"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." - Function and Concept
There is an absolute metaphysical dichotomy between functions and objects and a parallel linguistic dichotomy between function-expressions and object-expressions. Frege says both that functions are unsaturated and that function-expressions are unsaturated, while both objects and object-expressions are saturated. Obviously, whatever unsaturatedness is, the way in which function-expressions are unsaturated is only analogical to the way in which functions are unsaturated; function-expressions are in need of supplementation by an object-expression, while functions are in need of supplementation by an object.
One of Frege’s greatest insights, which resulted in the development of the propositional calculus and later developments in formal semantics, was to extend this understanding of functional analysis from mathematics to logic and natural language. Frege states,
For example, the expression ‘Caesar conquered Gaul’ can be split up into the singular term ‘Caesar’ and the function-expression ‘conquered Gaul’. ‘Caesar’ designates a certain object and ‘conquered Gaul’ stands for a certain function, i.e. a function that takes an object x as argument an gives the True if x conquered Gaul and gives the False if it is not the case that x conquered Gaul. These type of functions, that is functions from objects to truth-values, Frege called “concepts”. Since concepts are just certain functions, they too are unsaturated.
"Such being the essence of a concept, there is now a great obstacle in the way of expressing ourselves correctly and making ourselves understood. If I want to speak of a concept, language, with an almost irresistible force, compels me to use an inappropriate expression which obscures…the thought. One would assume, on the basis of analogy with other expressions, that if I say ‘the concept equilateral triangle’ I am designating a concept…But this is not the case; for we do not have anything with a predicative nature…the [designatum] of the expression…is an object." - Comments on Sense and Designatum
Names and expressions of the form 'the F' are saturated and thus must designate saturated entities, i.e. objects. Concepts cannot be named. For example, if I tried to name my favorite concept, say, 'Conner', the name 'Conner' is a saturated expression and thus cannot designate an unsaturated entity. If 'Connor' designates anything it is an object and therefore cannot designate my favorite concept, even though I stipulated that it should.
It gets worse. Since expressions of the form 'the F' are saturated and must designate saturated entities, it follow that, in particular, expressions of the form 'the concept F' must designatesaturated entities and so cannot designate concepts. This leads Frege to endorse the strange sentence 'The concept horse is not a concept'. The first three words in the strange sentence compose a saturated expression and thus must designate an object; and so since 'the concept horse' designates an object, the concept horse is not a concept. Frege thinks that in this case we are “confronted by an awkwardness of language” that “cannot be avoided”.
In order to designate an unsaturated entity we must use an unsaturated expression. For example, 'x is a horse' designates an unsaturated entity, an entity that we are compelled to call “the concept horse”, but of course the expression is inappropriate, since it designates an object not a concept. Yet, since 'x is a horse' designates an unsaturated entity, there must be a unique entity it designates. The problem is that language does not allow us to say which entity. We cannot even say that 'x is a horse' designates the entity designated by 'x is a horse', without committing ourselves to 'x is a horse' designating an object, since any expression of the form 'the entity designated by A' must designate an object. Hence, in addition to denying that the concept horse is a concept he must deny the apparently analytic principle that for any expression A, A designates the entity designated by A.
Frege explains that identity can only be thought of as holding for objects, not concepts. When we say that object a is the same as object b, we are saying that a falls under all and only the concepts that b falls under. Whereas when we say that concept F is the same as concept G we are not saying that the concept F falls under all and only the concepts that G falls under, instead we are saying that all and only the objects that fall under F also fall under G.
Object-Sameness: a = b iff AF(Fa iff Fb)
Concept-Sameness: Fx ~ Gx iff Ax(Fx iff Gx)
In the Begriffsschrift the unsaturatedness of a concept is represented by an empty place where the object-expression must go. But since this place must be filled in order to have a well-formed expression, we could never have an expression that designates the concept itself flanking one side of an equality, e.g. an expression of the form 'F( ) = A' is not complete until we fill the space between the parentheses with an object-expression; but once we do this the left side of the equality will designate an object. It seems we are limited to either (i) making a claim about an identity between the extensions of concepts, i.e. ÊF(E) = ÊG(E), or (ii) making a universal claim about objects. Neither of these capture the relation we wish to state. Frege concludes that “the only recourse we really have is to say 'the concept F is the same as the concept G' and in saying this we have of course named objects, where what is intended is a relation between concepts.” What we want to say is (lambda x)[Fx] = (lambda x)[Gx], but Frege would insist that these lambda expressions must designate objects since they are saturated expressions (or perhaps we could regard functions as sets of ordered pairs of an element of the domain coupled with an element of the co-domain, but for Frege a set is an object, and thus cannot be a function. So we are left only with suggestions and allusions of what we are talking about, although what we really want to say cannot be said.
"I admit that there is a quite peculiar obstacle in the way of an understanding with my reader. By a kind of necessity of language, my expressions, taken literally, sometimes miss my thought; I mention an object when what I intend is a concept." - On Concept and Object
Frege maintains that he is relying on a reader who will meet him halfway, one "who does not begrudge a pinch of salt". If we concede to Frege that concepts can only be designated by incomplete expressions, while objects only designated by complete expressions, the obstacle we are dealing with immediately arises. If Frege is just introducing us to a formal language, then perhaps we should allow him this oddity. But Frege thinks that this obstacle is necessarily insurmountable. It is not just that German or the Begriffsschrift has this inherent limited expressivity, it is a necessary limitation of any linguistic system. Frege would insist that we cannot just stipulate that an expression like '(lambda x)[Gx]' designates that thing of which we cannot usually speak (i.e. that thing we would like to designate by the inappropriate expression 'the concept G'). Frege is gripped by this analogy between language and thought, where concept-expressions are in need of supplementation just as the concept itself is in need of supplementation. He thinks it is absolutely crucial that thoughts have an unsaturated component, so he transposes this commitment to the expression of thoughts in language: they too must be unsaturated.
We can agree that "not all parts of a thought can be complete; at least one must be unsaturated or predicative; otherwise they would not hold together", I see no reason to transpose this metaphor to the expression of thought.
Let '(lambda x)[Gx]' designates the unsaturated component of the thought usually expressed by the predicative aspect of the sentence 'a is G'. Given this stipulation we now have the means to express this same thought by the expression '(lambda x)[Gx](a)'. If we sever the parallel between language and thought, we can allow that the “saturated” expression '(lambda x)[Gx]' designates an "unsaturated" entity. Although it is clear that Frege would not accept this modification of his theory, it seems like this avoids the obstacle (or has the difficulty only been shifted?)