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In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce free variables by universal specification when required-- something you cannot usually do in mathematics. In most mathematical proofs, free variables are introduced either by a premise or by existential specification (instantiation).

In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce free variables by universal specification when required-- something you do not usually do in mathematics. In most mathematical proofs, free variables are introduced either by a premise or by existential specification (instantiation).

In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce free variables by universal specification when required-- something you cannot usually do in mathematics. In most mathematical proofs, free variables are usually only introduced either by a premise or by existential specification (instantiation).

In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce free variables by universal specification when required-- something you cannot usually do in mathematics. In most mathematical proofs, free variables are introduced either by a premise or by existential specification (instantiation).

In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce new free variables by universal specification when required-- something you cannot usually do in mathematics.

In mathematics, we might have a statement:

For all x in S, we have P(x)

where S is a set, the domain of quantification that may or may not be empty.

Classical first-order logic is a little weird in that every domain of quantification is assumed to be non-empty. It simply isn't necessary. I guess this assumption makes it possible to introduce free variables by universal specification when required-- something you cannot usually do in mathematics. In most mathematical proofs, free variables are usually only introduced either by a premise or by existential specification (instantiation).