We're told S believes p, hence S believes (p v q).

If someone objects, what you gonna do?

Consider:

S believes p
p is justified
p

Therefore S knows p

against

S believes p
p is justified
~p

Sam does not know p.

and against

S believes p
S believes p is justified
S believes S knows p

~p

In which case S is mistaken - all within the scope of S's belief.

...if Smith is rational and understands the conventions of formal logic, and believes that p, and entertains the proposition (p V q) in light of his belief that p, we should be surprised indeed if he does not acknowledge that he believes that (p V q) is true.

We should be perhaps even more surprised if he was asked which of the following was a more accurate rendition/account/representation of his believing the disjunction, and he did not immediately confirm the second...

"Either Jones owns a Ford or Brown is in Barcelona" is true.

OR

"Either Jones owns a Ford or Brown is in Barcelona" is true because Jones owns a Ford.

This is even clearer if we put all three on display...

To drive the point home, the same holds with Case I. We should be quite surprised if Smith was asked which of the following was a more accurate rendition/account/representation of his belief(s), and he did confirm the second...

Jones has ten coins in his pocket. Jones is the man who will get the job.

OR

The man with ten coins in his pocket will get the job.

We're told S believes p, hence S believes (p v q).

If someone objects, what you gonna do? — Banno

We're told that S has evidence of p, that S knows that p ∨ q follows from p, and so that S has evidence of p ∨ q. We're also told that S, recognising that he has evidence of p ∨ q, believes p ∨ q.

The relevant quote from Gettier:

Each of these propositions is entailed by (f). Imagine that Smith realizes the entailment of each of these propositions he has constructed by (f), and proceeds to accept (g), (h), and (i) on the basis of (f). Smith has correctly inferred (g), (h), and (i) from a proposition for which be has strong evidence. Smith is therefore completely justified in believing each of these three propositions.

That quote tells me nothing.

It tells you that Smith believes p ∨ q, by his own admission.

Trying to argue that Smith doesn't believe p ∨ q doesn't make sense. You might as well try to argue that Jones owns a Ford or that Brown isn't in Barcelona.

Sure. But should he? He does so for no other reason than Gettier's narrative security.

Sure. But should he? — Banno

He has evidence of p, and p ∨ q follows from p, and so he has evidence of p ∨ q.

Should I believe the following?

London is the capital city of England or pigs can fly.

I say I should. I have evidence that London is the capital city of England.

Exactly. Does he believe it independently, or as a result of his assuming that logic applies to the objects of belief statements. If the former, it's just for the sake of the narrative; if the latter, then... what?

Does it matter? He believes it to be true, he has evidence that it's true, and it's true. Just as with me and the proposition "London is the capital city of England or pigs can fly".

I'm wondering if this is one of the points at which you and Creative pass each other by.

An intersting notion. So Moore might believe he has a hand, and yet doubt it. Or Moore might know he has a hand, and yet doubt it. But not Moore might be certain he has a hand, and yet doubt it.

That might be right. — Banno

Exactly.

So far as I can see, it's best way to align this family of terms to reflect ordinary usage and to clean up the epistemologist's shop.

We can extend the treatment to "certainty": So long as we mean mere practical certainty or a feeling of sureness, but not absolute theoretical certainty, certainty is compatible with doubt.

More importantly, it just seems to be the case that minds like ours never or almost never attain absolute certainty, though many of us seem awfully sure of ourselves sometimes.

I've heard fallibilism is quite fashionable in the schools in our time.

More importantly, it just seems to be the case that minds like ours never or almost never attain absolute certainty.

This gives a weighted disjunction, (p(99%) v ~p(1%)). And that does not lead to (p v q). It's so simple it seems to be invisible to everyone, but as soon as it is possible that ~p, the damaging disjunction (p v q) cannot be made at all. — unenlightened

According to your calculus, does (p(99%) V ~p(1%)) imply ((p V q)99%)? Or how are your probabilistic weightings related to propositional logic?

How does the weighted disjunction address the Gettier cases? Does Smith believe ((p V q)99%), on your account?

IF p, then (p v q). That's valid, sound, true and contentless
— unenlightened

It's only sound if p is true. — Srap Tasmaner

No, it's always sound, because it already has your 'if' incorporated. I put it in capitals so you would notice. It is contentless because it does not claim that p is true. S wrongly makes the claim, that p is true, and then uses this formula to arrive illegitimately at (p v q).

The conclusion of an inference merits no more or less credence than what you grant your premises. If you're uncertain about your premises, then you should be just that uncertain about your conclusions. — Srap Tasmaner

Where do you derive this principle from? It isn't a law of logic. If I might use an analogy, the higher you want to build, the more secure you need to make your foundations. But suppose it is true...

