What is the significance of the order of quantifiers and how is it established?

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The order of quantifiers is significant because it determines the scope of the quantified variables in a logical statement. The order of quantifiers specifies which quantifier ranges over which variables in the statement.

For example, consider the following two propositions:

1. For all x, there exists a y such that x+y=2
2. There exists a y such that for all x, x+y=2

In proposition 1, the universal quantifier "for all x" ranges over all possible values of x, and the existential quantifier "there exists a y" ranges over all possible values of y that fulfill the equation x+y=2.

In proposition 2, the existential quantifier "there exists a y" ranges over all possible values of y, and the universal quantifier "for all x" ranges over all possible values of x that fulfill the equation x+y=2.

The order of the quantifiers is established according to the intended meaning of the logical statement. To determine the correct order, one must carefully consider the meaning of the logical expression and use appropriate mathematical, logical, or linguistic conventions.