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How can I translate the following into reasonable sentence ?

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You can let Rxyz stand for anything as long as it makes sense.

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For a mathematical example, see the "standard" epsilon-delta definition of limit :

for all epsilon, exists a delta such that, for all x ...

For a natural language example, see Jan von Plato, Elements of Logical Reasoning (2013), page 118 :

A sequence of quantifiers such as ∀x∃y∀z is quite natural.

There is for every person some problem on which every attempt will fail.

The three-place relation here is: person x fails on problem y in attempt z. This example also illustrates the choice of a domain. In this case the domain seems to contain, at least, persons, problems, and attempts. There is a threeplace relation that we can write as Fail(x, y, z), in which x is a person, y a problem, and z an attempt. These have to go in the right places in the relation.

The usual way to do it is to introduce predicates such as Person(x), Problem(y), and Attempt(z), and to write the formalization as in:

∀x∃y∀z [Person(x) and Problem(y) and Attempt(z) → Fail(x, y, z) ].


Note that

∀x∃y∀zRxyz

by Double Negation is equivalent to :

¬¬∀x∃y∀zRxyz.

Now, using the rules for quantifiers :

  • ¬∀ is equivalent to ∃¬

and :

  • ¬∃ is equivalent to ∀¬

we can "move inside" the inner negation, getting :

¬∃x∀y∃z¬Rxyz.

Thus, the two formulae are equivalent.


Thus, the equivalent version of :

There is for every person some problem on which every attempt will fail

can be read as :

There is no person such that for every problem there is an attempt which does not fail.

| improve this answer | |
  • I think you translated the first part correctly (∀x∃y∀zRxyz) = "There is for every person some problem on which every attempt will fail." but now how would you translate ¬∃x∀y∃z¬Rxyz ? By the way that's all I was asking for. (+1 for the extra stuff) – Questions Dec 8 '14 at 18:22

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