William M. Briggs

Statistician to the Stars!

Truth, Knowledge, Belief, & Gettier Problems

"Hey, you never know."

“Hey, you never know.”

Proof Isn’t All That

The first section can be skipped for those who know what necessary versus conditional truth is.

I recall an anecdote about John von Neumann which had a fellow asking von Neumann for the proof of some mathematical proposition. Von Neumann asked the fellow which of other several theorems the fellow might already know, and he mentioned two, whereupon von Neumann proved the proposition twice, along the two different paths. Implicit in the story is that he could have proven it upon the other paths as well.

We don’t know what this proposition was, so call it X. Since X is necessarily true, we can have knowledge of it, where knowledge, as some philosopher define it, is “justified true belief.” They’d say the justification comes from the proof and the belief comes from us as an act of our intellect.

But does truth come from the proof? Von Neumann showed there were many different ways of knowing a proposition was true, but the multiplicity did not add to the truth of X. X was just true, and always was, regardless whether anybody knew it or believed it. So there is a difference between the truth of some thing and our knowing it; or rather, there seems to be a difference in the justification of our belief the thing is true and its truth.

Let’s clarify. Take our old standby argument with premises E = “All Martians wear hats and George is a Martian” relative to the proposition Y = “George wears a hat.” Y given E is true; that is to say, we know that Y given E is true, that it follows. We may therefore believe Y given E, as a sort of joint proposition, say, Y-given-E. But Y by itself, sans E, is not a necessary truth. Neither is E by itself a necessary truth. But Y-given-E is. Y therefore is a conditional truth, given or accepting or believing or having faith that E.

A necessary truth is one which is true no matter what. Take non-contradiction. It cannot be true that Z = “X is true and so simultaneously is not-X true”. In other (and confusing words), not-Z is true. There isn’t any way to think that Z (except, as many do, by changing it so that Z is no longer Z, and then forgetting they made changes). Why is Z false? Who knows? God made it that way. Why is it true that W = “For every natural number r, r = r”? I have no idea. God made it that way. What is our justification for believing W? Faith? Or is it that we’re too light in gray matter to discover a proof—or, worse, a counter proof?

Actually, we do have reasons for believing not-Z and W. That we cannot think of how Z is true is a dandy reason for thinking it false, and all experience is that for every natural number r, r does indeed equal r. Induction supplies the rest. From our senses to the truth!

All this is just a sketch, which we needed for the real meat which follows.

Get Gettier

A man hears his wife say she bought him a lottery ticket and he thinks to himself, R = “I now have a chance to win”. Unbeknownst to him, his wife was teasing. We know this, his wife knows this, but the man does not. The man accepts his wife’s word, conditional on which he believes R. R given the premise “Wife bought ticket” is thus a conditional truth. A believable truth, too, given he accepts (unconditionally) his wife’s word. R is not necessarily true, however, as is obvious.

Now Edmund Gettier famously claimed there were situations in which a person has a justified true belief, yet that belief did not meet the test of knowledge. Our lottery situation isn’t quite what he had in mind, because everybody would agree that R is a conditional but not necessary truth. To make this a “Gettier problem”, let’s add the premise “The man’s mother bought him a ticket for the same drawing but told nobody”. It is clear that R is now true, say Gettier followers, and the man is should believe it, but his claim doesn’t rise to the level of knowledge because his accepting R is based on his believing something which is false in fact (his wife’s joke).

But R is still a conditional truth to us and to the mother, who know of her actions. R, being contingent, can never be a necessary truth.

Gettier “problems”, I think, are based on forgetfulness. We forget who knows what and we forget what question is being asked of the evidence. To the man, R is conditionally true based on one set of premises, and to us it is conditionally false based on one set of evidence (just the wife’s statement) true based on another set (adding the mother’s). R is never true is the necessary sense. Plus, there are any number of premises which can exist, and which can be believed, that make it conditionally true. Even conditioned on the premise, D= “I, the man in this example, bought my own ticket” R is still only conditionally and not necessarily true.

