I'm gonna be famous...

8-)

All you've did is criticise the tracking argument of epistemic closure, no?

So I don't see Smith as overstepping the bounds of reason and landing in a puddle of nonsense. I see him as a victim of chance. Something extraordinarily unlikely happens, and it will challenge his otherwise orderly process of belief formation. — Srap Tasmaner

Well let me ask you a question in return. If you have a reasonable belief p, and a reasonable unconnected scepticism q (say p - that aspirin is an effective painkiller, and q - that Bluebeard's treasure is buried on Easter Island), what is to be gained by forming the disjunction, (p v q) ? How does S advance his knowledge, or understanding or in any way profit from forming his disjunction? Does it enable a test of p, or the building of a deeper theory or something?

It's not a calculus, merely annotation. In propositional logic, "probably p" or "believed p" does not add up to p, but to (p v ~p) — unenlightened

(p V ~p) is mere tautology.

(p V ~p) is true, whether or not p.

probably p is an informative and definitive claim.

probably p can serve as a reasonable justification for further claims or actions, at least in some contexts.

probably p has its own formal implications:

IF (probably p) THEN (probably (p V q)).

I've never studied probabilistic logic, but I reckon the probability that (p V q) can't be any lower than the probability that p, and can't be any lower than the probability that q.

Here I think we are getting closer to what is going on in the Theaetetus. — Banno

It's been a while since I've looked at that dialogue. In my recollection, it's a conversation about knowledge, not certainty, though of course I might be mistaken. Perhaps you'll do me the favor of correcting my memory, by reminding me how the concept of certainty figures in that fine old legend.

Or is there some other way in which we're approaching Theaetetus?

One way we can be certain is when we take things as the bedrock of our discussion. In this sense, doubt is dismissed as not having a place in the discussion. — Banno

I wouldn't call that certainty, just a framing assumption.

We take claims for granted temporarily, for the sake of argument, for the sake of conversation, and thereby rule out whole regions of discourse for a while, to make room for the theme we've agreed to address.

A beautiful convention, without which there's little hope of progress in a diverse discursive community.

So, for example, this is not a discussion about the comparative benefits of diesel and petrol engines, and thinking it so is to misunderstand what is going on. Or, to use the all-pervasive example, one does not doubt that a bishop moves diagonally while playing chess. — Banno

That sounds right to me.

I might add "Ordinarily one does not doubt that a bishop moves diagonally while playing chess." For even such a simple rule might be doubted in some contexts: while learning the rules of the game, in a moment of confusion, in an altered state of consciousness, in a context of skeptical doubt....

The problem here is the philosopher's game of putting "absolute" in front of "certainty" and thinking that this means something. Outside of philosophy, minds like ours always or almost always certain. Few folk check that they have an arm before they reach for the fork. It's not the sort of thing that one doubts, outside the philosopher's parlour. — Banno

I take it "absolute certainty" means something along the lines of: 100% certain, beyond the possibility of doubt, beyond the possibility of error, not possibly false, indubitable in any discursive context whatsoever.... I agree the concept seems fanciful. I'm inclined to say that nothing is absolutely certain in this sense, and that the term is another one of those philosopher's fictions that make a laughing stock of their art when it's employed as anything more than a foil.

Thankfully practical certainty is the ordinary condition of human agents in the course of their ordinary affairs. So far as I can see, this only means that normally we're sure enough to act without question, without further checking, ascertaining, assuring, proving, testing, confirming, investigating.... Without any practical doubt, without any practical reason for doubt.

That natural confidence is often disrupted by circumstances. We learn by experience that our perceptual judgments and memories are fallible, that our calculations and inferences are fallible. Sometimes our expectations go unsatisfied or our plans go awry. Sometimes our conceptualizations turn out to have been confused or our interpretations turn out to have been biased. Sometimes what seems sure enough to us is doubted or denied by others; sometimes we doubt or deny what seems sure enough to others.

The care and method we employ in ascertaining the correctness of our judgments may vary along with our purposes and circumstances, including our assessment of the chance for error and our evaluation of the consequences of error.

Being practically certain, or feeling absolutely certain, is not the same as holding a claim that is absolutely immune to doubt.

Practical certainty is compatible with doubt -- in the study as well as in the marketplace.

