The following is an example of a valid argument form:
Person X has reputation for being an expert on Y.
Things said about Y by a person who has a reputation for being an expert on Y are likely to be correct.
Person X said Z about Y.
Z is likely to be correct.
That argument is not valid.
Not only is it valid it is trivially so. It does not even rely on the possibility of there being valid inductive arguments. I made it the most simple of deductions from supplied premises.
Valid arguments don’t become invalid with the introduction of additional information, but the argument you provided does.
Your problem here seems to be that you object to deducing a conclusion of ‘likely to be’ from a premise of ‘likely to be’. By very nature of uncertain information things that are merely likely do not always occur and yet this does not make reasoning about likely things invalid so long as uncertainty is preserved correctly. (The premise could possibly be neatened up such that it includes a perfect technical explanation with ceritus paribus clauses, etc but the meaning seems to be clear as it stands.)
There is no contradiction between an argument having arbitrarily high inductive strength (like the very best arguments from authority) and still being invalid.
If the argument was in the form of a deduction when only an induction is possible from the information then the appeal to authority is invalid. If the argument is a carefully presented inductive claim then it most certainly can be valid.
Not all arguments are deductions. Not all arguments that are not deductions are invalid.
The following is an example of a valid argument form:
Person X has reputation for being an expert on Y.
Things said about Y by a person who has a reputation for being an expert on Y are likely to be correct.
Person X said Z about Y.
Z is likely to be correct.
That argument is not valid.
Not only is it valid it is trivially so.
Jayson_Virissimo is talking about logical validity. The argument is not logically valid, because it is possible for “Z is likely to be correct” to be false, even if the other statements are true (for instance, add the premise “Z is incorrect”). Induction is not (in general) logically valid. It’s valid in other senses, but not that one.
Jayson_Virissimo is talking about logical validity.
Yes, we both are. We have gone as far as to accept a shared definition of logical validity and trace the dispute from there.
The argument is not logically valid, because it is possible for “Z is likely to be correct” to be false, even if the other statements are true (for instance, add the premise “Z is incorrect”).
This is simply false. The following premise:
Things said about Y by a person who has a reputation for being an expert on Y are likely to be correct.
… becomes invalid the moment there is in fact a “thing said ” that is not likely to be correct. That’s why I put it there! It is an instance of the class of premise “ALL G ARE W” and so just like all other premises in that class it is false if there is a G that is NOT W. it just so happens that ‘likelyhood’ is the subject matter here.
The above serves to make the premise in question rather brittle. While it does means that the whole argument can be treated as deductive reasoning (about the subject of likelyhoods) it is also means that there are very few worlds for which that premise is true and meaningful.
I interpreted your premise as:
(Things said about Y by a person who has a reputation for being an expert on Y) are likely to be (correct.)
as opposed to
(Things said about Y by a person who has a reputation for being an expert on Y) are (likely to be correct.)
If, as you seem to be agreeing, a thing cannot be “likely to be correct” and “incorrect” (as known by the same reasoner), then the premise reduces to “Things said about Y by a person who has a reputation for being an expert on Y are correct”.
I interpreted your premise as: (Things said about Y by a person who has a reputation for being an expert on Y) are likely to be (correct.) as opposed to (Things said about Y by a person who has a reputation for being an expert on Y) are (likely to be correct.)
The second was the intended meaning.
If, as you seem to be agreeing, a thing cannot be “likely to be correct” and “incorrect” (as known by the same reasoner), then the premise reduces to “Things said about Y by a person who has a reputation for being an expert on Y are correct”.
Given the ‘as known by the same reasoner’ clause wouldn’t that imply that it is ‘<...> cannot be known to be incorrect’? Either way it is clear that the encapsulation of the probabilistic parts is woefully inadequate here.
That which is said to be invalid in the text that you link to (things such as generalizing from anecdotes to make mathematically certain claims about a set) is not the same kind of reasoning as that which we are talking about here. Here we are talking about probabilistic arguments, about which you say:
Probabilistic arguments are necessarily invalid
That leaves us at an impasse. There is not really much more I can say if you pit yourself against what is a foundational premise of this site: That the correct way to reason from evidence is to use Bayesian updating. You have essentially dismissed the vast majority of all useful reasoning as invalid. I disagree strongly.
