You have p as a premise — Srap Tasmaner

I have "I believe p" as a premise. Does that not make a difference?

Right, I mean exactly that: if you believe that p, there's nothing contradictory about believing that ¬p→q and believing ¬p→¬q. It just means you get material implication.

I see. Thanks.

Let's look at how modus ponens is correctly applied to Smith's belief...

If Jones owns a Ford then either 'Jones owns a Ford' or 'Brown is in Barcelona' is true. Jones owns a Ford. Therefore, either 'Jones owns a Ford' or 'Brown is in Barcelona' is true.

What's the problem?

:-|

That's justified true belief about disjunction(the rules of correct inference). It's not believing the disjunction. That requires what I've been setting out...

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

The above holds good for every imaginable disjunction arrived at from belief that:((p) is true). There is never a problem of any kind. It's a solution. Believing a disjunction - for a rational person - is nothing more and nothing less than knowing what makes it true and believing that those conditions have been met.

It seems that some want to skip p3 and C1 and arrive directly at belief that:((p v q) is true) simply because the rules say that it follows from ((p) is true). As I've just shown in the last post, if Smith used modus ponens to arrive at belief, he would arrive at believing that either 'p' or 'q' is true. But Smith doesn't believe that either 'p' or 'q' is true. Thus, Smith wouldn't use modus ponens to arrive at believing that:((p v q) is true).

The point is that if I'm asked what would follow if ¬p then I would withdraw the disjunction rather assert q. — Michael

This is worth quoting again.

If someone tells Smith that as a matter of fact Jones does not own a Ford, what happens to my jars?

We dump all the reds out of "Jones" and leave just a blue; "Barcelona" still has 99 blues and the one red. Chances, drawing one from each jar, of getting at least one red? 0 + 0.01 - (0)(0.01), so just 0.01.

Yeah, that's a belief you're unlikely to hold.

In my last response, I ignored -- God knows why -- that in my model, we've got a probability for q, so some of that was crosstalk.

I am actually interested in what you were getting at with "Probably p, but if not then definitely q", the crossover from probable or partial belief to belief, assertion, placing a bet, answering a question on a test, or any other way of acting on a belief that commits you to accepting the consequences of so acting, positive or negative.

But I don't think it's relevant to Gettier's argument. We clearly can and do and probably should and must commit in this way. Smith does, and does so with some justification. That's all Gettier needs.

S believes {1. p , 2. (p v q)}
Reality {3. ¬p , 4. (¬p v q)}
Logic (p v q) & (¬p v q) → q

In a way, this is an ancient problem; Descartes was looking for certainty. Logic cannot cope with a false premise, it falls apart. But false beliefs happen to humans. We need to reason about reality and apply our beliefs to it, but logic only deals in certainties. Reality is never wrong, logic is never wrong, but S and possibly one or two other folk are sometimes wrong. And one wrong belief for a logician leads to explosion.

So S and the rest of us need to take account of our fallibility in reasoning about our beliefs.

Logic never asserts anything about reality. One might say that it it only ever asserts implications - "If (premises) then (conclusion)". Reality is pure infallible assertion "¬p, q ..." So there is never a conflict between them.

But S fallibly asserts p, logically concludes (p v "I am a monkey's uncle"), and ends up lost in the
jungle.

So he needs to keep hold of the fallibility of the assertion, and convey it through the argument in the same way that pure logic prefaces all its assertions with an "if".

So I'm trying to find ways of doing that, with "believably p" or "probably p". And conclusions logically derived from such fallible assertions then have to carry the logical caveat "If really p, then (p v q)"

The result of this is S's belief is not the bald (p v q) any more, but retains the condition (that p) attached to it. S doesn't believe (p v q) unconditionally as Gettier and others here claim, but the conditional, "If really p, then (p v q)".

And this means that he cannot then be said to have the belief Gettier needs him to have, to break the conception of knowledge.

S believes {1. p , 2. (p v q)}
Reality {3. ¬p , 4. (¬p v q)}
Logic (p v q) & (¬p v q) → q — unenlightened

The third step makes no sense in context.

You should have it as this, where B(r) is "Smith believes r":

1. B(p)
2. B(p ∨ q)
3. ¬p
4. q
5. ¬p ∨ q
6. B(p ∨ q) ∧ (¬p ∨ q)

The third step makes no sense in context. — Michael

Yes it does. It expresses the part of the story where Gettier infallibly tells us That Smith is wrong about Jones owning a Ford. It's like when God says "Let there be light". It is so, whether He has gotten around to giving you eyes or not. That's another teaching story, but it works the same way - the story is the story and you have to make sense of it the way it is.

Yes it does. It expresses the part of the story where Gettier infallibly tells us That Smith is wrong about Jones owning a Ford. It's like when God says "Let there be light". It is so, whether He has gotten around to giving you eyes or not. That's another teaching story, but it works the same way - the story is the story and you have to make sense of it the way it is. — unenlightened

The bit in bold is the bit that doesn't make sense:

(p ∨ q) ∧ (¬p ∨ q) → q

It's not that at all. It's:

B(p ∨ q) ∧ (¬p ∨ q)

There's no false premise. Smith's belief that p is false, but the premise "Smith believes p" is true.