In short, Gettier “problems” aren’t. This, incidentally, is one of the few cases where symbolic logic helps; I mean, being able to write the story down in symbols makes it much easier to see what goes where and who knows what, so that it is less easy to slip up.

Homework

I’m taking this example from Wikipedia, which (yes) does a good job explaining the set up.

The [justified true belief] account of knowledge is the claim that knowledge can be conceptually analyzed as justified true belief — which is to say that the meaning of sentences such as “Smith knows that it rained today” can be given with the following set of necessary and sufficient conditions:

A subject S knows that a proposition P is true if and only if:

  1. P is true, and
  2. S believes that P is true, and
  3. S is justified in believing that P is true

Recall von Neumann’s example and that X being true and anybody knowing X and the proof or belief of X are not the same thing. And also note that this definition mistakenly forgets to emphasize whether P is a conditional or necessary truth.

Here is a Gettier problem (also Wikipedia):

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 (by the rule of the transitivity of identity) 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.

What has gone wrong?

44 Comments

  1. I’m a little lost but I’ll try anyway:

    His belief that “the man who will get the job has 10 coins in his pocket” is not justified because it is true. Its truth is irrelevant to its justification.

  2. Briggs

    October 15, 2014 at 9:06 am

    Nick,

    Not quite. Try and make a parallel between the homework and the lottery example.

  3. Sander van der Wal

    October 15, 2014 at 9:12 am

    The belief that “Jones will get the job” was not a “justified true belief”. Jones did not already have the job, so Jones knew he did not have the job. The person who wanted to hire Jones did not know it either, because Jones could have refused the offer.

  4. The belief that the man who got the job would have 10 coins in his pocket started with the belief that Jones would get the job and Jones had 10 coins in his pocket. The belief that the “man who will get the job” was contingent on Jones being the one to get the job. The conclusion required both that Jones get the job and that the 10 coins were in Jones pocket. While it did turn out the person with the 10 coins did get the job, this was possible to conclude from the evidence.
    I think it goes something like:
    A then B
    B then C
    A then C
    In this case, A was not true but C was. C was true but not because of the logic involved.

    I would disagree that “W = “For every natural number r, r = r” is true because God made it that way. We made it that way by definition. Same for the Z proposition “Z = “X is true and so simultaneously is not-X true”” The Z proposition leads to the philosophical conundrum of how could God make a rock to heavy to lift, based on the idea that God can do anything, ie is omnipotent. One either has to exempt God from logic or conclude that we created logic to match our world, not God’s.

  5. “this was Not possible to conclude”

    (Really must improve proofreading skills—making myself a note which I will promptly lose!)

  6. Sheri, I would agree with you, and I think I goes further. Given the conclusion A->B, B->C and A->C and knowing A was false but C was true – then we can see that B->C is not needed. I.e. it could be that Jones does not have 10 coins but 15 and Smith only thinks he had 10.

  7. Theldoria: So A and B could have false, because Smith erroroneously believed Jones had 15. Back to the lottery example, right?

  8. I will tell you what is wrong: That fact that Jones has 10 coins to begin with. Why hasn’t he paid his “fair share”? Smith figured the rich just keep getting richer. Smith does not think his 10 coins make him rich, but because someone else does have, that makes Jones rich. Hence Jones will get the job because the rich always catch the breaks in life. It’s all about perception.
    So Smith gets the job but it is minimum wage. He must quit to maintain his better lifestyle on government assistance. Therefore it is as if Jones got the job to begin with because Jones will be hired when Smith quits. Thus proving Smith right that “Jones will get the job”. And all over 10 coins.
    As a man thinks in his heart so is he… Our lives will conform to the words we speak.

