That argument is a simple and valid deduction (with an implied premise of ‘rudimentary probability theory’). The conclusion cannot be (coherently) denied without denying a premise. This is what we are doing when we reason probabilistically (‘we’ referring to ’people while they are lesswrong thinking mode or something similar).
We can argue from first principles about logic and probability until the cows come home, but all it would take for me affirm your original critique of my position would be for you to supply an instance of an argument from authority in which it would be self-contradictory to affirm the premises and deny the conclusion.
It may come as a shock to your philosophy tutor from freshman year but it actually is possible to reason logically about probabilities.
We can argue from first principles about logic and probability until the cows come home, but all it would take for me affirm your original critique of my position would be for you to supply an instance of an argument from authority in which it would be self-contradictory to affirm the premises and deny the conclusion.
I am confused again. I just gave you an example of a valid probabilistic argument—the paragraph before the one you quote. I thought the instantiation to an argument from authority was made clear in the introduction. If it is not then let’s say:
A = “Bob says Gloops are plink”
B = “Gloops are plink”
This makes the argument:
IF all evidence available indicates p(Gloops are plink|Bob says Gloops are plink) = 0.85
AND all other available evidence about Gloops gives p(Gloops are plink) = 0.4
AND all evidence available indicates p(Bob says Gloops are plink) = 0.3
AND Bob says Gloops are plink
THEN available evidence indicates that the probability that Gloops are plink is approximately 0.64
Edit: Caspian noticed that the previous magic numbers were flawed. New magic numbers supplied!
I’m sure something similar could be a valid argument in favour of treating argument from authority as useful evidence, but I’m not finding it straightforward to check this argument above to see if it works.
‘available evidence’ in the last (THEN) line includes ‘Bob says Gloops are plink’ but ‘evidence available’ in the first three lines does not, right? Can the ‘all evidence available’ and ‘all other available evidence’ in the first three lines be taken to include all prior evidence known before finding out ‘Bob says Gloops are plink’? If so, the first three premises are contradictory—Bob says Gloops are plink 0.7 of the time, and almost all of that time he is correct, so p(Gloops are plink) > 0.4. If not, I need some further clarification of what probabilities are conditional on what evidence.
If so, the first three premises are contradictory—Bob says Gloops are plink 0.7 of the time, and almost all of that time he is correct, so p(Gloops are plink) > 0.4.
Thankyou and well spotted. Those completely arbitrary magic numbers don’t stand up to a second glance. In particular 0.7 is just silly. If you already know what the expert is going to say you barely need the expert to say it and so cannot be in a state of knowledge such that p(B) is so low. I’d better change them.
Okay, I think I’ve located the source of our disagreement (and it isn’t about validity at all). The term “probabilistic argument” is ambiguous. I have been using the term to refer to arguments that rely on inductive inference to move from the premises to the conclusion (in other words, the evidential link between premises and conclusion is less than perfect or the inductive probability is less than 1, since 0 and 1 are not probabilities). Alternatively, you seem to be using the term to mean an argument made up of statements that contain probabilities. Allow me to provide an example to illustrate the difference between the two notions:
Argument One
The probability of A is X.
The probability of B is Y.
A and B are mutually exclusive.
Therefore, the probability of A or B is X + Y.
Argument Two
X percent of F’s are G’s.
H is an F.
This is all we know about the matter.
It is X percent probable that H is a G.*
Argument One is valid, while Argument Two is invalid (although, it may be inductively strong depending on the size of X). According the my working definition, only Argument Two is a “probabilistic argument”, but (if my interpretation is correct) according to your working definition, both Argument One and Argument Two and “probabilistic arguments”. Does that sound about right?
Okay, I think I’ve located the source of our disagreement (and it isn’t about validity at all). The term “probabilistic argument” is ambiguous.
I think you’re right. Disagreement about the (potential) validity of Arguments from Authority is only a secondary outcome from what we consider Arguments from Authority to be.
