It's deductively valid to conclude that no ticket will win when you have evaluated whether each and every ticket wins or does not win, and all evaluate as not win. IE P1,P2,...,Pn derives P1 & P2 & ... & Pm when and only when m is less than or equal to n. IE, when each each ticket has been observed and evaluated as not winning (also a purely technical thing with the indices on the left and right of the sequent, must pick out the same tickets). If there are unobserved tickets, that makes m>n and it's no longer a deductively valid inference.

If there are unobserved tickets - or equivalently if the subject does not know the set of observed tickets is an exhaustive set containing a winner - the subject reasons differently. In the cases where the subject has no reason to believe there's not exactly one winner in all the tickets or they are given a subset of tickets they can evaluate how likely the subset is to contain any number of winners if they also know the probabilities that each ticket wins.

Ensuring that there is exactly one winner requires giving out the entire set of tickets and that the entire set of tickets contains one winner.

How the subject can reason depends heavily on their beliefs about the lottery and whether those beliefs are true; also whether the subject is equipped with the reasoning tools to evaluate probabilities in general. If the subject knows the mathematical structure of the lottery and their sub-sample of tickets they can assign probabilities. If they don't, they can't.

Note that the belief that 'I'll never win' is deduced from knowledge of the lottery's structure. A set of true enough empirical/contingent facts about the lottery set up and their mathematical implications. Rather than requiring the purchase of many tickets to furnish the belief with evidence.

Assuming a lottery of n tickets, the premises are:

P1. Ticket 1 won't win
P2. Ticket 2 won't win
P3. Ticket 3 won't win
...
Pn. Ticket n won't win

From this we can deduce:

C1. No ticket will win

It's a valid inference. — Michael

The conclusion doesn't follow. Validity is a component of deductive arguments not inductive arguments. I think the way one should look at this argument is the following:

As each ticket is bought it increases the probability of winning, so the conclusion that follows from P1, P2, P3...Pn does not lead to the conclusion that "No ticket will win." It leads to the conclusion that one or more tickets will win. In an inductive argument, as the number of supporting premises increases so does the strength of the conclusion (e.g., it increases the probability of winning). Obviously if you buy ten tickets it would be a weak inductive argument to say that the next ticket you buy will be a winner. However, if I buy 100 million tickets, then you strengthen the probability of justifying the conclusion that you will win.

Are you justified in believing that if you buy 100 million tickets that you will win? It depends, if the chance of you winning is 1 in 110 million, then you are justified. Even if you lose, you were still justified in believing you would win, because of the strength of the justification. Most of what we claim to know is based on inductive reasoning. Inductive reasoning doesn't require that you know with absolute certainty. Not that you're claiming this, I'm just making further points.

Let's say that 100 people have each picked out a ball. Given the high odds, I am justified in believing that Person 1 doesn't have the black ball. — Michael

Because you cannot rule out the possibility that any of these people have the ball you are not justified in believing that any one of them doesn't have the ball. Until you rule out the possibility that X is the case you are not justified in believing X is not the case. The justified belief is X is probably not the case. Likewise, with the lottery, if one has a ticket, the belief that this person will not win is not justified. That's why people buy tickets, the belief "I will not win" is not justified despite the low odds. However, "I will probably not win is justified".

In Gettier's original example, a person called Smith is applying for a job. Another person, Jones, whom is known to have 10 coins in his pocket, is applying for the job as well, and Smith (for a justified reason) believes Jones will get the job. This is the belief a. The conclusion b is that the person who gets the job has 10 coins in his pocket. What happens is that Smith himself gets the job, but also, although he didn't know this, had 10 coins in his pocket. — BlueBanana

To the above I wrote:

Uh... No. No. No.

Read the paper...

Then came this...

↪creativesoul Nitpicking, the case is practically the same. — BlueBanana

Uh, no it's not. The differences matter tremendously. That's why the charge of 'nitpicking' reveals neglect on your part to take note of them and why they matter.

No skin off my nose. I was enjoying the discussion you were having about fallibilism/infallibilism...

The conclusion doesn't follow. — Sam26

It does. If for each ticket "this ticket won't win" is true then no ticket will win.

Uh, no it's not. The differences matter tremendously. — creativesoul

What's really the difference? How do they matter? Smith justifiedly believes in one outcome and the other one happens.

Because you cannot rule out the possibility that any of these people have the ball you are not justified in believing that any one of them doesn't have the ball. Until you rule out the possibility that X is the case you are not justified in believing X is not the case. The justified belief is X is probably not the case. Likewise, with the lottery, if one has a ticket, the belief that this person will not win is not justified. That's why people buy tickets, the belief "I will not win" is not justified despite the low odds. However, "I will probably not win is justified". — Metaphysician Undercover

I disagree with your claim that justification requires certainty, as we discussed earlier. If someone shows me some ID and if the ID shows them to be 18 then I am justified in believing that they are 18, even though it isn't certain as the ID might be fake.