And here is where the logos differs from justification. Hanover brought this to mind elsewhere. When you learn that the cup is red (again), are you learning something about the cup, or something about the use of the word "red"? — Banno

I'm not sure how this coordinates with our discussion of certainty. In any case, it seems the answer depends on what you're ignorant of at the time you "learn that the cup is red".

If you already know what those words mean, then it may be you're color blind, or looking for red cups in the dark or in green light, and have finally determined that the cup in front of you is the red one you were looking for.

If you don't know what those words mean, but already have concepts of "red" and "cup", as one who doesn't speak English might, then you may only be learning the meaning of those English words, how they map onto your concepts and your native language.

If you don't have the concepts corresponding to the words "red" and "cup", then you may be acquiring those concepts for the first time, like a child, learning to pick out new sorts of things on the basis of perception, and learning to coordinate those things, and thoughts about those things, with the corresponding words in the English language.

Well, one hand washes the other. When you learn that r justifies p, you learn more than just that r materially implies p; you learn a new way of using "r" and "p". It does not automatically follow that, if r justifies p, it justifies p v q. — Banno

Do you mean to say it does not "automatically follow", in the head of every person who learns that r justifies p? In other words, not everyone who learns that r justifies p will immediately infer that r justifies (p V q)?

That seems like a psychological point, not a logical one. I suppose I agree with the psychological point. But I would expect that, at least once a person has acquired a grasp of the form of basic propositional logic, he will be disposed to assent to the claim "r justifies (p V q)" as soon as he is disposed to assent to the claim "r justifies p".

It's a good first step; but so much more is involved. Consider the ancient distinction that splits knowing into knowing how... and knowing that..., and then pretends that knowhow has no place in philosophy.

Until it was pointed out that philosophers ought know how to use words.

To paraphrase, there are ways of knowing that are not exhibited in statements, but shown in what we do.

These are missing from Gettier. — Banno

Perhaps we might say Gettier's puzzles are neutral with respect to the distinction. They seem directly concerned with knowing-that, not knowing-how. Nonetheless, his problematization of justified true belief as a model for knowledge would have implications for justified true beliefs about know-how, and justified true beliefs involved in know-how.

It's tempting to suppose that models of knowing-that and models of knowing-how could be reconciled by subsuming either one into the other, and that perhaps neither has priority. If that can be done, it seems the way is open to a unified account of both sorts of knowing.

The carpenter has a special sort of know-how. He also knows that he has this know how. He believes that he can construct and repair various sorts of wooden object. His belief is justified -- not necessarily by sentences, but by deeds and memory of deeds that may be expressed in sentences if need be. He can demonstrate the skill, and he can instruct others. His demonstration and instruction may, but need not, involve speech.

Perhaps Plato shows his bias as a member of a class of privileged Athenian teachers whose reputation and livelihood depend on their expertise in literacy and rhetoric, when he suggests that true belief doesn't count as knowledge unless it's accompanied by a linguistic account. Or perhaps he doesn't mean that the account must be spoken, but only that it must be expressible in speech -- a rational understanding, a reasonably informed grasp of the matter that could, but need not, be expressed in language by a competent speaker. Or perhaps he glosses over this distinction without recognizing it, due to the richness and ambiguity of the term logos, or his own habits and biases.

By a more recent convention, we model beliefs and justifications as "propositions" and "propositional attitudes" expressed in language, but I see no reason to insist, and good reason to deny, that all beliefs and justifications have an essentially linguistic form. So far as I know, these conventions typically take for granted that the propositions do not have an essentially linguistic form, though of course we give them linguistic expression when we speak about them.

IF (probably p) THEN (probably (p V q). — Cabbage Farmer

Yes, it certainly seems on the face of it that a deduction from probable p inherits (at least) the same probability.

IF (Improbably~p) THEN (improbably ~p v q)

This seems to be just as valid, only less probable.

IF (probably p) THEN (probably (p v ~q).
IF (Improbably~p) THEN (improbably ~p v ~q)

And these too. Does that worry you at all? It worries me.

In 1961 Henry Kyburg pointed out that this policy conflicted with a principle of agglomeration: If you rationally believe p and rationally believe q then you rationally believe both p and q. Little pictures of the same scene should sum to a bigger picture of the same scene. If rational belief can be based on an acceptance rule that only requires a high probability, there will be rational belief in a contradiction! To see why, suppose the acceptance rule permits belief in any proposition that has a probability of at least .99. Given a lottery with 100 tickets and exactly one winner, the probability of ‘Ticket n is a loser’ licenses belief. — SEP

Epistemic paradoxes.