  9. The following clear exposition of the Gettier case by Massimo Pigliucci (one of my favorite bloggers; see also http://scientiasalon.wordpress.com/ ) springs to mind!

    http://rationallyspeaking.blogspot.com/2013/07/progress-in-philosophy-gettier-case.html

  10. Please explain: God does not have a body (per Aquinas) AND Jesus has a body (per Holy Eucharist) AND Jesus is God (per Nicean creed)

  11. Briggs

    October 15, 2014 at 3:58 pm

    Hans,

    Let’s see what kind of a student you are. The answer to your question is contained here, in full, from Q1 to Q43. Be sure not to commit the If-I-Don’t-Understand-The-Answer-The-Answer-Must-Be-Wrong-Therefore-I-Must-Be-Right Fallacy. That’s the same fallacy used by Philosophy 101 students everywhere. Also don’t fall pray to the Internet’s favorite, If-I-Can’t-Understand-It-In-Less-Than-One-Minute-It-Must-Be-Wrong fallacy, also beloved of Probability 101 students.

    Steve,

    Your answer is only partial. You failed to account for Jones’s federally protected victim category, which Smith knew about in advance, hence his initial judgement. Jones is planning on suing.

    JH,

    At least Pigliucci is consistent in getting things wrong. Pragmatism? No thanks. Take it from me, a qualified expert.

    Everybody else,

    From what viewpoint are each of Smith, Jones, and we looking? Which premises do we each condition on? Who knows what and when? And what does, to everybody, “the man” mean?

  12. Hans: That’s why I don’t include God in the logic thing. We made logic up and it does not apply to God. (Briggs is free to disagree, of course.)

  13. I agree with Sheri, at least if I am understanding her correctly. The way I would put it is that Jones’ belief is not justified in the “logical” sense, and that one reason people get confused by these examples is because they interpret “justified” as meaning something more like “excusable” or “blameless”.

    In Pigliucci’s article linked to by JH, Sheri’s response is I think captured in what he calls the “no false premise” condition, but the only case he proposes to counter that clearly *does* contain a false premise – namely that when the man saw what he thought was a dog he was in fact seeing a dog.

  14. No Sheri the believers made God up, so logic doesn’t apply to God, as for any character in literature. Dogma overrules logic.

  15. Briggs. If i can’t understand it, then it is false reasoning, or a dogma.
    The right answer was: “you must believe it.”
    You cannot prove God, “you must believe it”

  16. Hans: And you know they made this up how? Where you there when the idea of God originated? (I’m guessing that you think the idea of God was like the Big Bang—came out of nowhere?) Did you witness the lack of cruxifiction of Christ? Do you have a time machine that we don’t know about? Please, let me in on your secret source of irrefutable hard evidence that God was made up by people.

    Dogma overrrides logic in more than one direction, it seems. (Pretty arrogant that if you can’t understand it, it’s false reasoning or dogma. Do you have a card with your certification as the “dogma determiner extraordinaire?)

    (Yes, you do annoy me.)

  17. Briggs, I’ll take a mild exception to two of your statements:
    First, “X is true and so simultaneously is not-X true”.
    I believe that quantum logic violates this principle.. see
    http://plato.stanford.edu/entries/qt-quantlog/ and Andrej Grib’s article in “quantum cosmology and the laws of nature”
    http://www.ctns.org/books.html click on the book “Quantum Cosmology…” and then to the right on the chapter summary by Grib.

    Second, referring to necessary truths “God made it that way.”
    I don’t think most theologians or philosophers would agree with you. According to books I’ve been reading about Free Will and Divine Providence, God’s Foreknowledge, there are three classes of truth or knowledge: ” Natural Knowledge, necessary (truth) independent of God’s free will (e.g. 2+2 =4); Middle Knowledge, contingent (truth) independent of God’s free will (might happen, but won’t); Free Knowledge, contingent, dependent on God’s free will (will happen).”
    This quote was taken from “Divine Providence–The Molinist Account” by Thomas Flint.

  18. I screwed up… your quote “X is true and so simultaneously is not-X true” would not hold according to the law of non-contradiction, but can hold in quantum mechanics.

