# Does this idea have a name: "A is a type of B, but B is not a type of A"

This question has been touched on in other questions but not answered in a way that fully answers my own question. Like here:

Argument "a is b" but "b is not a" valid?

What is the name of the fallacy characterized by "All A are B; therefore all B are A"?

To put my idea in everyday terms it is "Fleas are a type of parasite, but parasites are not a type of flea". That is, there is a super-set (parasites) and a sub-set (fleas); a hierarchy. Does the statement have a name of any sort? (I'm not looking for a fallacy.)

• I can only think of taxonomy at the moment. en.wikipedia.org/wiki/Taxonomy Feb 25, 2019 at 2:02
• Are you asking for the name of confusing "if A then B" for "if B then A"? That is called affirming the consequent, a.k.a. fallacy of the converse. Feb 25, 2019 at 5:00
• @Bread - It is part of what makes taxonomies useful.
– ALT
Feb 25, 2019 at 23:06
• @Conifold - Not the name of a fallacy, just the name of the idea: “A is a type of B, but B is not a type of A”.
– ALT
Feb 25, 2019 at 23:07
• It is related to the idea of proper subset: if all A-s are B-s, but not all B-s are A-s, then A is a proper subset (subtype, subclass) of B. The set of those B-s that are not A-s is called the set difference of B and A, denoted B \ A, a.k.a. relative complement. Feb 28, 2019 at 4:52

It does not have a name, but a symbol in logic. The symbol is ⇒. For example, you could say:

A⇒B is true but B⇒A is false.

This is called material implication.

• Also, you could think of rectangles and squares. All squares are rectangles but not all rectangles are squares. Feb 25, 2019 at 4:14
• @ALT A⇒B is the material implication, "but B⇒A is false" is not a part of it. In fact, it may well also be that B⇒A is true. Every bachelor is an unmarried man, and every unmarried man is a bachelor. Feb 25, 2019 at 23:15
• @Conifold, in this particular instance B⇒A cannot be true.
– ALT
Feb 27, 2019 at 2:31
• @ALT Then your idea is not named "material implication". Feb 27, 2019 at 8:09
• @Conifold, hmmm, then what is it called??
– ALT
Feb 27, 2019 at 23:39

∀x(Ax->Bx)

∀y(By->Ay)

Is an invalid deduction since we can have two sets A&B such that A⊆B, but B⊄A.

For instance, if we let A= {1} and B={1,2}, we fulfill the condition above.

Let A={set of all cars}, and B={set of all things with 4 wheels}, then A⊆B, but B⊄A.

That is, every car has 4 wheel, but not every 4-wheeler is a car!

That said, there is no particular name for this formal fallacy in Predicate-Logic, but it has a corresponding fallacy in Propositional-logic:

P->Q

Q->P

This called affirming the consequent.