Well let me ask you a question in return. If you have a reasonable belief p, and a reasonable unconnected scepticism q (say p - that aspirin is an effective painkiller, and q - that Bluebeard's treasure is buried on Easter Island), what is to be gained by forming the disjunction, (p v q) ? How does S advance his knowledge, or understanding or in any way profit from forming his disjunction? Does it enable a test of p, or the building of a deeper theory or something? — unenlightened

Yes! It appears to be totally unmotivated, doesn't it? At the very least, it violates Grice's "Be relevant" maxim. It even seems to edge toward the logically tenuous mental gymnastics we associate with conspiracy theories. If this is science, it's pretty bad science, right?

I see the main issue as coincidental confirmation, so I'm ignoring Gettier's agenda most of the time. I imagine Smith learning that "Jones owns a Ford or Brown is in Barcelona" is true, but not learning what makes it true, and thus treating it as confirmation of his hypothesis that Jones owns a Ford. I have also imagined Smith not forming the disjunction at all, but simply making a test that he thinks is of Jones owning a Ford but is actually a test of "Jones owns a Ford or Brown is in Barcelona".

Suppose that's what happens, but then through other channels Smith discovers Jones does not own a Ford.* He might forget all about it, but if he is a good scientist, that unexplained positive will bug him. To get to the bottom of that, he'll have to be able to form this goofy disjunction.

Think about the discovery of cosmic microwave background radiation, how it went and how it could have gone. Suppose, contrary to fact, there is no CMB, and Penzias and Wilson were looking for it. They get this noise, check their equipment out, and think they've found it, but the source of the noise was actually pigeons nesting in their dish. What actually happened is the opposite: they weren't looking for it, checked their equipment, chased off the pigeons, and it was still there. They determine its characteristics as best they can, but have no idea what it is until someone tells them about the prediction that the Big Bang would cause such a thing.

Brown being in Barcelona is the pigeons in the first scenario and the CMB in the second. It's the unknown unknown. And when there are unknown unknowns, you can mistake noise for signal and signal for noise. To suss out what's going on you may eventually have to form odd disjunctions involving pigeons and the Big Bang.

That's my big picture version of what's going on. I think Gettier's examples are outlandish in order to make any claim of knowledge implausible, but he could have made them simpler. For example:

I leave my keys on the table, you mistakenly grab them on your way out, realize you have my keys rather than yours and put them back. I have no idea my keys ever moved. Do I know where my keys are? It's luck: I "know" but only because you put them where I put them, not because they stayed there. You chose the salient location for my keys, but you might not have, and I have no idea I'm relying on our shared rationality ...

My hunch is that Smith's disjunctions are unacceptable in part as a matter of linguistics, and insofar as that supports our communal rationality, he is violating a norm of some kind. They can also be criticized as you have done here, as being unmotivated, even pointless. In fact, I mentioned this about a week ago: if a disjunction is part of an argument from cases, what is the result both of these produce that could eliminate the disjunction? (Compare the CMB story, where pigeons and Big Bangs both result in noise.)

I have no idea-- finally answering your specific question-- and I think the lack of apparent rationale is why we are inclined to reject them. My thought when I brought this up before is that it explains our feeling that these disjunctions are arbitrary. And I suspect we could even measure that: how unlikely would an event be that could be caused (or enabled?) either by Jones owning a Ford or Brown being in Barcelona? I think most of us would guess pretty dang unlikely. (But keep in mind the Connections TV show!)

I think this is the neighborhood where most discussion of Gettier sets up shop. Is there something about the conditions (beyond Smith's control) that makes this not knowledge? Is it something in Smith's behavior, some norm of rationality he has violated?



* Coincidentally, I have driven three different Fords for several years each without owning any of them.

About the methodological approach...

Logic cannot account for truth(as correspondence to fact/reality). Rather, it presupposes it by virtue of assuming the truth of it's premisses, and aims to preserve it by virtue of establishing and following 'the rules' for 'correct' inference.

All thought/belief presupposes it's own correspondence to fact/reality somewhere along the line. Thought/belief consists - in part - of the presupposition of truth(as correspondence). Because thought/belief consists of the presupposition of truth, and neither logic nor formal notation can account for truth, then neither logic, nor formal notation can properly account for thinking/believing that something or other is true, believing a disjunction notwithstanding.

