We possess concepts and we combine them into thoughts. Those are deceptively easy words to say. What is this “possessing” of concepts? Somehow concepts are stored in the mind, unconsciously, but not in the form of use or mention: we are not using all of our concepts at any given moment, and we don’t store them by mentioning them in mental quotation marks (which itself would involve using a meta-conceptual concept analogous to using a quotation name of a word). Possessing a concept is not like possessing fingers or frontal lobes: concepts are not possessed in the way bodily parts are. They are more like memories (though not exactly so), but the nature of memory is puzzling too: how do memories exist in the mind? But the question I want to probe here concerns not possession but combination: What is involved in combining concepts?
We don’t even have any bad theories to refute and ridicule. You might point to other uses of “combine” and compare the case of concepts to them. We combine ingredients to make a cake: but that operation is nothing like combining concepts to make a thought—it is not performed with the hands and there is no mixing. What about combining words into a sentence? Here we must tread carefully. If we just mean uttering words in temporal succession, then we know what that is, but it clearly isn’t what happens when we think by combining concepts. There is no uttering and we don’t just string concepts into a temporal sequence—they have to be properly combined to form something meaningful. If we mean combining words in the language of thought, then we have a special case of the problem: what is this combining? The pieces have to fit together, constitute a whole, and produce a proposition: how does the mind achieve this—by what process or mechanism? How, for example, are simple mental acts of predication generated? An individual concept is somehow hoisted into consciousness at the same time as a general concept and the two are somehow brought into juxtaposition. But what is this juxtaposition? It can’t be just that they exist side by side, spatially or temporally; they have to be combined. What is the mental glue? What is the mode of connection? A whole is assembled from parts, but what kind of assembly is it?
We can imagine dualist or materialist theories of conceptual combination. The dualist theory is apt to be mainly negative: conceptual combination is not any kind of physical combination. It is not the joining together of extended things into a more extended thing, like pieces of Lego. Rather, the immaterial mind enables concepts to link up in a quasi-magical way, as only an immaterial mind can. The trouble with this is that it is not an explanation; and surely we don’t want the puzzle of conceptual combination to require dualism for its solution. The materialist view will maintain that combinations in the brain underlie conceptual combination—as it might be, the co-excitation of distinct neural networks. No doubt there exist underlying physical complexes in the brain, but it is hard to see how they could constitute and explain the combination of concepts. They exist at the wrong level of analysis; we should be able to say something about concepts as such that articulates what is involved in their combination. What is it about a concept that enables it to slide so smoothly into a linkage with another concept? What properties does it have that explain its combinatorial powers? There are theories about the referential powers of concepts (such as causal theories), but what theories are there about the power of concepts to hook up with each other? Concepts can combine with certain other concepts but not with others: what is the difference? You can combine the concept John with the concept house to get the concept John’s house, but you can’t combine John with Mary to get John Mary or house with planet to get house planet. Concepts can accept or reject potential partner concepts depending on their inner nature. They can repel or attract other concepts.
We might now try to take a leaf out of Frege’s book: he said that some concepts are saturated and some are unsaturated.  An unsaturated concept contains a space for a saturated concept, thus saturating it. This is no doubt an obscure doctrine, though not without some intuitive pull, but the question is how to apply it to psychological processes. Concepts (senses), for Frege, are abstract non-psychological entities, so his notion of saturation applies at that level: but how does it apply at the level of concepts in the psychological sense (“ideas” in Frege’s terminology)? In what sense is a psychological entity like my concept house “unsaturated”? This seems like metaphor or mumbo-jumbo (choose your poison).
We don’t experience the mode of joining that concepts undergo or engineer, so we can’t observe how the combination works. It is this secret joining that allows for the famed productivity and infinity of possible thoughts (and meaningful sentences), but it is quite opaque to introspection or any other mode of observation. If concepts didn’t combine, thought would be impossible, even quite limited thought. If concepts lost their ability to combine, through some sort of brain ailment, thought would stop dead in its tracks. The glue is at least as important as the items glued. But the glue doesn’t reveal itself—it is hardly as if concepts have sticky ends! Even metaphors are thin on the ground here; no possible theory suggests itself. One’s feeling is that joined concepts are a bit like people holding hands—there is a part that is designed as a gripping or hooking device. But this is absurd fantasy or pointless poetry not the beginnings of a theory. Alternatively, one speaks of synthesis: in conceptual combination a synthesis of concepts is formed. That sounds right enough, but again it is hardly a theory, more like a reformulation of the problem. For what is it to synthesize concepts? Complex concepts have parts that are brought together, but how are they brought together? What is this “bringing together”?
We know what combining physical objects is—spatial aggregation—but what is combining the units of thought? In Frege’s terms, what is the combination of senses (now construed psychologically)? Senses look outward to references, but they also look sideways to other senses—those that they can join with. It is written into a sense what it refers to, but it must also be written into it what it can combine with—with this but not with that. And it must be possible for senses to lock together into complex senses for the duration of a thought and then dissolve apart when the thought is over. Some operation splices one sense or concept to another, but then separation reasserts itself. There is a concept-combining device that moves concepts from where they are stored in the mind and forms strings of them displaying internal unity, and then disassembles these strings into their dormant isolated constituents. They are not combined in their stored form, being isolated units, though they are quick to enter into combinations; the combinatorial device imposes on them a kind of brief marriage with other concepts, quickly leading to divorce. Concepts thus flow in and out of combinations with other concepts; the puzzle is how they get cemented together for the duration. What is the composition of the conceptual glue? How do concepts find each other?
Let me try to make the problem vivid by adapting Brentano. He introduced the idea of intentionality as a (non-physical) relation between a mental entity and something that exists outside of it but which is somehow its object: the mental entity is “directed towards” the object, intrinsically connected to it. The relation is somewhat obscure but it seems real enough—thoughts are obviously about things. Let’s introduce the idea of concept-to-concept intentionality, whereby a concept “refers” to any concept with which it can combine. It is written into a concept what kinds of combination it accepts and what it rejects. Furthermore, when a concept is acting as part of a combination it has this kind of horizontal intentionality vis-à-vis the concepts combined with it. There is a relation R such that the concept has R to the concepts combined with it. The concept thus both points outward beyond concepts and also inward to other concepts: it is combinatorial as well as referential. It has a kind of double intentionality. And it needs both aspects if it is to do its job as a constituent of thought: it needs two sorts of relation—inter-conceptual and extra-conceptual. Both are admittedly puzzling and evidently sui generis, but I find the inter-conceptual relation even more elusive and perplexing than the extra-conceptual. In virtue of what do concepts combine? Hume spoke of causation as the “cement of the universe” and found it puzzling; concepts have their “cement of the mind” and it too is puzzling. We don’t even have inadequate theories of it. Indeed, it is far from easy to make the problem visible.
 Strictly, for Frege some concepts (“senses”) stand for saturated entities (“objects”) and some stand for unsaturated entities (“concepts” in Frege’s technical sense): but I am not concerned with the details of Fregean exegesis here.