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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.)

Thanks in advance.

ALT
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2 Answers2

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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.

Math Bob
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    Also, you could think of rectangles and squares. All squares are rectangles but not all rectangles are squares. – Math Bob Feb 25 '19 at 04:14
  • @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. – Conifold Feb 25 '19 at 23:15
  • @Conifold, in this particular instance B⇒A cannot be true. – ALT Feb 27 '19 at 02:31
  • @ALT Then your idea is not named "material implication". – Conifold Feb 27 '19 at 08:09
  • @Conifold, hmmm, then what is it called?? – ALT Feb 27 '19 at 23:39
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∀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.

I hope that answers your question!

Bertrand Wittgenstein's Ghost
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