The attribution and/or recognition of causality is among the most rudimentary forms of thought/belief formation. So, it is of no real surprise that some folk would believe that if we force our thoughts to follow some logical structural form then doing so would cause us to arrive at true belief, as long as we start with true premisses.

It doesn't work that way unless we've gotten the rules right. If we can follow all the rules and still arrive at something that we all find unacceptable, then something is wrong with the rules.

:-|

The positive note here is that we can use ordinary language as a means to set out thought/belief processes that attempt to strictly adhere to disciplined convention. That's what I've done here, and in doing so, shed some much needed light upon the fact that believing a disjunction consists of more than Gettier or anyone else since, has properly accounted for.

The rules say that disjunction follows from a belief that p is true. The rules say that if that belief is true then so too is the disjunction. The rules say that if I follow them and arrive at believing that disjunction, then I have arrived at a belief that based upon good reason. I mean, after-all it's based upon the rules. The rules say that that is a well-grounded belief. It even encapsulates the ground within parentheses as a means for showing it.

However...

The rules do not require that that ground be properly taken account of when we are reporting upon that belief. Hmmmm...

Funny that. Look at all the trouble that that oversight has gotten us into.

When the ground is properly accounted for there is no problem, because believing a disjunction is nothing more and nothing less than believing it is true because p. So, it's a problem regarding how believing a disjunction has been taken into account for over half a century.

For anyone who wants to argue the semantics of 'or', I point you towards the solution. Fill it out, and then get back to me.

For anyone else who thinks/believes that I've only touched upon disjunction, I ask that you present a case that satisfies JTB but we shouldn't count it as knowledge, and in doing so, make certain that you take proper account of the ground by virtue of stating that the conclusion has been reached and/or is drawn because the ground(is true)...

Then look at the results.

All you've did is criticise the tracking argument of epistemic closure, no? — Posty McPostface

I think I've done quite a bit more than that. It will take while to sink in though. Paradigm shift is long overdue.

Delusions of grandeur...

I'm gonna be famous.

I only addressed the OP, where tracking failed. I didn't read the whole thread so I could have missed the memo. Maybe write a paper in academic rigour?

I'm working on it.

X-)

If you want, you could start at page 40 or so, and let me know what you think. Weaknesses, etc.

The man with ten coins in his pocket will get the job because Jones has ten coins in his pocket, and Jones is the man who will get the job.

Don't think I can trump what has already been said. Anyway, looking forward to reading a paper on it.

Surely someone better versed in the logic of probability than I could help us clear up this confusion.

Yes, it certainly seems on the face of it that a deduction from probable p inherits (at least) the same probability.

IF (Improbably~p) THEN (improbably ~p v q)

This seems to be just as valid, only less probable. — unenlightened

I'm not sure it's valid.

I take it the probability of (~p V q) will be very high when the probability of q is very high, regardless of the probability of ~p. Thus the improbability of ~p does not adequately inform our judgment about the probability of (~p V q). For the improbability of a disjunction requires the improbability of both of its terms, while the probability of a disjunction requires merely the probability of at least one of its terms.

Given an infinite number of such cases selected at random (cases in which it's improbable that ~p, but we have no idea whether it's probable that q), it may be reasonable to expect that (~p V q) will be false more often than (p V q) will be false. But it seems to me a statistical judgment of this sort is not the same thing as an inference in one particular case from the improbability of ~p to the improbability of (~p V q).

I recall von Mises makes a similar point in Probability, Statistics, and Truth, but I don't have my hard copy and haven't managed to find the passage online today.

IF (probably p) THEN (probably (p v ~q). — unenlightened

This one has the same basic form as

IF (probably p) THEN (probably (p V q)).

Moreover, I take it the two claims are consistent. So we might say, further:

IF (probably p) THEN ((probably (p V q)) AND (probably (p V ~q)))

IF (Improbably~p) THEN (improbably ~p v ~q) — unenlightened

I'm not sure this is valid either, for the same reasons as the first formula above.

Perhaps it may help to compare: If I roll a twelve-sided die (p) and a four-sided die (q) together, what is the probability that at least one of the two dice lands on a value greater greater than 4?

And these too. Does that worry you at all? It worries me. — unenlightened

I have yet to see reason for concern.