  19. OK, quantum logic doesn’t necessarily violate the law of non-contradiction, rather has different distributive laws ….I looked at Grib’s paper again.

  20. Briggs,

    “Take it from you, a qualified expert!” Hahahaha. Thanks for the laugh.

    (Please read Footnote [3] in Massimo’s post. )

    Have you ever heard the Chinese idiom “A blind eats dumplings”?

  21. Sheri, of course somebody first contemplated the existence of a God. But orthodox Roman Cathocism is a non sequitur. Here is a list of Roman Catholic Truths:
    Mary ascended with her body into heaven: denied by protestants
    Consecrated bread is the body of christ: denied by protestants
    Priests shall not marry: denied by protestants and eastern orthodox
    Jesus is God: denied by muslims
    Jesus is the messiah: denied by jews
    there is a final judgment: denied by buddhists
    There is only one God: denied by hinduists

    It’s the same bait and switch exercise you see when alarmists claim that co2 is a greenhouse gas implies a forthcoming thermageddon.

  22. Sheri,

    Did you witness the lack of cruxifiction of Christ?

    Did you witness the non-Advent of Lord Krishna?

    Dogma overrrides logic in more than one direction, it seems.

    A universal human trait, he says dogmatically.

    Yes, you do annoy me.

    I didn’t get the feeling he was trying to appease you.

  23. Hans: That someone first contemplated the existence of God is not self-evident, though you are claiming it is. It is equally probably that God first revealed Himself to man. Logically equivalent and equally provable or not provable, as the case may be.
    I was not addressing Roman Catholic faith, just the existence of any god.
    It’s not bait and switch–no one dropped in one idea and shoved another in it’s place. Maybe a better analogy? (If you are trying to discuss one particular religion versus the existence of any god, then it’s you who are baiting and switching. My comments are generic.)

    Brandon: First question makes zero sense.
    Second, kind of cute, but still.
    Third, please don’t speak for Hans. He’s quite capable of doing so himself. (That was one of your “head ’em off at the pass” moves anticipating Hans would tell me I was getting angry with him. I apparently misjudged Hans’s response.)

  24. Sheri, Let’s look at a biblical account of God revealing Himself to man:
    “God told me to kill my son, but a passer-by intervened” (Genesis 22)
    Now how would you interpret this, if this would happen today? (Yes it does happen all the time today) Would you say: “That is God speaking”?

    Monotheistic religions evolved from polytheistic religions, did you know that Yahweh used to have a wife called Asherah?

    I’m not attacking you, just your thoughts 😉

  25. Hans: I made it very clear I am not here to debate a specific religion. My statement was concerning the fact that not believing in God is virtually identical to believing, in the sense that neither can be proven to be true nor disproven. Both are taken on faith that the history involved in the beliefs and the beliefs are not verifiable in any way.

    Specific religions have problems, yes. So does a great deal of science, or so called science. And politics and economics and virtually all of life. I am not going to debate a specific religious belief with you, at least not on this blog. It generally comes down to a “did you know” fest and yes, I generally I do know. It’s also pointless, since you have faith in your science and that there is no God based on your intrepretation of history. Nothing I type will matter anyway. A complete waste of time and we both know it. All I’m saying is that both atheism and belief in God rely on things that can never been shown to be true in the scientific sense.

    I’m not attacking you, just your thoughts? How am I not my thoughts? (Actually, you’re not attacking anything I believe with your religion quiz, so it really doesn’t matter to me. It’s just ridiculous to try and separate a person from their thoughts and actions—it’s who they are.)

  26. Sheri, Not-believing is not a belief. Consider the tooth-fairy. Not believing is the default. Believing is a state like pregnancy.

  27. Briggs

    October 16, 2014 at 1:23 pm

    Hans,

    Not-believing is a belief. Consider not believing in the Tooth-Fairy. That is the belief in “There is no Tooth Fairy.”

  28. Hans: You are still not understanding my point. You clearly stated that “the believers made God up”. You are explaining the origin of God, are you not? A belief that God is a myth. A belief.