And by “invalid” I mean that it would not be self-contradictory to affirm the premises and deny the conclusion.
It is a straightforward matter to construct arguments based on probabilistic reasoning (and, by extension, arguments from authority) that adhere to that criteria. They go something like:
IF all evidence available indicates p(B|A) = 0.95 AND all other available evidence about B gives p(B) = 0.4 AND all evidence available indicates p(A) = 0.7 AND A THEN available evidence indicates that the probability of B is slightly over 0.54
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).
It may come as a shock to your philosophy tutor from freshman year but it actually is possible to reason logically about probabilities.
Yes, I think I originally actually meant to put the actual A and B around the other way in the conclusion, which is how I did the actual math on the calculator. If A is the thing that has happened p(B|A) is the thing that belongs in the conclusion. Let me fix that. Thanks again.
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.
The terms “valid” and “invalid” have a precise logical meaning; that is the meaning Jayson_Virissimo intends, as they have said many times now.
I have no problem parsing Jayson’s claims. I would even repeat them if I wanted to guess the password of my highschool math teacher. However it is my assertion that the precise logical meaning has been applied incorrectly in this context. The problem is one of applying basic knowledge about logic without knowing enough about how to reason logically about probability.
As you are using them, you seem to mean “well-grounded, justifiable, effective, appropriate, and etc.”
That’s not the case? I’m surprised. I apologize for having misinterpreting you, but that really did seem to be what you were saying.
My claim, as unambiguous as I can make it, is that probabilistic arguments of the form presented here are valid such that to reject the conclusion but not one of the premises is it be inconsistent. I did not expect it to be a controversial claim to make in this context.
I don’t think it’s a question of “insufficient effort” really—the claim you made in this post was simply incorrect, and then you acted condescending towards people who didn’t “understand” it. This post seems to include a valid argument, but it’s a different type of argument from the ones you were talking about earlier in the thread.
This post seems to include a valid argument, but it’s a different type of argument from the ones you were talking about earlier in the thread.
That post is approximately the same argument as the one you consider incorrect. The first instance just didn’t make the reduction to “logical reasoning about probabilities” sufficiently explicit and used too much potentially ambiguous language.
I doubt tabooing the term “valid” would have helped. In my first reply to wedrifid I gave an explicit definition, a link to said definition (which includes citations), and an example. What more could you ask for?
It has generally been my experience, when a term proves problematic in discussion, that providing my definition for that term doesn’t work as well as either (a) agreeing to use the other person’s definition, when I understand it well enough to do so, or (b) not using the term.
Not only is it valid it is trivially so. It does not even rely on the possibility of there being valid inductive arguments. I made it the most simple of deductions from supplied premises.
Your problem here seems to be that you object to deducing a conclusion of ‘likely to be’ from a premise of ‘likely to be’. By very nature of uncertain information things that are merely likely do not always occur and yet this does not make reasoning about likely things invalid so long as uncertainty is preserved correctly. (The premise could possibly be neatened up such that it includes a perfect technical explanation with ceritus paribus clauses, etc but the meaning seems to be clear as it stands.)
If the argument was in the form of a deduction when only an induction is possible from the information then the appeal to authority is invalid. If the argument is a carefully presented inductive claim then it most certainly can be valid.
Not all arguments are deductions. Not all arguments that are not deductions are invalid.
Jayson_Virissimo is talking about logical validity. The argument is not logically valid, because it is possible for “Z is likely to be correct” to be false, even if the other statements are true (for instance, add the premise “Z is incorrect”). Induction is not (in general) logically valid. It’s valid in other senses, but not that one.
Yes, we both are. We have gone as far as to accept a shared definition of logical validity and trace the dispute from there.
This is simply false. The following premise:
… becomes invalid the moment there is in fact a “thing said ” that is not likely to be correct. That’s why I put it there! It is an instance of the class of premise “ALL G ARE W” and so just like all other premises in that class it is false if there is a G that is NOT W. it just so happens that ‘likelyhood’ is the subject matter here.