In 1961 Henry Kyburg pointed out that this policy conflicted with a principle of agglomeration: If you rationally believe p and rationally believe q then you rationally believe both p and q. Little pictures of the same scene should sum to a bigger picture of the same scene. If rational belief can be based on an acceptance rule that only requires a high probability, there will be rational belief in a contradiction! To see why, suppose the acceptance rule permits belief in any proposition that has a probability of at least .99. Given a lottery with 100 tickets and exactly one winner, the probability of ‘Ticket n is a loser’ licenses belief. — SEP — unenlightened

This is interesting, but suggests primarily that a rational believer would not thus lump together such beliefs without constraint. It seems to me the problem should be resolved by recharacterizing the judgment informed by a grasp of the odds. Kyburg, or the designer of the "policy" Kyburg criticizes, has smuggled irrationally expressed beliefs into a rational person's head.

Do you mean to suggest that this puzzle involving an irrational conjunction of inadequately expressed probabilistic judgments somehow informs our discussion about an inference in one particular case, from a probabilistic judgment that p to a probabilistic judgment that (p V q)?

Perhaps you would care to expand on the point. For one thing, does it matter that Kyburg's case involves conjunction, while our case involves inclusive disjunction? For another, what is there in our case corresponding to the contradiction generated by Kyburg's make-believe irrational believer? It's obvious where the contradiction lies in Kyburg's case, but so far you've given me no reason to suspect there's such a contradiction in the case at issue in our conversation. What have I missed?

It strikes me Kyburg's is another case in which probabilistic judgment over many instances is confused with probabilistic judgment in a single instance. In Kyburg's case the error's even worse, if the make-believe believer has leapt from the sound inference "Ticket n is (99:100) likely to lose" to the invalid non-probabilistic inference "Ticket n will lose". Whereas in our case, no one has been so foolish as to strip the probabilism off the claim. The cases also differ in that Kyburg's case has an explicitly mathematical form, whereas in our case Smith's probabilistic judgments do not have a clear mathematical form.

Epistemic paradoxes. — unenlightened

Is there something in this article especially relevant to our case?

Confusion and paradox are not the same.

It seems a safe bet that either my intuitions are confused and there is a paradox; or there is no paradox and your intuitions are confused.

I'll summarize my prima facie intuitions below. Let's kick them around, to see if they're consistent, and to see if we agree that they're reasonable. And to see if there are any typos...

Granting that we're judging a single particular case, with information about one term and no information about the other term of a disjunction:

VALID
(antecedent is sufficient to inform judgement that disjunction is probable)

IF (probably p) THEN (probably (p V q))
IF (probably p) THEN (probably (p V ~q))
IF (probably ~p) THEN (probably (~p V q))
IF (probably ~p) THEN (probably (~p V ~q))

IF (improbably p) THEN (probably (~p V q))
IF (improbably p) THEN (probably (~p V ~q))
IF (improbably ~p) THEN (probably (p V q))
IF (improbably ~p) THEN (probably (p V ~q))


INVALID
(antecedent is insufficient to inform judgment that disjunction is improbable)

IF (probably p) THEN (improbably (~p V q))
IF (probably p) THEN (improbably (~p V ~q))
IF (probably ~p) THEN (improbably (p V q))
IF (probably ~p) THEN (improbably (p V ~q))

IF (improbably p) THEN (improbably (p V q))
IF (improbably p) THEN (improbably (p V ~q))
IF (improbably ~p) THEN (improbably (~p V q))
IF (improbably ~p) THEN (improbably (~p V ~q))


Good exercise for a knucklehead like me.

It might help to make explicit:

IF (probably p) THEN (improbably ~p)
IF (probably ~p) THEN (improbably p)
IF (improbably p) THEN (probably ~p)
IF (improbably ~p) THEN (probably p)


And then put it together:

IF (probably p) THEN:
improbably ~p
probably (p V q)
BUT NOT NECESSARILY: improbably (~p V q)

IF (probably ~p) THEN:
improbably p
probably (~p V q)
BUT NOT NECESSARILY: improbably (p V q)

IF (improbably p) THEN:
probably ~p
probably (~p V q)
BUT NOT NECESSARILY: improbably (p V q)

IF (improbably ~p) THEN:
probably p
probably (p V q)
BUT NOT NECESSARILY: improbably (~p V q)

Of course we'd have to draw up different sets of tables for conjunction or exclusive disjunction.