I have been using the term to refer to arguments that rely on inductive inference to move from the premises to the conclusion (in other words, the evidential link between premises and conclusion is less than perfect or the inductive probability is less than 1, since 0 and 1 are not probabilities). Alternatively, you seem to be using the term to mean an argument made up of statements that contain probabilities.
I would not quite draw the line in the same place but it is perhaps best not to argue over the details.
Argument Two
X percent of F is G.
H is an F.
Probably, H is a G.
I agree that this in invalid (and my intuition agrees—I physically flinch if I imagine myself writing that). At the very least it needs an additional premise.
I think you’re right. Disagreement about the (potential) validity of Arguments from Authority is only a secondary outcome from what we consider Arguments from Authority to be.
I guess you’re right.
I would not quite draw the line in the same place but it is perhaps best not to argue over the details.
Sounds reasonable enough.
I agree that this in invalid (and my intuition agrees—I physically flinch if I imagine myself writing that). At the very least it needs an additional premise.
I added a premise and reworded the conclusion to match the standard formulation of the statistical syllogism here, but the argument form remains invalid (although, like I said earlier, it has the potential for high inductive strength depending on the size of X).
It’s unclear what it would mean to qualify the conclusion of a proof with “probably” as in your example, though. What does “Probably, H is a G” mean? Is it a (mathematical) statement about probabilities? Or is “probably” just a rhetorical qualifier to trick someone into thinking we’re allowed to conclude “H is a G”?
It’s unclear what it would mean to qualify the conclusion of a proof with “probably” as in your example, though. What does “Probably, H is a G” mean? Is it a (mathematical) statement about probabilities? Or is “probably” just a rhetorical qualifier to trick someone into thinking we’re allowed to conclude “H is a G”?
I agree that there is some ambiguity there regarding what ‘probably’ is supposed to mean when used that way and fortunately in this case it doesn’t even matter how we resolve that ambiguity. The probability that H is a G given known information could be less than 0.5 given information that the argument neglects to include. Without including (or implying) another premise it doesn’t matter much what definition of ‘probably’ we plug in!
...would be an invalid argument, H might not be a G.
Yeah, that’s why I said:
Argument One is valid, while Argument Two is invalid (although, it may be inductively strong depending on the size of X).
We are in agreement about the invalidity of Argument Two.
It’s unclear what it would mean to qualify the conclusion of a proof with “probably” as in your example, though.
Argument Two isn’t a proof. It is an argument form called the statistical syllogism. The statistical syllogism is induction, not deduction (like a mathematical proof would be).
Is it a (mathematical) statement about probabilities? Or is “probably” just a rhetorical qualifier to trick someone into thinking we’re allowed to conclude “H is a G”?
In my example, “probably” is meant to indicate that the conclusion does not necessarily follow from the premises (but that there is still an evidential link between the two). Induction is not simply rhetoric and it doesn’t involve any deception (although the problem of induction can be a real pain in the ass sometimes).
is an invalid argument, yes. My problem is that I don’t know what an argument like
Most Fs are Gs.
H is an F.
Probably, H is a G.
is even meant to mean.
Well, to digress a bit, the real problem is I’m not sure if any of this nonsense is actually getting to the heart of the issue, which is that probabilistic arguments aren’t really logical arguments at all. Not in the sense that they’re illogical or invalid or anything, but the whole system of bayesian reasoning just doesn’t really map 1:1 onto logic.
What I mean by this is that a logical brain, as one might design one, would have a small pool of statements, the belief pool, which it would add to as observations or deductions are made. A maze solving robot, for example, might have beliefs such as {at time t=0 I was at START, at time t=1 I was at (1,0), at time t=1 there was a wall on my left, …}. It would add to the belief pool as facts about the robot’s location and the maze are discovered, but never remove a statement from the pool, since the pool contains only certainties.
Logical arguments, like “If at any time there was a wall on my left and I was at position P, then the maze has a wall at configuration Q” are useful to this robot, since it can use them to fill its belief pool with such arguments’ conclusions. Moreover, a classification of arguments into valid and invalid is useful for this robot, so that it can ignore the ones which could result in introducing false statements into its belief pool.
