Does anybody actually make the daft claim that ordinary human reasoning follows patterns of formal logic? I've heard plenty of people make the claim that formal logic is the best way to reason properly (or at least that you should put your logic into valid argument claims), but nobody ever claiming that it fitted how humans generally operated.
Yes, several (mostly Dutch) people make that claim, and it's not a daft one. Don't confuse formal logic with classical logic!
Yes, several (mostly Dutch) people make that claim, and it's not a daft one. Don't confuse formal logic with classical logic!
I'm not sure what you are getting at here. Are you referencing Dutch booking? If so, I fail to see how that's relevant to Carinthium's question which is not about what is ideal reasoning but rather how humans actually reason.
No, I am literally referring to logicians in the present-day Netherlands. There are a bunch of people there who look into the use of (non-monotonic, highly non-classical) logics for modeling human reasoning, perhaps most prominently Michiel van Lambalgen.
Formal logic is a subset of arguments that are generally convincing to people. It's definitely not the best way to reason - that would be an ideal Bayesian - but it is a very good social standard for argumentation.
The term "formal logic" seems to be being used in several senses. I understand it refer to the branch of mathematics, but you seem to be using it to mean "rigorous reasoning". Most people would not understand a statement written in formal logic, according to the former definition, let alone find it convincing. And even with the latter meaning, there is much of formal logic that ordinary people find unconvincing, such as "The statement 'all crows are black' is logically equivalent to 'all non-black things are non-crows'; therefore, if I find something that isn't black, and it isn't a crow, that is support for the claim that all crows are black".
Edit: Warning, spoilers for the needle and disk puzzles, undisguised, in comments below
Solution to the needle and disk puzzles (rot13):
Obgu chmmyrf pbzr qbja gb "vf n enaqbz cbvag ba n fcurer zber yvxryl gb or jvguva rcfvyba bs gur rdhngbe be rcfvyba bs n cbyr", naq gur nafjre vf boivbhfyl gur sbezre, ohg va gur arrqyr chmmyr "arne n cbyr" pbeerfcbaqf gb "iregvpny", juvyr va gur qvfx chmmyr vg pbeerfcbaqf gb "ubevmbagny".
If I'm right, that didn't feel particularly counterintuitive.
I would guess that Johnson-Laird's point is something like this: If you imagine things clearly then the right answer is easy to find, but if you try to puzzle it out by pushing words or symbols around it's much harder and you're liable to be confused by gur snyfr flzzrgel orgjrra "ubevmbagny" naq "iregvpny".
Which is fair enough ... except that I have a weak visual imagination but solved the puzzles very easily. But, so far as I can tell by introspection, I didn't do it by pushing words or symbols around any more than by forming a realistic mental image of needles in space. More like a somewhat abstract kinda-spatial model of the situation, if that makes sense.
He introduces the puzzles with the words "To find out how good your imagery is try this problem". If he really means "imagery" then I think the problem doesn't do what he thinks. Without reading more of the book than I have (I just skimmed a few pages in the Google Books view) it's not obvious to me whether he means strictly visual imagery or merely somewhat spatial mental models.
I've also only read the preview, but I'd say spatial+dynamic and not imaginary-camera full-color visual.
Warning, spoiler ahead
A somewhat more mathematical version: assuming small number of radians n from either the pole or the equator.
For pins, the area of the (unit) sphere is pi*n^2 for the pole and 2pi*n for the equator, with the ratio n/2 for pole/equator area, which goes to zero as n goes to zero.
And as you said, for a disk "near the pole" means its normal is near the equator, so the relevant area is 2pi*n, "near the equator" means its normal is near the pole, so the result is the opposite.
An interesting aside is that disks and pins in 3D are actually Hodge duals (that's the reason torque or magnetic field can be treated as vectors sometimes).
Yes, that's how I visualized i, although it wasn't completely obvious to me why it was correct to do so. I suppose a more obvious, but unwieldy, visualization is marking two special spots (not colinear) along the perimeter of the disk and imagining how they could be animated within a sphere as the disk rotates.
That is clever, and I did not see it so thanks for the solution. I was going off on a tangent wondering if either vertical or horizontal configurations might be more or less stable due to air resistance. Although now that I think about it, I see how the ciphered solution answers the needle problem but not how it solves the disc problem, or rather not how it give the answer you says it gives. The disc problem has more symmetry so I can't see why one orientation would be preferred. I'm probably missing something.