As a rough definition I would say that my belief that X is true is justified if, given the evidence available to me, there is a high probability that X is true. In the case of the lottery example, given my understanding that the odds that any given ticket will win is 1/n, and given that 1/n is a very low probability, my belief that any given ticket won't win is justified.

It does. If for each ticket "this ticket won't win" is true then no ticket will win. — Michael

While it's true that for each ticket that ticket won't win, it's not true that if you buy P1, P2, P3,...Pn that no ticket will win. That's like saying if I buy every possible ticket, no ticket will win, again it doesn't follow.

... it's not true that if you buy P1, P2, P3,...Pn that no ticket will win. That's like saying if I buy every possible ticket, no ticket will win, again it doesn't follow. — Sam26

That wasn't my argument, though. My argument was that if P1 won't win and if P2 won't win and if P3 won't win ... and if Pn won't win then no ticket will win.

There are different degrees of certainty which are appropriate for the different fields of study. Aristotle explained this in his Nicomachean Ethics. Theoretical knowledge requires a higher degree of certainty than practical knowledge. What is being discussed in this thread is knowledge in theory, justified true belief. When you bring an example such as you have, saying I would be justified in believing X, in Z situation, you conflate practise with theory. You, knowing X, in Z situation, is an instance of practical knowledge, when justified true belief is concerned with knowledge in theory, epistemology.

The problem with introducing examples of practise into theory is that theory does not deal with the particularities of the various situations. You can never specify all the particulars of a situation of practise, in a theory. So in reality you cannot claim to be justified in your belief that the person is over 18, just because they showed you ID. The ID might be clearly fake, or the person might be a 5 year old with an older person's ID. Such examples, which attempt to introduce practise into theory are just examples of category mistake.

Perhaps the real issue of this thread is a failure to distinguish between knowing-how (practical knowledge), and knowing-that (theoretical knowledge). Such a failure will lead to the conclusion that there is no such thing as certainty because the skeptic can always find an example to make mistake possible.

That wasn't my argument, though. My argument was that if P1 won't win and if P2 won't win and if P3 won't win ... and if Pn won't win then no ticket will win. — Michael

If all your saying is that no one ticket will win, obviously that's true, but the argument doesn't appear to be saying that.

The argument is:

P1. Ticket 1 won't win
P2. Ticket 2 won't win
P3. Ticket 3 won't win
P4. There are 3 tickets
C. No ticket will win

(Except with more than 3 tickets, obviously).

What about the person who just knows they have the winning ticket?

Have you ever seen the game show "Deal or No Deal?" Fascinating exercise in probability which in its heyday sparked long math debates on the internet. You could always see someone on there turning down hundreds of thousands of dollars because they just knew they had picked the case with a million dollars in it.

Those people drove me nuts. Only time I was tempted to yell at the television. "No you don't! You don't know any such thing! It's random! Take the money!"

The more we talk about this example, the more I'm inclined to agree with @Metaphysician Undercover that you are no more justified in claiming that a given ticket will not win than you would be in claiming that a given ticket will.

In the lottery case, the urge to say "Just do the math" is almost overwhelming. 1 in 3.2 billion is a small number but it's not 0. It's just not, and saying that it is is just wrong.

What about the chair I'm sitting in? Is there a vanishingly small but non-zero chance it will disappear as I sit here, or turn into pudding, or whatever? Maybe? The difference is that knowing how lotteries and probability work, we know the mechanism responsible for uncertainty. Even if I'm a good Humean and accept that my knowledge of this chair is only probable, I think we should still call my beliefs about it justified because an unknown mechanism having unforeseeable consequences can have no place in our reasoning. Only probable, yes, but still justified because the only factors I have not taken into account are factors I cannot possibly know anything about.

Gettier cases frequently rely on coincidence. We all have lots of experience of coincidence, but no experience predicting the occurrence of a coincidence. Hence, the subjects in Gettier cases have beliefs that are indeed only probable, but still justified.

Then what of my example of someone showing me their ID? If it tells me that they're 18, am I justified in believing that they're 18? Unlike your example of the disappearing chair and like the lottery example, the mechanism responsible for uncertainty is known to us.

I would think it acceptable to say that my belief that they are 18 is justified because the probability that it's a fake is very low. Are you saying that such a belief isn't justified – indeed that I'm not justified in believing anything I'm told because it relies merely on the probability that I'm being told the truth? That strikes me as untenable.