@unenlightened

I'm still not sure what our discussion about probabilistic inference has to do with Gettier's paper. Is it your position that whenever rational people recognize that their reasonable expectations are grounded merely in "strong evidence", there is an essentially probabilistic deep-structure to their expectations, according to which their assertions like "Jones owns a Ford" cannot be evaluated as true or false, but must be reinterpreted as assertions like "It's sufficiently probable that Jones owns a Ford", or perhaps "I have good reason to claim that it's sufficiently probable that Jones owns a Ford"?

Along those lines, perhaps the beliefs we really "have" are beliefs about the probability of propositions, or about a given individual's being justified in making claims about the probability of propositions.... As a wholehearted skeptic, that strikes me as a line of thought worth pursuing, and I suppose there may indeed be something like this sort of structure underlying ordinary belief.

But how would that pursuit inform our view of Gettier's little essay?

At some point in the mesh of justifications and probabilistic judgments, a proposition becomes actionable, and its alternatives become negligible. Given his justifications and the probabilistic deep-structure of his expectations at the time we meet him, the rational Smith is prepared to act, until further notice, as if it were the case that Jones owns a Ford, as if it were the case that Brown is not in Barcelona, and thus as if it were the case that (EITHER Jones owns a Ford OR Brown is in Barcelona) is true. And Smith will have achieved this result by way of the same unhappy accident that we find in Gettier's paper, before and after all this additional trouble with probabilism.

Thus even the skeptic or probabilist may acknowledge that a traditional analysis of ordinary human beliefs in terms of "propositions held to be true" is a reasonable and useful simplification of the deep structure you've indicated.

So far as I can see, that brings us right back to the beginning: How, according to you, is recourse to probabilistic analysis of Gettier's puzzles relevant to our evaluation of Gettier's arguments and our assessment of the conception of justified true belief as a criterion for knowledge?

Surely someone better versed in the logic of probability than I could help us clear up this confusion. — Cabbage Farmer

Well, the use of "probably" and "improbably" isn't very useful. Try replacing it with percentages, where (for example) "probably" is ">= 75%" and "improbably" is "<= 25%". That then gives us:

If p >= 75% then ¬p <= 25%

Now if we consider a disjunction then this gives us:

If p >= 75% then p ∨ q >= 75%

So let's take each of unenlightened's examples:

IF (Improbably~p) THEN (improbably ~p v q) — unenlightened

If ¬p <= 25% then ¬p ∨ q ...?

Not enough info.

IF (probably p) THEN (probably (p v ~q).

If p >= 75% then p ∨ ¬q >= 75%

IF (Improbably~p) THEN (improbably ~p v ~q)

If ¬p <= 25% then ¬p ∨ ¬q ...?

Not enough info.

Well, the use of "probably" and "improbably" isn't very useful. Try replacing it with percentages, where (for example) "probably" is ">= 75%" and "improbably" is "<= 25%". — Michael

Thanks for crunching the numbers for us! I'm relieved to find that the math, at least, confirms my intuitions.

I was running with unenlightened's usage of the terms "probable" and "improbable". But I am, moreover, inclined to say that ordinary conceptions of probability, and of the logical form of probabilistic judgment, are conceptually prior to, or at least historically prior to and conceptually distinguishable from, mathematical models of probability and probabilistic judgment.

I speak analogously about the priority of ordinary number-concepts and numerical judgments, and about the priority of pre-numerical quantitative concepts and judgments (more/less, greater/fewer, bigger/smaller, lighter/heavier, faster/slower), as compared to the more refined conceptualizations obtained by the construction and analysis of sophisticated mathematical models of such concepts.

In our case, I would say there is a logic of probability that is prior to, or distinguishable from, mathematical models of probability. This is no mere academic point: It seems that most probabilistic judgments by ordinary humans do not have an explicitly numerical form, and in some cases it's not clear what numerical analysis could possibly be given to express a probabilistic judgment.

Accordingly, I take it there's some good sense to the sort of claims that unenlightened and I have been trafficking in.

One glaring thing we've neglected so far is an evaluation of equiprobables. My preference is to avoid expanding the set of values. So rather than three values (probable, equiprobable, improbable), I'd try lumping equiprobable into one of the other two terms. It seems more fitting to call a 50/50 chance improbable than to call it probable, so I'd start out by lumping the equiprobable with the improbable.

I haven't bothered to sort through the implications.