Consider q, that S has no reason at all to believe, except that Jones must be somewhere. Let's say q(0.01%)

Then q is (100 times) less likely than ~p.

The weight of incredibility of q exceeds the strength of credence of the premise, p on which it (p v q) rests.

What irks me about Gettier is that he appears to be assaulting a straw man. Who is it that believes knowledge is exactly justified true belief? — Banno

It often seems professional epistemologists count it their duty to construct and assault straw men. Consider their collective abuse of the moldy old straw man they call "the skeptic".

I'm not into theory-building, but I'm inclined to think that JTB stands as a fair account of our use of "knowing" language. That's not necessarily the same thing as an account of what knowledge is; but only something like a theoretical model or analysis of the conditions under which we count ourselves entitled or unentitled to assign "knowing" predicates to subjects, to say that S knows that p.

No, it's always sound, because it already has your 'if' incorporated. I put it in capitals so you would notice. It is contentless because it does not claim that p is true. S wrongly makes the claim, that p is true, and then uses this formula to arrive illegitimately at (p v q). — unenlightened

Here's an argument:
Everyone in this room is happy.
Steve is in this room.
∴ Steve is happy.

That's a valid argument, whether or not either of the premises are true. If both of the premises are true, then it is also a sound argument.

I have taken you to be saying that Sharon is only entitled to make such an inference as shown above if the premises are true. It is conceivable that the premises are true but Sharon does not know this, in which case she is entitled to make an inference that she does not know she is entitled to make. And so it may be. If Sharon does know that the premises are true, then she also knows she is entitled to make the inference.

I have argued that making such valid inferences is one of the ways Sharon will try to determine whether the premises are true, by further exposing them to confirmation or disconfirmation. In this example, she would determine that Steve is in this room and then try to determine whether Steve is happy. If he is, the universal premise is partially confirmed; if he is not, then the universal premise is partially disconfirmed, but may still survive a reformulation like "Everyone in this room but Steve is happy," or "Almost everyone in this room is happy."

On my approach, Sharon knows the argument is valid, so she can come to know whether "Everyone in this room is happy" is true by assuming it true, hypothetically, making an inference and thus a prediction about each person in the room, and then testing those predictions.

She could also make no such hypothesis, and no such predications, but just ask everyone, tally them up and find that everyone or everyone but Steve is happy, whatever. There's really not much difference in this case.

The difference is that my approach obviously scales up and allows the use of statistics and probability, besides recognizing that raw fact-gathering is not the only thing we care about. We also need to make predictions, so we need to be good at it.

The curious thing is, by the time Sharon knows she is entitled to make the inference, she no longer needs to.

Suppose now Sharon shares her results with a colleague, Carol. Carol notices Steve's name on the list of people in this room, and, knowing Sharon's results, infers that Steve is happy. But do we just say that Carol knows that everyone in this room is happy the same as Sharon does? Isn't it rather the case that Carol is taking Sharon's results as given, for whatever reasons good or bad? That she is assuming Sharon's result is correct? And then she could test it, by, for instance, asking Steve if he is happy, thus confirming or disconfirming Sharon's result.

This hypothesis-prediction-test-revision cycle seems eminently rational to me and depends on making valid hypothetical inferences of unknown soundness.

What here do you disagree with?

[Disjunctive syllogism stuff in a future post.]

What here do you disagree with? — Srap Tasmaner

Nothing much.

I'll just note that science, probability, and induction/abduction are what S does to arrive at his belief p. No quarrel with him there.

I'll just note that science, probability, and induction/abduction are what S does to arrive at his belief p. No quarrel with him there. — unenlightened

Yes absolutely.

Broad Agreement feels good.

This seems very relevant:

https://plato.stanford.edu/entries/closure-epistemic/#SkeAnt

Dretske and Nozick focused on a form of skepticism that combines K with the assumption that we do not know that skeptical hypotheses are false. For example, I do not know not-biv: I am not a brain in a vat on a planet far from earth being deceiving by alien scientists. On the strength of these assumptions, skeptics argue that we do not know all sorts of commonsense claims that entail the falsity of skeptical hypotheses. For example, since not-biv is entailed by h, I am in San Antonio, skeptics may argue as follows:

(1) K is true; i.e., if, while knowing p, S believes q because S knows that p entails q, then S knows q.
(2) h entails not-biv.
(3) So if I know h and I believe not-biv because I know it is entailed by h then I know not-biv.
(4) But I do not know not-biv.
(5) Hence I do not know h.
Dretske and Nozick were well aware that this argument can be turned on its head, as follows:

(1) K is true; i.e., if, while knowing p, S believes q because S knows that p entails q, then S knows q.
(2) h entails not-biv.
(3) So if I know h and I believe not-biv because I know it is entailed by h then I know not-biv.
(4)′ I do know h.
(5)′ Hence I do know not-biv.
Turning tables on the skeptic in this way was roughly Moore's (1959) antiskeptical strategy. (Tendentiously, some writers now call this strategy dogmatism). However, instead of K, Moore presupposed the truth of a stronger principle:

PK: If, while knowing p, S believes q because S knows that q is entailed by S's knowing p, then S knows q.

