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Let's suppose that intuitions are something halfway between, or a robust melding of, Kantian awareness/representation of particulars and then the "rational hunches" of more modern discourse. Let's assume that seeing a red sky "intuitively supports" judgments like, "The sky is red here and now," or, "I am experiencing a red sky."

Now, when there is talk of something being unintuitive, is this always the same as if to say that it's counterintuitive, or is there an epistemic difference between non- and anti-intuitive states/information? Like the phenomenal conservative might be more amenable to the one rather than the other, for example; or (it might be supposed) it is generally easier to justify non-intuitively justifiable claims than outright counterintuitive ones.

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    probably "φ is counterintuitive" means something like "B¬φ, or [B(ψ) and B(ψ → ¬φ)]", whereas "φ is unintuitive" is somehow weaker
    – ac15
    Commented yesterday
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    I'm not sure how to map it to the terminology, but I certainly think we experience two different states. Consider the famously counterintuitive example of the Monty Hall problem. The problem actively yields an incorrect intuition in many - we think the wrong answer (50/50 odds when you change doors) is the case. Whereas, there is another state where we just have trouble understanding, but we aren't thinking something is the case that is not - which happens all the time.
    – Adam Weisberg
    Commented yesterday

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The difference between unintuitive and counterintuitive is effectively the difference between "I didn't expect that" and "I expected something else entirely." The first is merely surprising, the second somewhat disconcerting. More philosophically, an unintuitive result forces us to expand the horizons of our understanding, while a counterintuitive result foces us to unlearn something: to alter the horizons of our understanding in more complex ways.

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  • There's a paper about epistemic horizons in physics that has been weighing on my mind for some time now, so your reference to such a concept on the philosophical level is apropos, here.
    – Kristian Berry
    Commented 13 hours ago
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Your post is a question for the dictionary. Without much ado the Cambridge dictionary states and exemplifies:

  1. counterintuitive:

  • Something that is counterintuitive does not happen in the way you would expect it to:

    It may seem counterintuitive to open a shop in the middle of a recession.

  1. unintutive:

  • not easy to use or learn, especially without any special training or practice:

    At first the game can seem somewhat unintuitive, although it is actually quite easy to learn.

  • not what you would expect based on beliefs or feelings, although maybe true in fact:

    In this age of the internet, it seems unintuitive that sales of cookbooks have actually increased.

Apparently, there is no sharp discrimination how to use these two words.

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    I asked more about an epistemic than a semantic difference. Do you have any resources on the epistemology of intuition that would be helpful?
    – Kristian Berry
    Commented yesterday
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    @KristianBerry The following book has a chapter on mathematical intuition link.springer.com/chapter/10.1007/978-0-8176-8394-8_2
    – Jo Wehler
    Commented yesterday
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    @KristianBerry, that word "epistemic" gets used a lot here, but really just means relating to knowledge. Would you please clarify the shortcomings of this answer, and how mixing in "epistemic" might clarify your confusion over the terms? Because I don't really understand what you are asking otherwise
    – Michael Hall
    Commented yesterday
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    @MichaelHall I just mean, is there a difference in evidence or justification between the types of claims?
    – Kristian Berry
    Commented yesterday
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    @KristianBerry, ok but what do you mean by "claims"? The terms are subjective descriptors you might apply to a result depending entirely on what your personal preconceived notions were.
    – Michael Hall
    Commented yesterday
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Counterintuitive statements or examples involve a contradiction based on experience. For example, most nuts ( when used with bolts) involve clockwise motion to tighten the nut. A nut that loosens with clockwise motion is counterintuitive.

Unintuitive statements or examples can't be understood based on individual experiences. Quantum mechanics is unintuitive because the effects it describes don't match our natural (unaided by technology) observations.

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  • Just adding examples. Unintuitive: to open an automatic door you need to stand and wave. Counterintuitive: in order to open a tri-state door the handle needs to be set horizontal from being pressed down instead of being pushed down from being horizontal.
    – user3819867
    Commented yesterday
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It has always been my understanding that counter-intuitive and unintuitive are not the exact same.

If something is counterintuitive, it behaves the opposite of what you expect. If something is unintuitive, it does not do what you expect. Everything that is counterintuitive is also unintuitive, but not everything unintuitive is counterintuitive.

Let's say that we are looking at changes in Y to see how they affect X. Let's also say that X and Y are measured values that could go up, down, or stay the same. My intuition is that X must increase as Y increases.

If X doesnt change when Y is increased, or doesn't change as much as I predicted, that result is unintuitive to me, it did not do as I predicted. If X decreases when Y is increased, it is both unintuitive and counterintuitive. It doesn't just fail to follow my predictions, it does the opposite of what I predict.

The area would become more grey in qualitative situations where we might have different opinions on what are behavioral opposites, but in the case of numbers increasing/decreasing/staying the same, it should be clear when something is counterintuitive and when it is just unintuitive.

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There's an x that needs to be made sense of. There's an explanation E for x
Scenarios

  1. You apply your intuition to x and it draws a blank: Unintuitive
  2. You apply your intuition to x and you get an explanation D that doesn't jibe with E: Counterintuitive

Boils down to not getting the stick (unintuitive), as opposed to getting the wrong end of the stick (counterintuitive).

Do nefer things.

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Examples may help.

Intuitive user interface: things are the way you expect. Click the checkbox to check it and turn the option on.

Counterintuitive user interface: things are exactly the opposite of how you expect. Click the checkbox to check it ...and turn the option off.

Unintuitive user interface: things are basically unguessable. If checked, the checkbox turns the option on... but clicking the mouse does nothing: email the front desk to check the checkbox.

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I'll take my shot at explaining the difference.

"unintuitive" - usually hints at the wrong, unintelligent presentation of a subject

"counterintuitive" - points to a legitimately difficult subject that can cause confusion.

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Unintuitive:

Prior knowledge does not provide a logical foundation for this result.

Counterintuitive:

Prior knowledge suggests a result contrary to this one.

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