You can’t really do the same thing with probabilities. The closest thing to a representation of probabilistic reasoning in logic is the mathematical deduction of statements about conditional probability, with conclusions like P(A | evidence XYZ) = 0.462. When you encounter new observations you use them by trying to generate theorems of the form P(X | all previous evidence + the new evidence) = Y, whereupon you can then plug X and Y into your expected utility calculations or whatever.
In this system an argument like “X% of F are G, and H is an F, so H is probably G” isn’t really an argument where you can then import the resulting conclusion into your “belief set”, because there isn’t any such thing. If the argument means anything at all, it’s as an informal derivation of P(H is G | all relevant evidence) after informing the reader that X% of F are G, and H is an F, assuming that the reader doesn’t have any other relevant evidence. It wouldn’t make sense to say that this argument is invalid since H might not be a G, because it’s not asserting that H is G, it’s asserting that P(H is G | relevant evidence) = y.
We can argue from first principles about logic and probability until the cows come home, but all it would take for me affirm your original critique of my position would be for you to supply an instance of an argument from authority in which it would be self-contradictory to affirm the premises and deny the conclusion.
Also, what’s with the snark?
I am confused again. I just gave you an example of a valid probabilistic argument—the paragraph before the one you quote. I thought the instantiation to an argument from authority was made clear in the introduction. If it is not then let’s say:
A = “Bob says Gloops are plink”
B = “Gloops are plink”
This makes the argument:
IF all evidence available indicates p(Gloops are plink|Bob says Gloops are plink) = 0.85
AND all other available evidence about Gloops gives p(Gloops are plink) = 0.4
AND all evidence available indicates p(Bob says Gloops are plink) = 0.3
AND Bob says Gloops are plink
THEN available evidence indicates that the probability that Gloops are plink is approximately 0.64
Edit: Caspian noticed that the previous magic numbers were flawed. New magic numbers supplied!
I’m sure something similar could be a valid argument in favour of treating argument from authority as useful evidence, but I’m not finding it straightforward to check this argument above to see if it works.
‘available evidence’ in the last (THEN) line includes ‘Bob says Gloops are plink’ but ‘evidence available’ in the first three lines does not, right? Can the ‘all evidence available’ and ‘all other available evidence’ in the first three lines be taken to include all prior evidence known before finding out ‘Bob says Gloops are plink’? If so, the first three premises are contradictory—Bob says Gloops are plink 0.7 of the time, and almost all of that time he is correct, so p(Gloops are plink) > 0.4. If not, I need some further clarification of what probabilities are conditional on what evidence.
Thankyou and well spotted. Those completely arbitrary magic numbers don’t stand up to a second glance. In particular 0.7 is just silly. If you already know what the expert is going to say you barely need the expert to say it and so cannot be in a state of knowledge such that p(B) is so low. I’d better change them.
That didn’t solve the main problem, I think I found what it was. I’ll reply to the other post.
Okay, I think I’ve located the source of our disagreement (and it isn’t about validity at all). The term “probabilistic argument” is ambiguous. I have been using the term to refer to arguments that rely on inductive inference to move from the premises to the conclusion (in other words, the evidential link between premises and conclusion is less than perfect or the inductive probability is less than 1, since 0 and 1 are not probabilities). Alternatively, you seem to be using the term to mean an argument made up of statements that contain probabilities. Allow me to provide an example to illustrate the difference between the two notions:
Argument One
The probability of A is X.
The probability of B is Y.
A and B are mutually exclusive.
Therefore, the probability of A or B is X + Y.
Argument Two
X percent of F’s are G’s.
H is an F.
This is all we know about the matter.
It is X percent probable that H is a G.*
Argument One is valid, while Argument Two is invalid (although, it may be inductively strong depending on the size of X). According the my working definition, only Argument Two is a “probabilistic argument”, but (if my interpretation is correct) according to your working definition, both Argument One and Argument Two and “probabilistic arguments”. Does that sound about right?