Imagine poking a needle through the middle of a disk, perpendicular to the disk. Then the needle is horizontal when the disk is vertical, and vertical when the disk is horizontal. So the fraction of vertical needles is going to be the same as the fraction of horizontal disks, and the fraction of horizontal needles is going to be the same as the fraction of vertical disks.
Yes, the way I finally understood this was imagine a horizontal disk whose circumference the equator. It can spin on one axis (through the center of the disk from north pole to south) while remaining horizontal. However a vertical disk oriented along a great circle can spin on two axes while remaining vertical, one from the north pole to the south pole crossing a diameter of the disk and one through the center of and perpendicular to the disk.
The key is that there are three different dimensions being collapsed into two "orientations". With a needle, two dimensions are "horizontal", and one is "vertical". With a disk, two are "vertical", and one is "horizontal". Part of the issue with this problem is that people don't generally have an explicit definition of "vertical" and "horizontal". Ask them whether they know what they mean, and they'll say "sure", but ask them to give a definition, and they'll flounder.
I was going off on a tangent wondering if either vertical or horizontal configurations might be more or less stable due to air resistance
Vertical configurations are more stable, if I'm thinking about the problem correctly.
The question looks unclear to me because it's not clear if they're asking about statics or dynamics. If you look at the dynamics of the system, it seems clear that needles being vertical is an attractor, and we should expect randomly initiated particles to be more likely to be in the attractor as time goes on.
But suppose we neglect air resistance. Imparting a random force to the needles and a random attitude to the needles are different, because the first implies to me that most needles will have some angular momentum, and thus we need to look at paths on a sphere rather than points on a sphere, whereas the second makes shminux's interpretation the obvious one.
"random force" is surely odd phrasing, but that's what they used in their experiment. If the coins aren't affected by air (which, oddly, they didn't specify, but I suggested they shouldn't be), and haven't hit anything, then we can imagine most "random force" distributions to choose uniformly a random orientation (because the random distribution goes high, and enough time has passed to do so, even if the starting orientations are all the same). Most importantly, if we assume we can answer the question, then initial orientation can't matter because it wasn't specified.
I haven't read this book, but it is awell-known problem that this theory of mental models isn't formally well-developped and therefore not particularly predictive. Johnson-Laird apparently misunderstands what formal logic is when he claims that the phenomena in question show that it cannot be the basis for human reasoning. He doesn't seem to be aware that you can build all sorts of interesting non-monotonic logics that do unexpected things; and the first paragraph of your citation doesn't indicate that he has become any more competent in this regard.
You may be interested in this recent paper on the Erotetic Theory of Reasoning.
Thanks for the recommendation.
It does seem that Johnson-Laird is good at telling convincing-sounding stories and short on proving/predicting much (and probably his field).
So I'll read it for the collection of interesting experiments and over-confident speculation.
I find those two puzzles to be rife with ambiguities. First, a "force" is not going affect the attitude; only a torque will do that, and as another poster has noted, there's a bit of a difference between "random force" and "random attitude". There's also the question of just what "random" means. The word "random" just means that there is some probability distribution; it does nothing to tell us what it is. Finally, there's the issue of starting orientation. If the needles start out vertical, then the most likely result is that they will end up vertical.
I know - all of us over-precise folk had the same frustrations. But if you allow the problem to be "the things are in a (uniformly) random configuration ('attitude' - nice word choice!) and haven't bounced or even started in some known orientation", it's a fun problem to think about. A uniform orientation seems appropriate since it's maximum entropy given the word-problem constraints.
In the Google books preview, I see the author spends some time claiming that we build iconic visual/spatial representations and that a lot of our thinking isn't verbal or available to verbal introspection (fairly uncontroversial to me).
Depending on what is meant by 'spacial,' the mental life of those blind from birth will address this claim.
Good guess. That's the subject of some experiments mention in the linked preview (it's about making inferences from verbal descriptions of scenes) - when the descriptions are spatial, sighted-from-birth are faster; everybody's the same on visual but non-spatial attributes.
I haven't read it, but from that brief description it sounds like they'll be focusing heavily on stuff like the Wason Selection Task. So, there are definitely experiments that can be done in this general field- but I don't know how Mental Model Theory interprets those experiments, whether it proposes new interesting ones, or how it differs from other theories in this field.