The above serves to make the premise in question rather brittle. While it does means that the whole argument can be treated as deductive reasoning (about the subject of likelyhoods) it is also means that there are very few worlds for which that premise is true and meaningful.
I interpreted your premise as: (Things said about Y by a person who has a reputation for being an expert on Y) are likely to be (correct.) as opposed to (Things said about Y by a person who has a reputation for being an expert on Y) are (likely to be correct.)
If, as you seem to be agreeing, a thing cannot be “likely to be correct” and “incorrect” (as known by the same reasoner), then the premise reduces to “Things said about Y by a person who has a reputation for being an expert on Y are correct”.
Is this really what you intended?
The second was the intended meaning.
Given the ‘as known by the same reasoner’ clause wouldn’t that imply that it is ‘<...> cannot be known to be incorrect’? Either way it is clear that the encapsulation of the probabilistic parts is woefully inadequate here.
No, but it does seem to be the implication.
No it cannot.
That which is said to be invalid in the text that you link to (things such as generalizing from anecdotes to make mathematically certain claims about a set) is not the same kind of reasoning as that which we are talking about here. Here we are talking about probabilistic arguments, about which you say:
That leaves us at an impasse. There is not really much more I can say if you pit yourself against what is a foundational premise of this site: That the correct way to reason from evidence is to use Bayesian updating. You have essentially dismissed the vast majority of all useful reasoning as invalid. I disagree strongly.
You are correctly restating my claim. The vast majority of all useful reasoning is invalid. And by “invalid” I mean that it would not be self-contradictory to affirm the premises and deny the conclusion.
It is a straightforward matter to construct arguments based on probabilistic reasoning (and, by extension, arguments from authority) that adhere to that criteria. They go something like:
IF all evidence available indicates p(B|A) = 0.95
AND all other available evidence about B gives p(B) = 0.4
AND all evidence available indicates p(A) = 0.7
AND A
THEN available evidence indicates that the probability of B is slightly over 0.54
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).
It may come as a shock to your philosophy tutor from freshman year but it actually is possible to reason logically about probabilities.
Instead of:
you mean:
Right?
Yes, I think I originally actually meant to put the actual A and B around the other way in the conclusion, which is how I did the actual math on the calculator. If A is the thing that has happened p(B|A) is the thing that belongs in the conclusion. Let me fix that. Thanks again.
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.The terms “valid” and “invalid” have a precise logical meaning; that is the meaning Jayson_Virissimo intends, as they have said many times now.
As you are using them, you seem to mean “well-grounded, justifiable, effective, appropriate, and etc.”
Really this all could have been avoided if you all had just taboo’d the offending terms.
I have no problem parsing Jayson’s claims. I would even repeat them if I wanted to guess the password of my highschool math teacher. However it is my assertion that the precise logical meaning has been applied incorrectly in this context. The problem is one of applying basic knowledge about logic without knowing enough about how to reason logically about probability.
That isn’t actually the case.
That’s not the case? I’m surprised. I apologize for having misinterpreting you, but that really did seem to be what you were saying.
My claim, as unambiguous as I can make it, is that probabilistic arguments of the form presented here are valid such that to reject the conclusion but not one of the premises is it be inconsistent. I did not expect it to be a controversial claim to make in this context.
I don’t think it’s a question of “insufficient effort” really—the claim you made in this post was simply incorrect, and then you acted condescending towards people who didn’t “understand” it. This post seems to include a valid argument, but it’s a different type of argument from the ones you were talking about earlier in the thread.
See my reply to you in that context.
That post is approximately the same argument as the one you consider incorrect. The first instance just didn’t make the reduction to “logical reasoning about probabilities” sufficiently explicit and used too much potentially ambiguous language.
I doubt tabooing the term “valid” would have helped. In my first reply to wedrifid I gave an explicit definition, a link to said definition (which includes citations), and an example. What more could you ask for?
It has generally been my experience, when a term proves problematic in discussion, that providing my definition for that term doesn’t work as well as either (a) agreeing to use the other person’s definition, when I understand it well enough to do so, or (b) not using the term.
Is your experience different?
And please see here for the most relevant reply (to a comment declaring an equivalent definition.)