely that Jill would be able to dismiss this counter-evidence, and continue to retain a strong belief that N was dead. Conditions 4a and 2 are now undermined. In this case, Jill does not know 'Leader N has been assassinated' because there exists evidence q that would significantly weaken Jill's belief that p.

The pattern of the Jill and the newspaper case is the same as the Henry and the barn example. In these cases, S believes a true p, based upon relevant premises (satisfying condition 3) and strong evidence (satisfying 4a), but because of the existence of undermining evidence (i.e., the existence of barn facades, contradictory media reports) neither Jill nor Henry knows p. Most philosophers agree that neither party has knowledge in these cases. The PE definition explains why this is so. The following three cases are examples that challenge the viability of condition 4b.

Example #2: Does S Know that Tom Grabit Stole the Book?
This case was suggested by Lehrer and Paxson (1969). In this case, a student S visually observes Tom Grabit steal a book from the library. S is very familiar with Tom Grabit because Tom attends several of S's classes, and S clearly sees the theft. S has a true belief p that 'Tom Grabit stole the book' based upon relevant and strong perceptual evidence. Conditions 1, 2, 3, and 4a are all satisfied. But unknown to S, Tom Grabit's mother, Mrs. Grabit, being aware of the theft accusation against her son, states that it could not be Tom who stole the library book, because Tom was thousands of miles away, and that it must be Tom's mischievous identical twin brother, John, visiting from out of town must be guilty of the theft. Given that S doesn't know that Mrs. Grabit made this statement, does S still know p? Many people, including Harman and myself are inclined to think that S no longer knows that Tom stole the book, because this undermining testimony, brings reasonable doubt that Tom is guilty, even if S is (in fact) unaware of this undermining evidence.

However, Harman adds the following embellishment: Almost everyone except S knows that Mrs. Grabit is a chronic pathological liar who is in the habit of defending her son, and that Tom has no identical twin brother. Given this fact, Harman states that Mrs. Grabit's testimony should not undermine S's true belief, and S really does know that Tom stole the book. Harman contends that S knows that Tom stole the book, even when there is misleading undermining evidence that S hasn't considered.

I entirely disagree. It is my intuition that if S did not know that Mrs. Grabit was a liar and that Tom did not have an identical twin, then this unconsidered testimony may prevent S from knowing that Tom stole the book. For anyone with the background information about Mrs. Grabit's habitual lying, her testimony would not be undermining evidence, and they would know Tom stole the book (if they had witnessed the theft). But for S, this unconsidered evidence might override S's initial premises, so S might not know p. Harman's blanket intuition that Mrs. Grabit's deceptive testimony should not undermine S's belief that Tom stole the book is without merit if we consider S's background knowledge.

Footnote: But, of course, if it is later found out by S that Mrs. Grabit is a chronic liar, or there exists no identical twin, then these additional premises added to S's stock of premises, would allow 4a and 4b to be satisfied, and S would again know that Tom stole the book.

Footnote: Keith Lehrer (2000, p. 159) shares Harman's intuitions and denies that inaccessible misleading evidence (Mrs. Grabit's false testimony) should undermine the student's knowledge that Tom stole the book.

Example #3: Does S Know that J Purchased a Ford from the Local Dealer?
An example by David Annis (1973) illustrates the widely-held intuition that unknown misleading evidence should not be allowed to defeat a person's strong personal justification for believing a true p that is based upon relevant reasons. Suppose that S1 is told by his friend S2 that he just "recently bought a new Ford truck from a local dealer." S1 knows that S2 is reliable and honest. It is also true that S2 bought the truck from the local Ford dealer. However, suppose that another person, S3, overhears S2's statement and decides to ask the salesman how much did S2 pay for his new truck? The salesman S4, who is generally reliable and honest, responds to S3 that "S2 did not purchase a Ford truck from them." S3 believes the dealer, and says nothing to S1 or to S2. According to Annis' story, the reason that the salesman asserted that S2 did not purchase a Ford from them was that the salesman confused S2 with another person S5, who was also in the building the same day, but didn't make a purchase. This undermining testimonial evidence that S1 is unaware of, is simply a matter of confusion and an honest mistake made by the salesman S4.

In this situation there exists undermining evidence that 'the dealer denies S2's purchase' not considered by S1 that might significantly weaken S1's belief. If S1 wouldn't be able to dismiss this contradictory evidence, then S1 would fail to know that S2 bought the truck from the local dealer. Although S1 has strong truth-connecting reasons for believing p (i.e., S2 bought a Ford from local dealer) and S1 isn't aware of this undermining evidence, S1 may not know p. For many philosophers, this is an unacceptable result. It is objected that 4b falsely denies knowledge to S1, and so the PE definition fails.

While it may be unfortunate that S1 doesn't have knowledge that S2 bought the truck from the local dealer, and S1 has only a true belief based upon strong and relevant evidence, this seems to be how it is. We need to admit that sometimes in cases where there exists conflicting (and misleading) facts that we are unaware of, we do not know. Misleading evidence can come in the form of lies and hoaxes, as well as unintentional misinformation, irrelevant facts, and so on. The circumstantial ignorance of misleading or conflicting evidence can prevent S1 from having knowledge in otherwise normal situations. If an unconsidered true proposition q would lead S1 to doubt p, given S1's existing evidence for believing p, then S1 does not know p. Although there are many ways a true belief can be undermined by the contingent existence of misleading or conflicting evidence, in most ordinary situations, there will not exist inaccessible misleading undermining evidence that would lead S1 to doubt a belief held upon strong evidence.