Has anyone read "How We Reason" by Philip Johnson-Laird? He and others in his field (the "model theory" of psychology/cognitive science) claim that their studies refute the naive claim that human brains often operate in terms of logic or Bayesian reasoning (probablistic logic). I gather they'd say that we are not Jaynes' perfect Bayesian reasoning robot or even something resembling a computationally bounded approximation to it.
I'm intrigued by this recommendation:
... formal logic cannot be the basis for human reason. Johnson-Laird reviews evidence to this effect. For example, there are many valid conclusions that we never bother to draw because they are of no practical use to us. We also make systematic errors in reasoning that we would not make using logic. The content of logic problems used in research studies greatly affects their difficulty; it would not if logic were the primary process. We use knowledge to help us imagine possibilities and then evaluate the possibilities for consistency with other evidence.
Constrained by the span of short term memory, the strength of our general intellectual abilities, and our level of expertise, we construct and manipulate mental models of the problems we reason about. "...[F]rom the meanings of sentences in connected discourse, the listener implicitly sets up a much abbreviated and not especially linguistic model of the narrative ... Where the model is incomplete, material may even be unwittingly invented to render the memory more meaningful or more plausible." We can manipulate these models in a number of ways, including updating them with new information, combining two models when appropriate, searching for confirming evidence or information, and using counterexamples to challenge a model's validity.
Mental model theory explains a number of systematic errors human beings make when reasoning. For example, the difficulty of reasoning problems is related to the number of models that must be held simultaneously in memory to work through them. And we exhibit a recurring bias to use a single model to reason about situations that have more possibilities than we can keep track of. We oversimplify. Consistent with model theory, we have difficulty reasoning with information about what is false about a situation.The real key to human rationality is our ability to recognize and grasp the implications of counterexamples
It seems like an interesting read, but I'd like to know if the research field is a scientific one, i.e. that their stories aren't just pleasing, but can predict, or at least explain real phenomena.
In the Google books preview, I see the author spends some time claiming that we build iconic visual/spatial representations and that a lot of our thinking isn't verbal or available to verbal introspection (fairly uncontroversial to me).
I liked the two related imagination-puzzles:
1. I have thousands and thousands of very thin needles, which I hold in a bundle in my hands. I throw them up into the air, imparting a random force to each of them. They fall to the ground, but, before any of them hits the ground, I stop them by magic in mid-air. Many of the needles are horizontal or nearly so, and many of them are vertical or nearly so. Are there likely to be more needles in the first category, more needles in the second category, or do the two categories have roughly equal numbers?
[and the same thing but for very thin circular disks - let's assume they're also dense, so the air isn't a factor]
2. I have thousands and thousands of very thin circular disks, which I hold in a bundle in my hands. I throw them up into the air, imparting a random force to each of them. They fall to the ground, but, before any of them hits the ground, I stop them by magic in mid-air. Many of the disks are horizontal or nearly so, and many of them are vertical or nearly so. Are there likely to be more disks in the first category, more disks in the second category, or do the two categories have roughly equal numbers?
He claims that for spatial propositions where we can imagine a picture that's more or less equivalent ("the cabinet is behind the piano" [as we face the keys]), the negation of that proposition can't be so pictured (in direct correspondence) because ... where would you put the cabinet? You could imagine all the alternative places it could be (presupposing that there is a specific piano and specific cabinet). You could imagine something "not cabinet" behind the piano (a cabinet with a red x, a cabinet repelling field?). He suggests an or(p1,p2,....pn) of images where we imagine the cabinet to be (that aren't behind the piano). I'm not sure what we can conclude from this. We already know that negation is tricky - linguistically, and mentally. Maybe I like to imagine someone telling me "no, you're wrong to say X" - to use a non-visual system.
He explains that inferences about (written) non-spatial visual relations (light/dark clean/dirty) take longer to process than spatial ones, that the spatial and visual word inference word problems had different fMRI hot spots, that congenitally blind people weren't faster on spatial queries (i.e. were slower on average than non-blind, but didn't suffer any additional penalty on the visual ones). I suppose this could be taken as weak evidence that we can perform "logical" inferences with some sort of spatial-relationship processing, and that perhaps non-spatial attributes take longer to translate (even though they refer to visual qualities like light/dark).
I'm leaning toward buying the book, since the writing is pleasant. But I thought first I'd ask if anyone could recommend for the quality of research in this field.