Unlike K, PK underwrites Moore's famous argument: Moore knows he is standing; his knowing that he is standing entails that he is not dreaming; therefore, he knows (or rather is in a position to know) that he is not dreaming.

According to your calculus, does (p(99%) V ~p(1%)) imply ((p V q)99%)? Or how are your probabilistic weightings related to propositional logic? — Cabbage Farmer

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)

So what about the disjunctive syllogism?

If I assign to A a probability of r, and to B a probability of s, what probability should I assign to ~A & B? (That is, to ~A & (A v B).) That would be (1 - r)s. Since the probability of A v B is r + s - rs, it's also pr(A) + pr(~A & B). If pr(A v B) = 1, then if pr(A) goes to 0, pr(B) = 1. So there's no weirdness treating the usual disjunctive syllogism as a special case of standard probability.

I don't have Smith assigning a probability of 1 to Jones owning a Ford, and I don't have him assigning a probability of 0 to Brown being in, say, Barcelona. Those are assumptions of mine that I think are defensible from the text-- and from life-- but there's certainly room to argue otherwise.

So what should Smith's view be of the possibility that Jones does not own a Ford but Brown is indeed in Barcelona? Given probabilities of 0.90 for the Ford and 0.01 for Barcelona, he should assign a probability of 0.001 to Barcelona but no Ford. As it should be, since pr(Ford & Barcelona) = 0.901. So that's at least consistent.

But it has to be admitted that what I'm doing here is not-- what should we call it?-- "simply" inferring one belief from another. I allow Smith to form the prediction that Ford or Barcelona based on his hypothesis that Jones owns a Ford, but then in order to assign probabilities to it and to the disjunctive syllogism (to its premises actually, since he already has a prior for Barcelona), he does the math.

Thus I never see Smith being in the position of saying, "Probably A, but if not then definitely B."

Where do you derive this principle from? It isn't a law of logic. If I might use an analogy, the higher you want to build, the more secure you need to make your foundations. — unenlightened

Yeah, I have no justification for that (my thing about the credence you give a conclusion). I think it's a reasonable rule of thumb, something like Hume's saying that "the wise man proportions his belief to the evidence." For instance, in the case at hand of addition, the likelihood of A v B is higher than the likelihood of A, but that's because I'm smuggling in a prior for B.

As a matter of fact-- and this gets to your second point-- if A entails B, then the likelihood of B is at least as high as that of A. (If all F are G, there are at least as many G as F.)

So while there's intuitive support for the general idea of firm foundations and less and less certainty the farther your chain of inference carries you from those foundations, you have to be careful. If your theory as a whole is thought of as just a big conjunction of all of your current beliefs, and if some of those are less than certain, then all of them being true is less likely than some of them taken alone, because when you multiply the independent ones, their product is necessarily smaller. Sure. But our theories are more complicated than big conjunctions. There's a lot of dependence, entailments, conditional probabilities and disjunctions in there.

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.

We can extend the treatment to "certainty": So long as we mean mere practical certainty or a feeling of sureness, but not absolute theoretical certainty, certainty is compatible with doubt. — Cabbage Farmer

it just seems to be the case that minds like ours never or almost never attain absolute certainty. — Cabbage Farmer

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

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.

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"? 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.

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.

Cabbage Farmer wrote:

The truth conditions of (p V q) are met as soon as the truth conditions of p are met. Or as soon as the truth conditions of q are met. Or as soon as the truth conditions of (p AND q) are met.

It seems this is the point you're neglecting, which has sent you off on a search for red herring.

I'm actually quite aware of that. However, the focus is squarely upon the necessary and sufficient conditions for believing a disjunction. So, it's not that I've neglected that point. Rather, it seems irrelevant to the focus.

Smith believes that p.

Smith does not merely believe that there are abstract inferential relations between any pair of propositions and their disjunction. He believes the truth condition for a particular disjunctive claim has been satisfied. He believes that Jones owns a Ford...

Agreed. Where is that being properly accounted for aside from the conclusion of my solution?

