To Mock a Mockingbird
Raymond Smullyan
2025-26 book club!
Meeting notesBirds:
| Mockingbird | Mx = xx In a forest with a Mockingbird which satisfies the composition condition, every bird A is fond of (A∘M)(A∘M) |
| Kestrel | Kxy = x |
| Identity | Ix = x |
| Lark | Lxy = x(yy) In a forest with a Lark, every bird A is fond of (LA)(LA) |
| Warbler | Wxy = xyy |
Compositors
| Name | Behaviour | Minimal 🅱️ form | Roadkill form |
|---|---|---|---|
| Bluebird | Bxyz = x(yz), Bxy = x∘y | B | B¹ |
| Dove | Dxyzw = xy(zw) | BB | B² |
| Blackbird | B₁xyzw = x(yzw) | BBB | B³ |
| Eagle | Exyzwv = xy(zwv) | B(BBB) | B⁷ = B¹¹ |
| Bunting | B₂xyzwv = x(yzwv) | B(BBB)B or BB(BBB) | B⁸ = B¹² |
| Dickcissel | D₁xyzwv = xyz(wv) | B(BB) | B⁴ |
| Becard | B₃xyzw = x(y(zw)), B₃xyz = x∘y∘z | B(BB)B | B⁵ |
| Dovekie | D₂xyzwv = x(yz)(wv) | BB(BB) | B⁶ = B¹⁰ |
| Bald Eagle | Êxy₁y₂y₃z₁z₂z₃ = x(y₁y₂y₃)(z₁z₂z₃) | EE = B(BBB)(B(BBB)) | left as an exercise (possible to figure out from the rest of the table) |
Beatboxing birds
| Name | Behaviour | Notes |
|---|---|---|
| Cardinal | Cxyz = xzy | Swaps arguments |
| Thrush | Txy = yx | Backward function application |
| Robin | Rxyz = yzx | Rotates |
| Finch | Fxyz = zyx | Reverses |
| Vireo | Vxyz = zxy | Rotates the opposite of a robin |
| Cardinal once removed | C*axyz = axzy | Cardinal but skips over its first argument. R*, F*, and V* are defined analogously. |
| Cardinal twice removed | C**abxyz = abxzy | Cardinal but skips over its first two arguments (or alternatively, Cardinal once removed, once removed). R**, F**, and V** are defined analogously. |
Definitions:
| Fond | A is fond of B if AB = B |
| Egocentric | A is fond of itself (ie AA = A) |
| Agree | A and B agree on a bird x if Ax = Bx |
| Agreeable | A is agreeable if given any B, there exists a bird x on which A and B agree |
| Compatible | A and B are compatible if there exist x and y where Ax = y and By = x |
| Happy | A is happy if it is compatible with itself |
| Normal | A is normal if it is fond of at least one bird NB: in a forest with a Mockingbird and where the composition condition holds, every bird is normal |
| Fixation | A is fixated on B if for every bird x, Ax = B |
| Hopelessly egocentric | A is hopelessly egocentric if it is fixated on itself (ie for every bird x, Ax = A) |
| Left cancellation law for kestrels | if Kx = Ky then x = y |
| Commute | x and y commute if xy = yx |