I think what you're doing is conflating Smith's argument and Gettier's argument.

Smith's argument is:

1. p
2. p ⊨ p ∨ q
3. p ∨ q

He recognises it as valid, he believes the premises, and so he believes the conclusion.

Gettier's argument is:

1. Smith believes p
2. Smith believes p ⊨ p ∨ q
3. Smith believes p ∨ q
4. ¬p
5. q
6. p ∨ q

Smith's belief 3 is true because of 6.

The bit in bold is the bit that doesn't make sense:

(p ∨ q) ∧ (¬p ∨ q) → q

It's not that at all. It's:

B(p ∨ q) ∧ (¬p ∨ q) — Michael

I don't understand your notation. What's B?

Smith believes that

Gotcha. But that's not what I'm doing there. I've got three different voices, belief, reality and logic. Logic does not believe anything but tautologies of it's own construction. Reality is infallibly what it is, and Belief has to reach an accommodation with them or talk nonsense.

Belief has to reach an accommodation with them or talk nonsense. — unenlightened

A false belief isn't nonsense. It's just false. If you have a cat and I believe that you don't have a cat then the principle of explosion doesn't come into play (as you suggested earlier).

It's just the case that:

1. A believes p
2. ¬p

There's only an issue when we have one of these situations:

1. A believes p
2. A believes ¬p

1. p
2. ¬p

Nothing like this shows up In Gettier's reasoning.

Strike 'nonsense', replace with 'bullshit'.

So Smith has the false belief p, which Gettier accepts. But he also has the true belief p ∨ q. So I don't actually understand your criticism.

So I don't actually understand your criticism. — Michael

That has been my justified belief for some time. ;)

S doesn't believe (p v q) unconditionally as Gettier and others here claim, but the conditional, "If really p, then (p v q)". — unenlightened

My version of the story goes like this: Smith puts the odds of Jones owning a Ford at 9-1, and the odds of Brown being in Barcelona at 1-99, so the odds of at least one being the case are a little better than 9-1. Then what? He makes a prediction, and he acts on that prediction. If you never actually place your bet, you don't get paid. If you never actually test a hypothesis, you don't learn anything.

Gettier tells us that Smith accepts (g), (h), and (i). Suppose Smith could ask someone who knows. If he started with (h), he would get the answer he expected, and continue to believe (h). But if he also asked about (g), and then about (i) too, he would not get the answers he expected.

Once he gets "false" for (g), he knows Smith doesn't own a Ford, right? The result for (i) confirms this, and now Smith will be strongly inclined to believe that Brown is in Barcelona.

But it's not that simple. There could be another factor here: can Smith tell the difference between Brown being in Barcelona, on the one hand, and his informant telling him the truth about (h) but lying about (g) and (i)? Can Smith tell the difference between testing "p v q" and testing "(p v q) & (z v x)"? There's nothing for it but to keep forming hypotheses and keep testing.

And so must we. We have to actually plump for "p v q" and get on with it, and be prepared to revise our beliefs as we go. That's why Gettier is useful: it could always turn out to be the truth of q reinforcing your belief that p. Science is hard.

Now, how do you test "If really p, then p v q"?

Now, how do you test "If really p, then p v q"? — Srap Tasmaner

My view is that you don't test it because it's a matter of pure logic. Science doesn't formulate unconnected disjunctions and then try and establish which arm is true. Nor for that matter does the average Smith. Though logic permits the move, it's not a useful, revealing move as far as I can see.

The sort of thing science busies itself with is connected disjunctions - "the glass contains water or vodka" - and then we get the hydrometer out. "The glass contains water or my neighbour has a beard" is not something it is sensible to consider, never mind claim as a belief, or try and test.

The sort of thing science busies itself with is connected disjunctions - "the glass contains water or vodka" - and then we get the hydrometer out. "The glass contains water or my neighbour has a beard" is not something it is sensible to consider, never mind claim as a belief, or try and test. — unenlightened

Then what if Smith has strong evidence to suggest that it's water, but in fact it's vodka? He has a justified true belief that "the glass contains water or vodka".

Say he buys a bottle of Evian from Tesco, but some prankster in the factory has switched the contents.

Then what if Smith has strong evidence to suggest that it's water but in fact it's vodka? He has a justified true belief that "the glass contains water of vodka". — Michael

If Smith has a justified belief that the glass contains water, why would he want to think, claim or believe that it contains water or vodka? He wouldn't, he would think claim and believe it contains water - wrongly. The circumstance where he would perhaps form the disjunction is if he saw that it contained a clear, colourless liquid, and that someone had sipped from it (not white spirit then), but didn't know exactly what liquid. If he thinks he knows what is in the glass, he has no reason to think the disjunction. And then it is as arbitrary and pointless as an unconnected disjunction.