  29. Sheri,

    The Krishna comment was meant to illustrate what you wrote later:

    A complete waste of time and we both know it. All I’m saying is that both atheism and belief in God rely on things that can never been shown to be true in the scientific sense.

    Amen, sister. We agnostics are the only scientific ones … except those of us who believe in AGW. Amirite? I’m right.

    You’re right, I should not speak for others and usually don’t. Comment retracted.

  30. Brandon: Okay, I kind of understand the Krishna comment. Yes, agnostics are the only scientific ones concerning the existence of God. As for AGW, well, let’s not go there! 🙂

  31. Sheri, you believe that all Gods exept one are myths. We only differ one god. Why don’t you believe in the other Gods?

    Briggs, non-believing is the default state, the tabula rasa. Unbelieving is the state of enlightenment when you finally grasp the logical fallacies of the believers. I am patient, you’ll get there in the end.

  32. The existence of a tooth fairy is a poor example because it is hard to find an adult who is in the state of non-belief as opposed to belief in the negation, but I believe it is wrong to confound those two states in general. For one thing, non-belief includes the state of acknowledged ignorance – such as that which I currently have with regard to the Riemann Hypothesis.

    If a “belief” is defined as a level of certainty that is absolute, then I don’t think I have any, but if it is defined as an opinion which is held with sufficient confidence to cause action then I have many.

    With regard to God (or gods) I can believe (with a fairly high degree of conviction) in the literal falsehood or meaninglessness of every proposition that has ever been made considering it’s (or their) nature or intentions without disbelieving in the possibility that those propositions have some valid figurative interpretation relative to some aspect of our experience.

    But let’s come back to Briggs’ seemingly favourite topic. I do not believe in AGW (at least of an extent that will certainly cause major economic and humanitarian disasters), but am sufficiently impressed by some of the arguments for its likelihood that I consider significant preventive action to be more prudent than inaction or denial. In other words, although I don’t believe in the proposition “catastrophic AGW will happen”, I do believe (albeit without complete conviction) in the proposition “If the probability of catastrophic AGW were computable then then the expected harm from ignoring it would exceed the expected cost of any measures yet being seriously considered to counteract it.”

  33. Montague (C. M. Boyd)

    October 16, 2014 at 3:03 pm

    Ah! This is the topic I wrote on in a philosophy class last year; I couldn’t believe (I could believe, actually) that wikipedia didn’t have this most basic and obvious solution… I mean, this is easy to explain to highschoolers, if not middleschoolers or younger children.

    For the belief to be justified it must hold something false to be true, thus failing the true belief requirement. For the true belief to not entail a false belief as a premise, it would have to be an unjustified belief. Ergo, the Gettier problem does not present a (truly) justified, true belief.

  34. Montague,”For the belief to be justified it(?) must hold something false to be true”
    sounds as if you are saying the *belief* must entail a falsehood, but I think what you are trying to say is that, in a Gettier problem, the purported *justification* always involves a false premise (an so is not in fact a valid justification).
    I am surprised that you didn’t see this in Wikipedia as it is famous as the “false lemma” objection.

  35. …or did you mean to say that “For a belief to be justified it must NOT hold something false to be true”? and so that, in a Gettier problem, where the belief always turns out to be false, it cannot have been properly justified. That is fair enough, but then the point of the problems is to explain exactly how the purported justification was wrong.

  36. Montague (C. M. Boyd)

    October 16, 2014 at 3:43 pm

    You’re right, both about my poor english, and what I intended to say. My apologies.