No I'm inclined to agree with you, because I take the justification to be a practical thing. It's a matter of applying your knowledge of circumstances in the way members of your epistemic community do. In retail, for instance, you exercise slightly more caution accepting large bills than you do as the customer at a bank because it's a known fact that people try to pass counterfeit bills at stores. If you have no reason to think someone's ID is fake, and if members of your epistemic community would as a rule see no reason to think the ID is fake, then I think you're justified in assuming the ID is genuine, even though you know there is a non-zero chance that it's not.

Then what makes the ID example different to the lottery ticket example? We're justified in believing that the ID isn't fake because the probability that it is is high, but we're not justified in believing that the lottery ticket won't win even though the probability that it will is high?

I think you're being too casual with the word "probability" there, is all.

If there are 500 million government-issued ID cards in the US, and an additional, say, 200,000 fake IDs, does that mean the odds of any given ID being a fake are about 3.8%? No, of course not. Because IDs don't randomly appear. The driver's license in my wallet came directly from the Georgia Department of Transportation; I don't have to wonder if it might be a fake.

But if you're the owner of a bar, and some kid presents you with an ID showing that they're of legal drinking age, then you might have a responsibility to look closely at that ID. As with counterfeit, that's a situation where people are known to present a fake ID and we know why they do it.

EDIT: dropped some 0's, but never mind that.

The argument is:

P1. Ticket 1 won't win
P2. Ticket 2 won't win
P3. Ticket 3 won't win
P4. There are 3 tickets
C. No ticket will win

(Except with more than 3 tickets, obviously). — Michael

This argument I can agree with, but it's more complicated than that. Here we're talking about what's probably the case, and the inductive argument above is weak, so the conclusion that no ticket will win follows. However, if my argument is based on P1, P2, P3...P4 (P4 being the total number of tickets bought, viz., 1.2 x 10^8), out of a possible number of possibilities of 1.5 x 10^8, then what conclusion do you think follows? It certainly isn't that C. No ticket will win. It's then probable (my e.g.) that you have a winning ticket based on the number of tickets you've bought in relation to the total number of possible combinations.

The question then arises, "Are you justified in believing that you have a winning ticket?" The answer is, you are justified in believing that you have a winning ticket, i.e., it's based on what's probably the case. Much of our knowledge is like this, I can say, "I know..." based on what's probably the case, not what's necessarily the case. It's another use of the word know, this somewhat connects to what's already been talked about in this thread.

Part of the problem with Gettier is that there seems to be a disconnect between how we define knowledge (JTB for e.g.), and a claim to knowledge, as you know they are very different animals.

This argument I can agree with, but it's more complicated than that. Here we're talking about what's probably the case, and the inductive argument above is weak, so the conclusion that no ticket will win follows. However, if my argument is based on P1, P2, P3...P4 (P4 being the total number of tickets bought, viz., 1.2 x 10^8), out of a possible number of possibilities of 1.5 x 10^8, then what conclusion do you think follows? It certainly isn't that C. No ticket will win. It's then probable (based on my e.g.) that you have a winning ticket based on the number of tickets you've bought in relation to the total number of possible combinations. — Sam26

I'm not sure how this relates to my point, though.

The argument is:

P1. Ticket 1 won't win
P2. Ticket 2 won't win
P3. Ticket 3 won't win
P4. There are 3 tickets
C. No ticket will win

(Except with more than 3 tickets, obviously). — Michael

All I'm saying is that the above argument follows, but the conclusion would change based on the number of tickets bought. Originally you included P4 as Pn, which would mean a range of possible numbers, and its this range which would change the conclusion. The conclusion that no ticket will win depends on the range of Pn, and if the range is high enough, then your conclusion would be false. It's true, given this e.g., but it may be false given your other example. If this doesn't relate to your point, then I'm not sure what you're saying.

All I'm saying is that the above argument follows, but the conclusion would change based on the number of tickets bought. Originally you included P4 as Pn, which would mean a range of possible numbers, and its this range which would change the conclusion. The conclusion that no ticket will win depends on the range of Pn, and if the range is high enough, then your conclusion would be false. It's true, given this e.g., but it may be false given your other example. If this doesn't relate to your point, then I'm not sure what you're saying. — Sam26

This was my original argument:

Assume a lottery of n tickets.

P1. Ticket 1 won't win
P2. Ticket 2 won't win
P3. Ticket 3 won't win
...
Pn. Ticket n won't win

From this we can deduce:

C1. No ticket will win

It's exactly the same as the above, except there are n tickets rather than 3. The reasoning is the same regardless of the number of tickets.