Once you've tested it with the numbers you can substitute back in the ordinary terms:

If p is probable then ¬p is not probable

If p is probable then p ∨ q is probable

If ¬p is not probable then ¬p ∨ q is ...? Not enough info.

If p is probable then p ∨ ¬q is probable

If ¬p is not probable then ¬p ∨ ¬q is ...? Not enough info.

So far as I can see, that brings us right back to the beginning: How, according to you, is recourse to probabilistic analysis of Gettier's puzzles relevant to our evaluation of Gettier's arguments and our assessment of the conception of justified true belief as a criterion for knowledge? — Cabbage Farmer

Ok, let's go back to the beginning. Gettier's claim is that B(p) -> B(p v q) for all q. My complaint about this is that the logic conserves truth, but belief is not truth. Now the difficulty with all the analysis above is that it separates the probable and improbable in order to then apply the rules of logic. The reason for bringing up probability was to try and get at the difference between belief and truth.

Now my suggestion has been that the way to express this is as a disjunction, (p v ~p), which we can annotate with percentages to illustrate the inclination of the belief, thus: (p75% v ~p25%). In this way the belief and the doubt are kept in one expression that can be asserted as true of necessity, and that truth can thence conserved by logical operations.

So then the Gettier disjunction becomes ((p75% v ~p25%) v (q1%)), or S could make the conjunction, ((p75% v ~p25%) & (q1% v ~q99%)). The point being that to make the disjunction with an arbitrary q, the first term must be true100%.

Gettier's claim is that B(p) -> B(p v q) for all q. — unenlightened

No it's not. His claim is that J(p) → J(p ∨ q). He then says that Smith believes p ∨ q.

I don't think it affects my argument; the implication conserves truth, and justification is no more truth than belief is.

I don't think it affects my argument; the implication conserves truth, and justification is no more truth than belief is. — unenlightened

It preserves justification as well. If there are good reasons to believe p then there are good reasons to believe p ∨ q. How could it be any other way?

It doesn't make much sense to say that I'm justified in believing that London is the capital city of England but not justified in believing that London is the capital city of England or pigs can fly (or for a less silly example, that London is the capital city of England or unenlightened has brown eyes).

"Why wouldn't there be?" is not a justification for your assertion. I have already presented in great detail considerations as to why there wouldn't be, and it is your job to provide some justification, not rhetorical questions. but to repeat myself as demanded, there wouldn't be because logic conserves truth, and justifications are not the same as truths, or guarantees of truth.

If p is true then p ∨ q is true. Therefore, if there is strong evidence that p is true then there is strong evidence that p ∨ q is true. If one has strong evidence that p ∨ q is true then one is justified in believing that p ∨ q is true.

Smith has strong evidence that p is true and so strong evidence that p ∨ q is true and so is justified in believing that p ∨ q is true.

If p is true then p ∨ q is true. Therefore, if there is strong evidence that p is true then there is strong evidence that p ∨ q is true. — Michael

That argument is invalid.

That argument is invalid. — unenlightened

It is if you accept the principle that evidence is closed under entailment (or, more specifically, under disjunction introduction), which I do. Although clearly you don't. I honestly don't know how to go about convincing you of it. It's so obvious to me as to be pretty much axiomatic (like disjunction introduction itself).

So asking me to prove that evidence of p is evidence of p ∨ q is like asking me to prove that p ∨ q is true if p is true. All I can do is repeat examples in the hopes that eventually it'll get through.

I have evidence that "I have hazel eyes or unenlightened has brown eyes" is true. That evidence is evidence that I have hazel eyes; namely, what I see in the mirror.

Are you saying that I don't have evidence that "I have hazel eyes or unenlightened has brown eyes" is true? That strikes me as absurd.

It is if you accept the principle that evidence is closed under entailment (or, more specifically, under disjunction introduction), which I do. Although clearly you don't. I honestly don't know how you go about convincing you of it. It's so obvious to me as to be pretty much axiomatic (like disjunction introduction itself). — Michael

No. The argument you presented is invalid. How you go about convincing me is to present a valid argument. I accept the premise, but I reject the conclusion, until you present a valid argument.

No. The argument you presented is invalid. — unenlightened

It's valid if the principle of closure under entailment is correct.

If you have a valid argument but you have a hidden premise, disclose the hidden premise and present the valid argument. I accept your premise, but your argument is invalid as stated. make the valid argument.