Note: This version of the statistical syllogism can be found on page 81 of Harry Gensler’s Introduction to Logic: 2nd Edition.
I think you’re right. Disagreement about the (potential) validity of Arguments from Authority is only a secondary outcome from what we consider Arguments from Authority to be.
I would not quite draw the line in the same place but it is perhaps best not to argue over the details.
I agree that this in invalid (and my intuition agrees—I physically flinch if I imagine myself writing that). At the very least it needs an additional premise.
I guess you’re right.
Sounds reasonable enough.
I added a premise and reworded the conclusion to match the standard formulation of the statistical syllogism here, but the argument form remains invalid (although, like I said earlier, it has the potential for high inductive strength depending on the size of X).
Most Fs are Gs.
H is an F.
Therefore H is a G.
would be an invalid argument, H might not be a G.
It’s unclear what it would mean to qualify the conclusion of a proof with “probably” as in your example, though. What does “Probably, H is a G” mean? Is it a (mathematical) statement about probabilities? Or is “probably” just a rhetorical qualifier to trick someone into thinking we’re allowed to conclude “H is a G”?
I agree that there is some ambiguity there regarding what ‘probably’ is supposed to mean when used that way and fortunately in this case it doesn’t even matter how we resolve that ambiguity. The probability that H is a G given known information could be less than 0.5 given information that the argument neglects to include. Without including (or implying) another premise it doesn’t matter much what definition of ‘probably’ we plug in!
Yeah, that’s why I said:
We are in agreement about the invalidity of Argument Two.
Argument Two isn’t a proof. It is an argument form called the statistical syllogism. The statistical syllogism is induction, not deduction (like a mathematical proof would be).
In my example, “probably” is meant to indicate that the conclusion does not necessarily follow from the premises (but that there is still an evidential link between the two). Induction is not simply rhetoric and it doesn’t involve any deception (although the problem of induction can be a real pain in the ass sometimes).
Damn, you edited your comment >.<
We are in agreement that
is an invalid argument, yes. My problem is that I don’t know what an argument like
is even meant to mean.
Well, to digress a bit, the real problem is I’m not sure if any of this nonsense is actually getting to the heart of the issue, which is that probabilistic arguments aren’t really logical arguments at all. Not in the sense that they’re illogical or invalid or anything, but the whole system of bayesian reasoning just doesn’t really map 1:1 onto logic.
What I mean by this is that a logical brain, as one might design one, would have a small pool of statements, the belief pool, which it would add to as observations or deductions are made. A maze solving robot, for example, might have beliefs such as {at time t=0 I was at START, at time t=1 I was at (1,0), at time t=1 there was a wall on my left, …}. It would add to the belief pool as facts about the robot’s location and the maze are discovered, but never remove a statement from the pool, since the pool contains only certainties.
Logical arguments, like “If at any time there was a wall on my left and I was at position P, then the maze has a wall at configuration Q” are useful to this robot, since it can use them to fill its belief pool with such arguments’ conclusions. Moreover, a classification of arguments into valid and invalid is useful for this robot, so that it can ignore the ones which could result in introducing false statements into its belief pool.
You can’t really do the same thing with probabilities. The closest thing to a representation of probabilistic reasoning in logic is the mathematical deduction of statements about conditional probability, with conclusions like
P(A | evidence XYZ) = 0.462. When you encounter new observations you use them by trying to generate theorems of the formP(X | all previous evidence + the new evidence) = Y, whereupon you can then plug X and Y into your expected utility calculations or whatever.In this system an argument like “X% of F are G, and H is an F, so H is probably G” isn’t really an argument where you can then import the resulting conclusion into your “belief set”, because there isn’t any such thing. If the argument means anything at all, it’s as an informal derivation of
P(H is G | all relevant evidence)after informing the reader that X% of F are G, and H is an F, assuming that the reader doesn’t have any other relevant evidence. It wouldn’t make sense to say that this argument is invalid since H might not be a G, because it’s not asserting that H is G, it’s asserting thatP(H is G | relevant evidence) = y.