1. Dimensions of the Conceptual Universe

First-order concepts refer to percepts, which refer to objects; they derive their semantic value [meaning] from that which they reference and their relationship to other references.

There are numerous dimensions to the conceptual universe. Of key significance are the first conceptual dimensions to be formed; their study tells us a great deal about cognition because there are few complexities to deal with (at least when compared to the study of language use in adults).

Conceptual space is discrete: in other words, concepts are atomic. However, since concepts can be formed out of other concepts, there is a sense in which this is not the case (atomic things cannot be composed of other things). Conceptual space is different than perceptual space, in that concepts are not spatial. In some contexts, however, concepts may be treated in terms of the space (or the dimensionality) of the percepts that they reference.

This book has been building a case for the identification of the psychological notion of concepts with the mathematical notion of sets. Although most of the technical details of this association are left to the last section of the book, several basic similarities are worth pointing out. One is that sets and concepts can be both atomic and composed of other things. In other words, sets are singular entities, even though they may be composed of multiple entities: set braces have the power to unify that which they contain. Concepts have exactly that same quality: they are single entities themselves (which makes them atomic), although they may be defined as a collection of other things (which makes them non-atomic). In psychology, concepts are also known as generalizations or unitizations (behavioral psychologists in particular avoid associating with the word concept, since its definition is often imprecise). The dual nature of both concepts and sets is understood in this book primarily in terms of reference: references themselves are singular, however they may refer to multiple entities in the referenced domain.

Concepts unify the perceptual data on one side of a decision boundary.

A percept represents one side of a (perceptual) decision boundary. A concept, as it represents that percept either directly or indirectly, also entails a conceptual counterpart in virtue of its perceptual counterpart. This forms the basis of logical negation: if we know the concept of apple, we can form the concept of not-apple.

Without knowledge of the larger context in which a concept is defined, however, the logic of negation is somewhat illogical. This is made clear in a famous example by the philosopher Carl Hempel: “All ravens are black” . First, note that this is a conjecture about objects (as opposed to being essential to the definition of a raven): it does not say that ravens are necessarily black. To make it clear that we are talking about the extensions of these concepts (i.e. actual ravens as opposed to the abstract concept of ravens), we might also say “the collection of all raven things is a part of the collection of all black things” . In any case, because this statement is about the world instead of language, we cannot necessarily determine its truth or falsity. As we cannot know the truth conceptually ( a priori ), we must determine the answer in physical space ( a posteriori ).

Hempel asks which things constitute evidence for the statement that all ravens are black. It is not too difficult to see that every black raven that we encounter provides some amount of evidence for this statement. Hence, if we encounter a large number of ravens, we are led to believe in the universal applicability of Hempel's statement. However, Hempel encourages us to consider an additional statement which is logically equivalent to “all ravens are black” : “all non-black things are non-raven things” .[66]

Given these features of inductive learning, however, every thing that we encounter which is a non-black non-raven adds evidence to the thesis that all non-black things are non-ravens, and therefore to the thesis that all ravens are black. However, this (logically equivalent) induction feels paradoxical for many people. It does not sit well with most people that a red bicycle adds support to the hypothesis that all ravens are black.

This finding may be partially explained by noting that the population of non-black non-ravens is much larger than the population of black ravens, so the support for the conclusion that is lent by examples from the larger population is limited to a similar extent. Non-raven things say relatively little about raven things, since the former set is so immense and diverse relative to the latter. However, if we imagine a small world in which only birds existed, encountering a non-black non-raven would provide more evidence for the hypothesis that “all ravens are black” .

A picture is worth a thousand words.

The discussion of the conceptual universe in this book focuses on its role in logical thought. However, direct analysis of concepts and thinking is difficult, in part because it relies on subjective report (where a large amount of subjectivity enters into the picture). Since language is easy to analyze relative to thought, and because of the strong correlation between them, this chapter focuses on words instead of directly on concepts. Unfortunately, there is at least one aspect of cognition which is unamenable to this treatment: intuition.

We take it for granted that studying the structure of languages can tell us a lot about the structure of thought. This assumption is related to a stronger, much-debated hypothesis, which is that conceptual thought is not significantly different than mental speech . Historically, theorists believing in this hypothesis (called the Language of Thought hypothesis) mapped what we knew about the operation of symbols directly onto the operating principles of the brain. This mapping produced biological models which were found to be largely untenable. However, to assume that the deep structure of a sentence corresponds to some internal (cognitive) representation of a thought seems undeniable.

Intuition is a kind of thinking which does not seem to be obviously perceptual or conceptual. Since intuition is so rich, it is tempting to associate it with perception as opposed to conception (since conception is often relatively dichotomous). However, intuition seems to be able to both use and transcend the limitations of the conceptual mind. So instead of classifying intuition as either rational or irrational, we classify intuition as multi-rational.

The way that syntax is used to combine words into sentences implies a single semantic hierarchy, as opposed to multiple hierarchies. In other words, although we might make a pun and thereby provide two or more conceptual hierarchies for a given sentence, the normal mode of (conceptual) understanding is to understand in only one way, or from one point of view. Intuition, by contrast, is a mass of connections and associations which are linked together in many ways. For the intuitive mind, everything is related to everything else.

When we intuit something, it is difficult to say how we arrived at that intuition: an intuition is too rich of a phenomenon to be described within the relatively narrow confines of several dozen symbols. Even though we cannot say exactly what an intuition is in a small number of words, it certainly is possible to say what is occurring in an individual who experiences an intuition: the subjective experience may not be able to be easily described, but it is possible to describe an intuition from an objective perspective. The experience is ineffable; its mechanism is effable.

An intuition is capable of grasping in an instant what it takes volumes of books to say. Again, it is multi-categorical, as opposed to noncategorical: it allows multitudes of associations to arise, as opposed to selecting only one or several. To categorize something is to make relevant a certain feature in virtue of which it is categorized; while that is certainly useful in some contexts, it simultaneously makes numerous other features of that object irrelevant. For example, an apple may belong to the thing-with-seeds category, but if we seize on this aspect of the apple too strongly, we will neglect its other aspects. If we bring only the seedy quality of the apple to the foreground, we forget that it is good to eat, or that it can be carved into tiny statues. For the apple-as-intuited, however, all things are relevant information. Intuition is capable of experiencing something in its entirety, at least in as far as that thing is known.

An intuited something is not either this or that: it is not something taken out of context, or understood in terms of its membership in only one or several categories: it is that thing as it relates to everything else. As a result, the expression of an intuition is a nontrivial task. This may account for the fact that intuitions are related to other intuitions, as opposed to being defined directly. These relationships are expressed as metaphor: for example, “Flesh is like grass” invites us to compare two concepts without being explicit about the numerous relations between them. There are countless implications and unstated associations in metaphor; so many that we rarely (if ever) enumerate all of them. So it stands to reason that we comprehend metaphor with intuition.

[66] These statements are logically equivalent because they express the same underlying structure. They rely on exactly the same dichotomies, so conclusions about the thing on one side of a decision boundary entail conclusions about the thing on the other side.