Question

If \(\forall x\exists y P \left( {x, y} \right)\)is true, does it necessarily follow that \(\exists x\forall yP \left( {x, y} \right)\) is true?

Short answer

Both\(\forall x\exists y P \left( {x, y} \right)\)and\(\exists x\forall yP \left( {x, y} \right)\)are different.

Step-by-step solution

Introduction

Consider\(\forall x\exists y P \left( {x, y} \right)\) to be true.

Proof using meanings

If\(\forall x\exists y P \left( {x, y} \right)\)is true then \(\exists x\forall yP \left( {x, y} \right)\)is not true.

Since \(\forall x\exists y P \left( {x, y} \right)\)is true means “for every x, there exists y such that\(P \left( {x, y} \right)\)is true”.

\(\exists x\forall yP \left( {x, y} \right)\)is true means “there is some x, for every y such that \(P \left( {x, y} \right)\)is true”.

So both are different.