What about the chair I'm sitting in? Is there a vanishingly small but non-zero chance it will disappear as I sit here, or turn into pudding, or whatever? Maybe? — Srap Tasmaner

I so far find the concept of possibility to be very obnoxious. To me it seems to be equivocated all over the place within realms of philosophy but, like so many others, I haven’t been able to satisfactorily make peace with it on a philosophical level—i.e., to figure out how it is equivocated.

To illustrate (via what to me a semi-humorous example): Is it possible to be struck by lightning during the time when bit by a shark while holding a winning lottery ticket in one’s pocket? This form of possibility to me is a different beast than that of a vanishing chair. I can’t come up with any contradictions involved in such a thing happening. Yet, without crunching numbers, the improbability of this first mentioned occurrence is so extreme as to make the occurrence utterly noncredible. So here I conclude it to be a justifiable but noncredible, existential possibility. There’s so many of these that it’s not worth mentioning.

With the vanishing chair, however, I’m venturing that contradictions between believed truths would need to occur for this possibility to be valid. But this would falsify at least some of these believed truths (a set likely including laws of nature, etc. together with that of the vanishing chair possibility). Therefore, such conceivable possibilities might likely be invalid due to contradictions.To me this comes close to the BIV hypothesis being upheld as valid possibility by some; though not by me and I presume not by most others.

In addressing a lottery ticket on its own, that there is some possibility of winning a lottery—as MU has argued—is justifiable (a lot more so than the possibility of lightning + shark + lottery ticket, since the former is more probable than the latter). But then we venture into the likewise nebulous world of credibility. Whether or not the possibility of winning a lottery ticket is credible will depend on the character of the individual. Las Vegas is all about people finding such possibility credible.

Here, you can have a series of premises stating "it is credible that ticket n might (/will?) win".

Don't know how the altered premises would work out. I mainly wanted to draw some distinction between possibilities which we appraise to be validly noncontradictory and those conceivable possibilities which stand a good chance of being contradictory if enquired into deeply enough, this apropos the vanishing chair example.

I think I'm largely with you here, with the proviso that we do have experiences that falsify our beliefs sometimes. (And of course science sets out to produce just such experiences.) So while self-contradiction might rule out a possibility, contradicting some belief or beliefs of ours does not.

And I come back also to the sense of an epistemic community with shared norms of evidence and rationality. Those are, similarly, not absolute.

So while self-contradiction might rule out a possibility, contradicting some belief or beliefs of ours does not. — Srap Tasmaner

I'm in agreement. While this isn’t a formal argument, one could I think devise an argument against BIV along these lines for example: BIVs are a possibility of what ontically is resulting from all first-hand experiences of what ontically is; yet the BIV hypothesis contradicts the reality of all first hand experiences in fact being of that which is ontic; hence, either BIV or not-BIV where BIV results in logical contradictions and not-BIV does not. Therefore, not-BIV is justified whereas BIV is not. Anyways, something along these lines would make, I believe, possibilities such as that of BIVs invalid. (if there are rebuttals to this informal argument, I'm not going to try to more formally uphold it)

Then what makes the ID example different to the lottery ticket example? We're justified in believing that the ID isn't fake because the probability that it is is high, but we're not justified in believing that the lottery ticket won't win even though the probability that it will is high? — Michael

The ID example is different from the lottery example. One consists purely of odds, the other there is a person and ID to be judged. What justifies "the person is over 18" is the time of birth until now. Strictly speaking, the ID does not justify, it's a substitute, a representation of the time of birth, which serves the purpose, in practise.

The criteria for justification is specific to the particular situation at hand, and the person judging, and that is why I could not answer creativesoul's request for the criteria for justification. You use the ID, and appeal to the "odds" that it is correct, in an attempt to justify your claim to the time of birth, just like you use odds in your attempt to justify your claim to "this ticket will not win". Whether or not your claim is actually justified is a matter of the discretion of those who judge your argument. But there is more to judging ID than an appeal to odds, and since you have not produced the person, and the ID, so that we can judge the authenticity, we cannot judge your justification on this matter. We are familiar with the odds in lotteries, and some participants here think that your claim "this ticket will not win" is justified, I do not. All you demonstrate here is a difference between those who buy lottery tickets and those who do not.

P1. Ticket 1 won't win
P2. Ticket 2 won't win
P3. Ticket 3 won't win
...
Pn. Ticket n won't win

From this we can deduce:

C1. No ticket will win — Michael

All the premises are justified, but that doesn't mean they're true. You forgot

P(n+1). One of the P(1,2,...,n) is false.

I know.