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) about what makes this particular disjunction true))
C1. ((p v q) is true because...(insert belief statement(s) corresponding to the prior "if" in p3))

Cabbage Farmer wrote:

What I'm suggesting is that

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

is already enough to give a truth condition for (p V q). Or in other words:

(p V q) is true if p.

I've already touched on this, but in this post I'm arguing against the notion that p3 and C1 aren't necessary for believing a disjunction, and that we could simplify my solution by replacing them with ((p v q) is true if (p) is true). As before, belief that:((p v q) is true if (p) is true) IS p3 in the solution. So, it's clearly necessary, however it's also insufficient. Believing a disjunction requires more than belief that:((p) is true, deducing (p v q) from belief that:((p) is true, and thinking about the truth conditions of (p v q). That's as far as ((p V q) is true if (p) is true gets us. That's as far as p3 gets us. Believing the disjunction takes another deduction, or as you said earlier...

...Smith entertains the proposition (p V q) and makes a rational judgment.

That is precisely what's been missing in every account but my own. That rational judgment is the conclusion itself...

((p v q) is true because (p))

As previously skirted around, when formalized, that judgment and what it rests it's laurels on is not absent. To quite the contrary it is contained within parenthesis to tell the reader what grounds the conclusion and looks like this... (from1,2) etc.

So then, why is it left absent in the account of what believing a disjunction requires?

We're told that S has evidence of p, that S knows that p ∨ q follows from p, and so that S has evidence of p ∨ q. We're also told that S, recognising that he has evidence of p ∨ q, believes p ∨ q. — Michael

It's what we're not told that matters, because that's what's missing, and that's the problem.

Gettier says nothing at all about Smith deducing that (p or q) is true. Gettier's formula is missing a necessary deduction. That's been adequately argued for without subsequent refutation nor even due attention from you.

Believing a disjunction requires and is entirely exhausted by the following...

p1. ((p) is true)
p2. ((p v q) follows from (p))
p3. ((p v q) is true if...(insert belief statement(s) about what makes this particular disjunction true))
C1. ((p v q) is true because...(insert belief statement(s) corresponding to the prior "if" in p3))

I supposed I'm baffled by the fact that some people would rather continue traveling down a path that Gettier paved with an utterly inadequate criterion for what counts as believing a disjunction.

I mean, there it is. That's what it takes, and nothing less...

Examine it. Check it out. Fill it out. Imagine any disjunction you like that follows Gettier's formula, and plug in the values accordingly. You'll never arrive at anything that constitutes being a problem for JTB. That holds good for any and all disjunctions deduced from believing P. There are no exceptions...

The irony...

There is no stronger justificatory ground for believing that Gettier got it wrong.

I know he did.

X-)

Something interesting happens when we remove the bits about justification and Gettier's commentary regarding the rules. Those are nothing more than distractions. We can then look at what's left in terms of Smith's thought/belief...




Smith believes the following:

p1.Jones owns a Ford.




Smith constructs the following:

Either Jones owns a Ford, or Brown is in Boston.
Either Jones owns a Ford, or Brown is in Barcelona.
Either Jones owns a Ford, or Brown is in Brest-Litovsk




Smith believes all three are entailed by Jones owns a Ford.

p2."Either Jones owns a Ford, or Brown is in Boston" follows from Jones owns a Ford.
p2."Either Jones owns a Ford, or Brown is in Barcelona" follows from Jones owns a Ford.
p2."Either Jones owns a Ford, or Brown is in Brest-Litovsk" follows from Jones owns a Ford.




Therefore Smith believes all three.

C."Either Jones owns a Ford, or Brown is in Boston" is true.
C."Either Jones owns a Ford, or Brown is in Barcelona" is true.
C."Either Jones owns a Ford, or Brown is in Brest-Litovsk" is true.



Really now???

p1.Jones owns a Ford.


p2."Either Jones owns a Ford, or Brown is in Boston" follows from Jones owns a Ford.
p2."Either Jones owns a Ford, or Brown is in Barcelona" follows from Jones owns a Ford.
p2."Either Jones owns a Ford, or Brown is in Brest-Litovsk" follows from Jones owns a Ford.


C."Either Jones owns a Ford, or Brown is in Boston" is true because Jones owns a Ford.
C."Either Jones owns a Ford, or Brown is in Barcelona" is true because Jones owns a Ford.
C."Either Jones owns a Ford, or Brown is in Brest-Litovsk" is true because Jones owns a Ford.

Which more closely represents modus ponens when the ground is spoken aloud?

Ground is important no?

What is a justified true belief if not a well-grounded true belief?

Well-grounded false belief won't do.

Groundless false belief even less so.