If it were pragmatically important for S1 to know that S2 bought the truck from the local Ford dealer, S1 could verify the proposition with the accountant of the dealer S6, and perhaps uncover the reason why the salesman S4 mistakenly reported false information to S3. The explicit investigation for additional evidence (and eliminating false or irrelevant evidence) is a practical requirement for obtaining knowledge when it is important for S1 to know p.

Example #4: Sam and the Failed Hoax: An Airplane Flight is Not Interrupted.
Suppose that Sam has a scheduled plane flight to fly from Atlanta to Los Angeles at 11AM on a given day aboard a major commercial carrier. Sam wakes up at 7AM and thinks 'There will not be mass panic aboard my flight because of a bomb threat today.' He believes this because his airline is a reliable carrier, the security presence is strong, and it is unlikely that there would be a bomb threat or a hoax.

But suppose, too, that on that morning, the crazy marketing department at radio station KRZ in Dallas has decided to implement the socially irresponsible decision to promote their new shock radio show, by employing a disc jockey to board the same flight in Atlanta, and loudly scream a false bomb threat while the plane is in flight. The plan is to divert the plane with an emergency landing to Dallas to promote the radio show with sensational media coverage. Given that the stunt will likely be successful, since security cannot detect a verbal hoax, Sam's belief that 'There will not be mass panic aboard my flight because of a bomb threat today' is clearly undermined and may be false. He does not know that he and others won't be fearful later in the day.

But what happens, if by chance, the disc jockey takes a taxi to the Atlanta airport, and on the way to the airport the taxi breaks down with a flat tire. As a result, the disc jockey is unable to board the plane on time and is left behind. In this situation, does the unexpected breakdown of the taxi, now give Sam knowledge in his motel room that he will not panic from a bomb threat? Although conditions 1, 2, 3, and 4a are all satisfied Sam's successful and unimpeded flight to Los Angeles, Sam did not know that he wasn't going to be fearful. Condition 4b is violated. If Sam was to become aware of the planned KRZ radio stunt, either before (7AM) or after the flight (7PM), Sam would admit that his belief was not knowledge, but was a matter of luck. The planned hoax (even if it fails) prevents Sam from knowing that there would not be panic onboard. Unconsidered undermining evidence inaccessible to S, can prevent S from knowing p (in some cases).

Example #5: Does S Know that his Daughter is Attending Class?
Suppose that on a Friday afternoon a father, S, is diligently working at his place of employment and thinks p: his usually trustworthy teenage daughter is presently attending her Friday high school classes. She is conscientious and gets good grades. But S later hears from another father at work that q: a majority of the students in that high school have just participated in a Friday-school-skip-day and are now at the beach.

Does S know that his daughter is in class, given the undermining fact q? Is q a proposition that is irrelevant, and misleading to S's belief that p? Is q a defeating proposition for S? No matter whether S has epistemic access to proposition q, we cannot say that S knows or doesn't know p (his daughter is at school) without assessing S's reasoning and his external circumstances (i.e., the PE conditions). Whether q (a majority of students are participating in Friday skip day) is a misleading fact, and irrelevant to belief p isn't within S's ken, no matter whether S has access to proposition q or not.

But we can say this. Whenever there exists a true evidential proposition q that is (actually) misleading, there are two possible contingent outcomes: Either S has a personally justified true belief based upon relevant reasons, but not knowledge, if S would be unable to dismiss the importance of q as worthy of concern and doubt. Or otherwise, if S was made aware of evidence q, S might dismiss q as misleading and irrelevant, given the strength of a whole body of other evidence (e.g.,his daughter's studiousness) and retain knowledge. By dismissing q, S doesn't lose strong personal justification (and knowledge) that p (his daughter is at school) just because there exists undermining evidence q (Friday skip day) that S is (or isn't) aware of.

Conclusion
With a no-defeaters condition 4b it is maintained that the intuitions of Harman, Lehrer, Dretske, Pappas and Swain are mistaken when they claim that misleading evidence that S is unaware of, should never prevent S from having knowledge. The undermining evidence of a lying mother, a salesman's mistaken testimony, the intended bomb scare of a radio prankster, and the existence of a school skip day might undermine S's personal justification if considered as part of S's evidence, leading to a potential loss of knowledge. This potential loss of knowledge is of great concern to critics of the no-defeaters condition. Critics continue to insist that it is 'inappropriate' or 'unfair' that knowledge is denied in situations where S believes a true proposition based upon truth-connecting evidence. It is claimed that the no-defeaters condition severely reduces what is intuitively taken as knowledge and makes it too difficult to attain knowledge.

In response, it is admitted that the Tom Grabit case, the David Annis case, the air flight case, and the class attendance case are indeed cases of an "epistemic oddity" where odd misleading evidence that may not be considered (nor immediately acccessible) may prevent S from possessing knowledge. But this is the price of existing (true) counter-evidence that S would not be able to resolve. Despite that there are scenarios where a true belief based upon relevant evidence could be undermined by the existence of inaccessible misleading evidence, it can be taken for granted that in most situations, there simply does not exist such misleading evidence that might lead S to doubt p. These cases (#1-#5) are cleverly constructed as thought experiments about the hypothetical existence of misleading evidence. In everyday situations where S is unaware of some misleading evidence q, which might lead to a loss of knowledge, this often has no practical implications. S just has a justified true belief based upon relevant reasons, but not knowledge. The no-defeaters condition is a necessary condition for knowledge even if it sometimes has peculiar consequences.

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