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The "foundations" of Frege's analysis of language are in his articles :

Relevant for your question is the first one; see :

or :

Following a detailed analysis of mathematical formulae :

The two parts into which a mathematical expression is thus split up, the sign of the argument and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not. [page 141]

Frege generalize this analysis to natural language statements [page 146] :

Statements in general, just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or 'unsaturated.' Thus, e.g., we split up the sentence

'Caesar conquered Gaul'

into 'Caesar' and 'conquered Gaul'. The second part is 'unsaturated' - it contains an empty place; only when this place is filled up with a proper name, or with an expression that replaces a proper name, does a complete sense appear. Here too I give the name 'function' to what is meant by this 'unsaturated' part. In this case the argument is Caesar.

Previously [page 146], Frege writes :

We thus see how closely that which is called a concept in logic is connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value.

Thus, the Fregean analysis of the statement :

"the sky is blue"

must decompose it into two parts : the saturated part, i.e. the name of an object : "the sky", and the unsaturated one, i.e. the expression for a concept : "___ is blue".

In the "functional" syntax of Begriffsschrift (i.e. concept notation) :

blue(sky).

The "foundations" of Frege's analysis of language are in his articles :

Relevant for your question is the first one; see :

or :

Following a detailed analysis of mathematical formulae :

The two parts into which a mathematical expression is thus split up, the sign of the argument and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not. [page 141]

Frege generalize this analysis to natural language statements [page 146] :

Statements in general, just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or 'unsaturated.' Thus, e.g., we split up the sentence

'Caesar conquered Gaul'

into 'Caesar' and 'conquered Gaul'. The second part is 'unsaturated' - it contains an empty place; only when this place is filled up with a proper name, or with an expression that replaces a proper name, does a complete sense appear. Here too I give the name 'function' to what is meant by this 'unsaturated' part. In this case the argument is Caesar.

Previously [page 146], Frege writes :

We thus see how closely that which is called a concept in logic is connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value.

Thus, the Fregean analysis of the statement :

"the sky is blue"

must decompose it into two parts : the saturated part, i.e. the name of an object : "the sky", and the unsaturated one, i.e. the expression for a concept : "___ is blue".

The "foundations" of Frege's analysis of language are in his articles :

Relevant for your question is the first one; see :

or :

Following a detailed analysis of mathematical formulae :

The two parts into which a mathematical expression is thus split up, the sign of the argument and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not. [page 141]

Frege generalize this analysis to natural language statements [page 146] :

Statements in general, just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or 'unsaturated.' Thus, e.g., we split up the sentence

'Caesar conquered Gaul'

into 'Caesar' and 'conquered Gaul'. The second part is 'unsaturated' - it contains an empty place; only when this place is filled up with a proper name, or with an expression that replaces a proper name, does a complete sense appear. Here too I give the name 'function' to what is meant by this 'unsaturated' part. In this case the argument is Caesar.

Previously [page 146], Frege writes :

We thus see how closely that which is called a concept in logic is connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value.

Thus, the Fregean analysis of the statement :

"the sky is blue"

must decompose it into two parts : the saturated part, i.e. the name of an object : "the sky", and the unsaturated one, i.e. the expression for a concept : "___ is blue".

In the "functional" syntax of Begriffsschrift (i.e. concept notation) :

blue(sky).

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source | link

The "foundations" of Frege's analysis of language are in his articles :

Relevant for your question is the first one; see :

or :

Following a detailed analysis of mathematical formulae :

The two parts into which a mathematical expression is thus split up, the sign of the argument and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not. [page 141]

Frege generalize this analysis to natural language statements [page 146] :

Statements in general, just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or 'unsaturated.' Thus, e.g., we split up the sentence

'Caesar conquered Gaul'

into 'Caesar' and 'conquered Gaul'. The second part is 'unsaturated' - it contains an empty place; only when this place is filled up with a proper name, or with an expression that replaces a proper name, does a complete sense appear. Here too I give the name 'function' to what is meant by this 'unsaturated' part. In this case the argument is Caesar.

Previously [page 146], Frege writes :

We thus see how closely that which is called a concept in logic is connected with what we call a function. Indeed, we may say at once: a concept is a function whose value is always a truth-value.

Thus, the Fregean analysis of the statement :

"the sky is blue"

must decompose it into two parts : the saturated part, i.e. the name of an object : "the sky", and the unsaturated one, i.e. the expression for a concept : "___ is blue".