Science doesn't formulate unconnected disjunctions and then try and establish which arm is true. — unenlightened

That misses the point.

it could always turn out to be the truth of q reinforcing your belief that p. — Srap Tasmaner

Gettier just constructs an artificial example to show how this works. It happens when you think you're testing p but you're actually testing p v q v r.

Gettier just constructs an artificial example to show how this works. It happens when you think you're testing p but you're actually testing p v q v r. — Srap Tasmaner

Favour us with a less artificial example, you have my attention.

The fundamental issue I take with your account is that it sets up a bizarre situation where:

1. I know that "Jones owns a Ford or Brown is in Barcelona" is true if Jones owns a Ford,
2. I believe that Jones owns a Ford, and
3. I don't believe that "Jones owns a Ford or Brown is in Barcelona" is true.

I think that your "solution" to the Gettier problem is far more counter-intuitive than just accepting that knowledge isn't simply justified true belief.

Other Gettier situations – such as correctly believing that Max is a man because of false evidence that Max is a bachelor – also show issues with the JTB definition of knowledge, so trying to save it from the disjunction case seems like a wasted effort anyway.

trying to save it from the disjunction case seems like a wasted effort anyway. — Michael

You may be right. I'm just exploring, but the way you tell it, I don't see Max as much of a problem. But I agree that there is a problem with JTB anyway. That's why all this is interesting. Does Gettier 'get at' the problem? Do you or he have a solution? Epistemology is a bit of a mess, and it seems to have infected politics.

Using (p v q) as a means to represent Smith's belief is problematic. The content of Smith's thought/belief is precisely what's in question. Smith's believing Q cannot be adequately represented with belief that:((p v q) is true). Considering the truth conditions of (p v q) is required for knowing what (p v q) means. Rational people do not believe something if they do not know what it means. That's what rational people do. Smith is rational. As argued heretofore, Gettier's formula cannot take an account of Smith's consideration of the truth conditions of his particular disjunction(s).

To further show how the use of "p v q" is utterly inadequate, we all need to consider some other aspects of Smith's thought/belief. Smith must think about the rules of correct inference. The earlier post of Smith running modus ponens through his mind is a more than apt demonstration of the distinction between thought/belief about the rules. Belief that P and belief that Q are belief statements about the world while belief that:((p v q) is true)) is belief about the rules. Gettier claims Smith recognized the entailment, which is thinking about the rules of correct inference. Smith must think about the rules. Gettier's paper does not take proper account of this.

Who here would argue that thinking about the rules of correct inference is not required in order to arrive at believing Q ,when Q is derived from P, P entails Q, and the thinking/believing agent recognizes this entailment and the proceed to accept Q on the basis of P?

The process I'm setting out, when absent, results in a Smith that cannot believe a disjunction because without thinking about the rules, one cannot recognize the entailment.

Again, rational people do not assent to believing Q unless they understand what Q means, Q's following from P's notwithstanding. Rational people do not assent to believing Q simply because they believe P and some folk say that Q's following from P is necessary and sufficient for believing Q.

That is particularly germane when believing Q is believing a disjunction.

If it's true that one statement follows from(and/or is entailed by) another then it is so solely by virtue of the rules of correct inference saying so. One statement's following from another, in and of itself, neither warrants believing nor counts as believing the statement which is said to follow.

That is particularly germane when discussing believing a disjunction.

Various attempts have been made in recent years to state necessary and sufficient conditions for someone's knowing a given proposition.

To this there is only thing to state...

One must believe a proposition in order to know it. Smith does not believe that ((p v q) is true) except in the sense that "true" indicates being the result of correct inference. Gettier's formula does not have what it takes for Smith to arrive at believing Q.

You wrote:

If I believe that the statement "Jones owns a Ford" is true and written in this book, and if I believe that "Brown is in Barcelona" is also written in this book, then I believe that the statement "one of the two statements written in this book" is true.

How is that any different to saying "I believe that this or that statement is true"? Or "I believe that 'Jones owns a Ford' is true or 'Brown is in Barcelona' is true"? Or "I believe that Jones owns a Ford or Brown is in Barcelona"?

One of the two statements is believed. To state that one or the other is true is to believe that they both could be. It shows uncertainty where none exists. One of the two is believed.

You wrote:

I think what you're doing is conflating Smith's argument and Gettier's argument.

Smith's argument is:

1. p
2. p ⊨ p ∨ q
3. p ∨ q

He recognises it as valid, he believes the premises, and so he believes the conclusion.

Gettier's argument is:

1. Smith believes p
2. Smith believes p ⊨ p ∨ q
3. Smith believes p ∨ q
4. ¬p
5. q
6. p ∨ q

Smith's belief 3 is true because of 6.

This is interesting.

I'm curious. How exactly have you come to differentiate Smith's argument from Gettiers? Furthermore, does it even make sense to say that Smith has an argument? Seems to me that he clearly has a thought/belief process that is laid out by Gettier himself.

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

Is that an accurate rendition of what you're claiming Smith's 'argument' is?