Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and call it "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Call something like "star-shaped" a "pattern". An object can match a pattern.

Call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows many collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (with respect to some pattern that collections can match).

Likewise a drawing on the left shows a collection of collections, with some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call when the whole imitates its parts "meta", and call when the whole imitates its parts with respect to the way it imitates its parts "doubly-meta". Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Call something like "star-shaped" a "pattern". An object can match a pattern.

Call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows many collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (with respect to some pattern that collections can match).

Likewise a drawing on the left shows a collection of collections, with some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern". An object can match a pattern. Let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows many collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (with respect to some pattern that collections can match).

Likewise a drawing on the left shows a collection of collections, with some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern". An object can match a pattern. Let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows many collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (with respect to some pattern that collections can match).

Likewise a drawing on the left shows a collection of collections, with some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern". An object can match a pattern. Let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows many collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (with respect to some pattern that collections can match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern". An object can match a pattern. Let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows many collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern". An object can match a pattern. Let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

"Odd one out with respect to what property is the odd one out" would not fit left in this Problem: even though this example does seem doubly-meta, it is not doubly-meta in the right way. There is no odd one out with respect to the property of having an odd one out.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. Instead of having the form "X 'X __' ", this is more like "X [the _appearing in 'X_'] ". These examples could make for a different Bongard Problem.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern for collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern on collections).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more specific way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; is gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

Exhaustive list of all exhaustive lists of all ____.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's call something like "star-shaped" a "pattern" and let's call something like "all but one are [pattern]" a "pattern-parametrized pattern". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern of collections match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-parametrized pattern usually has to do with not all objects in the collection fitting the pattern. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Sorted ambiguously are examples that would fit left only by a stretched interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

Exhaustive list of all exhaustive lists of all ____.

X "X Y" vs. all are "X Y"; X Z.

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually getting more like ___"; gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Let's say "meta" means the whole imitates its parts, and "doubly-meta" means the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's say a "pattern" is something like "star-shaped" and a "pattern-parametrized pattern" is something like "all but one are [pattern]". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern collections can match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-dependent-pattern has to do with not all objects in the collection fitting a pattern. (See BP568, which is about solution ideas for BPs that are always overriden by a simpler solution.)

Shown sorted ambiguously above are some examples that would fit left only by particular interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (where every collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

"Exhaustive list of all exhaustive lists of all ____."

Right:

All are "all but one are ___"and all but one are black.

All are "every other is ___" and every other is solid polygons.

All are "gradually getting more like ___" and they are gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Let's say "meta" means the whole imitates its parts, and "doubly-meta" means the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's say a "pattern" is something like "star-shaped" and a "pattern-parametrized pattern" is something like "all but one are [pattern]". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern collections can match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

An unintended solution to this BP is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-dependent-pattern has to do with not all objects in the collection fitting a pattern. (See BP568, which is about solution ideas for BPs that are always overriden by a simpler solution.)

Shown sorted ambiguously above are some examples that would fit left only by particular interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (where every collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

"Exhaustive list of all exhaustive lists of all ____."

Right:

All are "all but one are ___"and all but one are black.

All are "every other is ___" and every other is solid polygons.

All are "gradually getting more like ___" and they are gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Let's say "meta" means the whole imitates its parts, and "doubly-meta" means the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's say a "pattern" is something like "star-shaped" and a "pattern-parametrized pattern" is something like "all but one are [pattern]". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern collections can match).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

Another solution to the BP with the above examples is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-dependent-pattern has to do with not all objects in the collection fitting a pattern.

Shown sorted ambiguously above are some examples that would fit left only by particular interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (where every collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

"Exhaustive list of all exhaustive lists of all ____."

Right:

All are "all but one are ___"and all but one are black.

All are "every other is ___" and every other is solid polygons.

All are "gradually getting more like ___" and they are gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Let's say "meta" means the whole imitates its parts, and "doubly-meta" means the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's say a "pattern" is something like "star-shaped" and a "pattern-parametrized pattern" is something like "all but one are [pattern]". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to various patterns); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern all the collections satisfy).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to various patterns).

Another solution to the BP with the above examples is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-dependent-pattern has to do with not all objects in the collection fitting a pattern.

Shown sorted ambiguously above are some examples that would fit left only by particular interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (where every collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

"Exhaustive list of all exhaustive lists of all ____."

"Odd one out with respect to what property is the odd one out" would not fit left in this Problem; there is no odd one out with respect to the property of having an odd one out. This example does seem doubly-meta, but not in the right way.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. Instead of having the form "X 'X __' ", this is more like "X [the _appearing in 'X_'] ". These examples could make for a different Bongard Problem.

"Odd one out with respect to what property is the odd one out" would not fit left in this Problem; there is no odd one out with respect to the property of having an odd one out. Instead of having the form "X 'X __' ", this is more like "X [the _appearing in 'X_'] ". It is doubly-meta in another sense.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. These examples could make for a different Bongard Problem.

"Odd one out with respect to what property is the odd one out" would not quite fit left in this Problem, although it does seem doubly-meta in some sense. There is no odd one out with respect to the property of having an odd one out. Instead of having the form "X 'X __' ", this is more like "X [the _appearing in 'X_'] ".

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. These examples could make for a different Bongard Problem.

"Odd one out with respect to what property is the odd one out" would not quite fit left in this Problem, although it does seem doubly-meta in some sense.

Still, there is no odd one out with respect to the property of having an odd one out. Instead of having the form "X 'X __' ", this is more like "X [the _appearing in 'X_'] ".

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. These examples could make for a different Bongard Problem.

Right:

All are "all but one are ___"and all but one are black.

All are "every other is ___" and every other is solid polygons.

All are "gradually getting more like ___" and they are gradually getting more like black triangles.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually getting more like "gradually getting more like ___".

Here is another way of putting it:

Let's say "meta" means the whole imitates its parts, and "doubly-meta" means the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is another way of putting it:

Let's say a "pattern" is something like "star-shaped" and a "pattern-parametrized pattern" is something like "all but one are [pattern]". An object can match a pattern. A collection of objects, with respect to a particular pattern, can match a pattern-parametrized pattern.

A drawing on the right shows a collection of collections. Every collection matches the same pattern-parametrized pattern (with respect to some pattern); furthermore the collection of collections matches that same pattern-parametrized pattern (also with respect to some pattern, a pattern all the collections satisfy).

Likewise a drawing on the left shows a collection of collections, and there is some noticeable recurring pattern-parametrized pattern. The collection of collections must match that pattern-parametrized pattern with respect to the pattern of matching that pattern-parametrized pattern (with respect to any pattern).

Another solution to the BP with the above examples is "not all groups share some obvious property vs. all do." It is hard to come up with examples foiling this alternative solution because the pattern-dependent-pattern has to do with not all objects in the collection fitting a pattern.

Shown sorted ambiguously above are some examples that would fit left only by particular interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (where every collection is a palindrome with respect to some property).

Here is a list of left example ideas that would be impossible to make:

"Exhaustive list of all exhaustive lists of all ____."