    I suppose I didn’t see it because the false lemma objection is somewhat glossed over in the early portion of the article, with a supposed rebuttal in this form of Gettier problem:

    “After arranging to meet with Mark for help with homework, Luke arrives at the appointed time and place. Walking into Mark’s office Luke clearly sees Mark at his desk; Luke immediately forms the belief ‘Mark is in the room. He can help me with my logic homework’. Luke is justified in his belief; he clearly sees Mark at his desk. In fact, it’s not Mark that Luke saw; it was a marvelous hologram, perfect in every respect, giving the appearance of Mark diligently grading papers at his desk. Nevertheless, Mark is in the room; he is crouched under his desk reading Frege. Luke’s belief that Mark is in the room is true (he is in the room, under his desk) and justified (Mark’s hologram is giving the appearance of Mark hard at work).” (wikipedia)

    What Wikipedia does not provide immediately thereafter is an obvious reply from the JTB objector, instead giving a convoluted response:

    “To save the “no false lemmas” solution, one must logically say that Luke’s inference from sensory data does not count as a justified belief unless he consciously or unconsciously considers the possibilities of deception and self-deception. A justified version of Luke’s thought process, by that logic, might go like this:

    1. That looks to me like Mark in the room.
    2. No factor, right now, could deceive me on this point.
    3. Therefore, I can safely ignore that possibility.
    4. “Mark is in the room,” (or, ‘I can safely treat that as Mark.’)
    And the second line counts as a false premise. But by the previous argument, this suggests we have fewer justified beliefs than we think we do.” (Wikipedia)

    The obvious solution is merely to point out that, rather than convoluting the issue with “Inception” style doubt-series, we can merely posit, from logical necessity, that Luke thinks it is Mark he sees, which is indeed a false premise, and of the very same sort we have already pointed out in the Farmer Brown and the Ten Coins problems, etc.

    It’s surprising, indeed, that Wikipedia doesn’t seem to take the JTB position seriously (or is that the fault of modern philosophers, whom it is the business of Wikipedia to chronicle?)

  37. I think the point is that no “justification” of a belief which is based on some perceived fact is free from the unproved assumption that the perception is accurate. By making that assumption we can think we know something but we might be wrong. (But I don’t know why philosophers seem to consider that a problem.)

  38. Sheri,

    Once I learned how to say “I don’t know” without flinching, being agnostic about God got somewhat easier. But it’s truly difficult to suspend belief — I do actually believe extant religions are man made, the Krishna comment is part of the supporting logic I use to justify my belief. OTOH, I lean more toward believing there is some sort of God than not. More of a hope than anything. Conclusion: I’m as human as anyone.

    AGW is the “best” of politics and religion, innit? I can’t help but share these two headlines from today:

    Feds: Don’t expect winter to be polar vortex redux

    AccuWeather predicting the polar vortex will pay us another visit this winter

    Seriously?!?

  39. Brandon: Sadly, I do think they are serious. 🙁

  40. Do you “seriously” take headlines as accurate summaries?(if so that really is sad)

    From the NOAA : ” It just won’t persist as much as it did last year, when extreme weather seemed to be stuck in place”
    and from AccuWeather:” it’s not going to be the same type of situation that we saw last year – not as persistent, not as long-staying as last year”

    Admittedly, the NOAA predictions are “slightly” warmer than the private ones, but both are saying that the “polar vortex” will probably swing down from time to time (as it usually does) but that it will also probably not “lock in” in a replay of what happened last winter.

    (and *neither* article attempted to make any connection, in either direction, between the PV and AGW)

  41. Mr. Briggs:

    None of your correspondents seem to recognize that all “knowledge” is contingent: built on a series of beliefs none of us ever examine very closely…

  42. @mysterian, I think I know that what seems true to you is wrong .

  43. Sheri,

    Sadly, I do think they are serious.

    It would be nice if journalists could get their headlines straight. Contradictory stories like that are infuriating.

  44. Alan,

    Indeed the bodies of those two stories ended up on the same page, but the headlines framed them differently. I don’t think it’s coincidence that first headline hails from San Francisco while the second one comes from Buffalo.

    One thing I’m sure of, if there is a significant cold snap this winter in the lower 48, we will see dueling headlines with global warming mentioned explicitly. Why such events should be necessarily incompatible with global warming is a question I can’t recall ever being asked.

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