9/5/2026
The fallacy, in English (illicit conversion)
'All cats are mammals' is true. It does not follow that 'all mammals are cats'. The inference is obviously invalid, yet the same move with something less concrete ('all F are G, so all G are F') is a standing trap. The moment the example is less obviously wrong than cats and mammals, trouble lurks.
A second fallacy, in English (existential import)
'All unicorns have horns' sounds like it should mean there's at least one horned thing. In modern logic it does not: a universal claim about an 'empty' category is vacuously true. Aristotle's logic assumed that categories are nonempty categories; ours does not. Modern logicians think that 'All unicorns have horns' is true. They also thing that 'All unicorns are horn-free'is true. Because, there are no unicorns. Careful now, they also think that 'It is not the case that all unicorns have horns.' is false. Why? because it is the negation of a true statement.
The diagram
Universe = {a, b, c}
C (Cat) = {a}
M (Mammal) = {a, b} (Cat is a proper subset of Mammal)
U (Unicorn) = { } (nothing at all is a unicorn)
Try it
Formula (c), `∀x(Cat(x)→Mammal(x))`, is true — every cat is a mammal. Formula (d), the converse `∀x(Mammal(x)→Cat(x))`, is false — *b* is a mammal but not a cat. Select (d) and **Deny** it, then watch the game find *b* as the individual that breaks the universal claim.
Separately: formula (e), `∀x(Unicorn(x)→Mammal(x))`, is true — vacuously, since nothing is a unicorn. But formula (f), `∃xUnicorn(x)`, is false. Endorsing both (g) and 'so something is a unicorn' is exactly the existential-import mistake.