What definition does contemporary analytic philosophy give of 'concept'? And what is the difference between a concept of something and a conception of something? Then what's the difference between concept, conception and theory? I'm trying to establish an epistemological priority like conception>concept>theory.
For the sake of concreteness let's consider the notion of fatherhood (I'm avoiding the use of 'concept' here because we'll be giving it a technical meaning). From experience we have a certain (probably) informal conception of fatherhood. We know, for example, that everyone has a unique father. We know that no one is his own father. We know that two persons sharing a father are siblings. And so on.
Once we have an informal conception of fatherhood we can choose (or devise) a logical framework (possibly equipped with a semantics or a proof theory) appropriate for the explication of the informal conception. We can choose, for instance, a language system that includes predicate symbols 'F', 'S'. Then we can let 'F(x, y)' be the explication of "x is a father of y", and 'S(x, y)' be the explication of "x and y are siblings". We can then capture all sorts of logical relationships between those predicates, corresponding to our informal conception of those relations. Here are some examples of that:
- Everyone has a unique father. ∀x ∃y : F(y, x) ∧ ∀z : F(z, x) → z = y.
- No one is his own father. ∀x : ¬F(x,x).
- Persons sharing a father are siblings. ∀x, y : ∃z [F(z,x) ∧ F(z,y)] → S(x,y).
This process of explication goes some of the way towards the clarification of the notion of fatherhood. The next step is the process of axiomatization, whereby the truths about fatherhood (or to be precise, about the relation F) are reduced to a number of axioms about F, from which, by means of certain rules of inference, the original body of knowledge about F can be 'restored'. It's important to note that there are lots of 'truths' about fatherhood, but axiomatization aims to capture the logical truths about F, i.e., the set of formulas that are true in any model (or interpretation) of the axioms of F. Now, axiomatization is not a trivial process for interesting cases, so a lot of thought goes into choosing the right subset of the truths about F to be the axioms. E.g., a few properties that could be among the axioms of fatherhood:
F is irreflexive: ¬∃x : F(x, x).
F is asymmetric: ∀x,y : F(x, y) → ¬F(y, x).
F is antitransitive: ∀x,y, z : [F(x, y) ∧ F(y,z)] → ¬F(x, z).
Those are just some possibilities among many others. Ideally, the set of axioms would meet certain conditions of independence, completeness (with respect to some semantics), and so on. A theory of fatherhood is just such a system of axioms for F closed under the rules of inference. Needless to say, everything is relative to a language. If we had started with a language that had function symbols (f, s) and '∈' but no explicit predicate symbols (F, S), we could either define predicates by putting restrictions on functions, or we could capture the truths about fatherhood using function symbols as follows:
(1) would be reformulated as: ∀x ∃y : f(x) = y ∧ ∀z : f(x) = z → z = y;
(2) would be reformulated as: ∀x : f(x) ≠ x;
(3) would be reformulated as: ∀x, y : ∃z [f(x) = z ∧ f(y) = z] → [y ∈ s(x) ∧ x ∈ s(y)]),
Both the explication/concept and the theory of the informal conception of fatherhood are intimately tied to an underlying language, and consequently will look different depending on the choice of that language. That's only my way of looking at things. Someone else thinking about the logic of fatherhood would probably choose a different language and come up with a different set of axioms.
For a treatment of explication as the process of moving from classificatory to comparative to quantitative concepts (we didn't talk about quantitative concepts above), look at the first chapter ("On Explication") of the following work:
Carnap, R. (1950) Logical Foundations of Probability.
Given that analytical philosophy is closely tied to analysis of how language is used, I would suggest that a close examination of the word concept and how it is used in philosophical discourse (as opposed to its ordinary usage). This does mean rather examining a large corpus of works, and one would need to decide whether one should look at solely analytical works, or also continental - or both.
I'm more familiar with continental work rather than analytical, so I'll start there. Kant had a notion of concept:
concept: the active species of representation, by means of which our understanding enables us to think. By requiring perceptions to conform to the categories, concepts serve as 'rules' allowing us to perceive general relations between representations.
The categories are the most general notion of concept, like quantity, quality, modality & relation. They are a priori, that is inate, and thus not concieved as such, that is we do not need any mental effort to use them, and were present from ones earliest age. Concepts perform a link between our perceptions and the categories, they are 'active' and thus require mental effort, and thus require concieving. It is a representation, because y idea of a book, is not the book itself, in fact there is no such thing, and nor is it any particular thing. This action of representation, or perhaps in language that is now more familiar, abstraction, is what 'enables us to think'.
Deleuze, defined philosophy as the production of concepts, and not the idea; but what about the relations between concepts, should it not also be a concept? It is, and we have produced our first one!
Deleuze also said that concepts are not ideas, that is not Platonic ideas which live in a Platonic world. Since Deleuze rejects the transcendental he must also rejects Platonic heaven.
To return to analytic philosophy, a key departure is Frege notion of a concept and object:
According to Frege, any sentence that expresses a singular thought consists of an expression (a proper name or a general term plus the definite article) that signifies an Object together with a predicate (the copula "is", plus a general term accompanied by the indefinite article or an adjective) that signifies (bedeutet) a Concept.
Thus "Socrates is a philosopher" consists of "Socrates", which signifies the Object Socrates, and "is a philosopher", which signifies the Concept of being a philosopher.
This was a considerable departure from the traditional term logic, in which every proposition (i.e. sentence) consisted of two general terms joined by the copula "is".
The distinction was of fundamental importance to the development of logic and mathematics. Frege's distinction helped to clarify the notions of a set, of the membership relation between element and set,
An actual quote from Freges On Concept and Object is clarifying:
a concept is the reference of a predicate; an object is something that can never be the whole reference of a predicate, but can be the reference of a [grammatical] subject” (Frege 1892, 198).
Here we see him make the link with the predicate calculus.