2. Higher-Order Concepts

Higher-order concepts refer to percepts-that-refer-to-concepts (i.e. symbols).

Ultimately, the meaning of concepts derives from percepts, and the meaning of percepts derives from objects. However, higher-order concepts may be constructed out of lower-order concepts: this construction is made possible by a symbolic calculus which involves both concepts and percepts. For example, while the concept of ice may be known directly to residents of Alaska, it may be impossible to know directly for residents of a hot country without refrigeration. However, residents of that hot country may still know what ice is, by its definition: solid, cold water. Although those residents do not have associated perceptual data (i.e. no direct experience with the referent of the concept), the concept can still be understood. A calculation of references, independent of the things referenced, can produce a result that can itself be both grasped intellectually and subsequently visualized.

This dichotomy between first and higher-order concepts is present in both language and thought, and has been known in many different contexts with different terminology: synchronic/diachronic, knowledge/news, a priori/a posteriori, synthetic/analytic, de re/de dicto, necessary/contingent, etc. Distinguishing between these two types of sentences is vital: to mistake one type of sentence for the other type leads to no end of confusion. For example, if one person were to say “criminals are bad” , another person might get upset. But how should we understand this sentence? Is it a statement about the word (or concept) “criminals” , and does it define that word? Or is it a statement about a certain kind of person (or object), and does it say something further about their moral (or immoral) character?

Sentences about concepts often take the following form: word is a definition (or subset is-a superset ). Although the copula “ is a ” is very common in this context, several forms of the verb “ to be ” , the verb “ means ,” and various other words can also be used. Some examples of this kind of sentence are shown below. When reading these sentences, you should interpret them as being definitional (i.e. about concepts), as opposed to contingent statements about the world.

  1. Apples are fruits.

  2. An apple is a fruit.

  3. Apples are red.

  4. An apple is a red thing.

In all cases, an object is defined by being a member of a certain class or by possessing a certain characteristic property. In sentences one and three, the phrase “ is a ” is not used, although the sentences can be transformed into sentences which do use the “ is a ” formulation: this results in sentences two and four, respectively. One way of viewing these transformations is simply as a transformation from plural to singular, although in the singular formulation there is the implication that these sentences could apply to any apple.

Statements two and four may be understood as conditional statements about individuals, e.g. “for every thing x , if x is an apple then x is a fruit” . In this case, the underlying conceptual structure is potentially quite different: instead of being a relation between concepts, it becomes a relation between objects. This process is called existential quantification : it turns relationships between concepts into relationships between objects. It is significant here because it demonstrates how logic reduces sentences about language to sentences about the world. From a cognitive point of view, however, this reduction is not necessary: in fact, in order to accurately model cognition, its universal application is incorrect. There are statements which are purely about language, as opposed to quantified things.[101]

Concepts of concepts create the potential for both great understanding and great confusion.

It is paradoxical for something to contain another thing, which in turn contains the original something. However, things can contain references, and those references can refer to things which contain their container. As a simple example, the world contains my head, and I imagine the world. Although it is not the case that both physically contain each other, there are valid points of view in which each contain the other.

The ability to defer the definition of references is one of the more powerful features of references, but it can lead to their misuse. Paradox is a fairly direct result of this misuse. The following phrase is a popular example of this misuse, which is known as the Liar's Paradox:

I am lying.

It seems rather innocuous at first, but it presents a tough question to answer: if we assume it is true, it becomes false; at which point, upon being negated, it becomes true once again. Clearly something has gone wrong here. Why is it that an attempt to answer this question is impossible, or at least leads to an infinite regress?

Probably the most striking thing about this sentence is that it is self-referential: it describes itself. Hence, a reasonable first step in the elimination of paradox eliminates self-reference. Unfortunately, the recognition of self-reference is confounded by the fact that it does not have to be immediate ; it can be a multi-step, circular phenomena, as illustrated by the two statements below:

1. The statement below is false.

2. The statement above is true.

The paradox in this case is more difficult to spot. The first statement is neither true nor false until we evaluate it. Under the assumption that it is true, we negate the truth of the second statement. This negated statement makes the assertion that the first statement is false, which contradicts our original assumption, and so on: the truth conditions do not converge.

The syntax of a given language dictates how to form concepts. Hence, if a sentence is syntactically well-formed, there is good reason to believe that we can form the concept that it was designed to communicate. However, even for sentences which are syntactically well-formed, a well formed concept is not guaranteed: consider Noam Chomsky's famous example, “Colorless green ideas sleep furiously” . Syntactically, we ought to be able to make sense of this; semantically, it turns out to be difficult, if not impossible. This sentence promises the formation of a concept which turns out to be meaningless at best: syntactically well formed sentences may still be semantically vacuous.

Sometimes this conceptual impossibility only becomes clear when we have unsuccessfully tried to form the concept. The paradox of the liar is subtle when it is first encountered; it seems consistent until we try to dereference its symbolic form. Unfortunately, this paradox is one of the easier forms of circularity to spot. For example, suppose you are learning the definition of a novel word, “nifity” . If we are told that “nifity” means “not infity” , we might conclude that we know what “nifity” means. However, if we have previously learned that “infity” means “not nifity” , we may not know what “nifity” means after all. In this case, although there is no paradox, the words are devoid of meaning.

[101] Note that the formulation is-a , due to the use of the indefinite article, expresses a subset relationship. By learning those categories of which a word is a subset, we come to understand the word. It may be significant that there is no convenient way to express the superset relationship. For example, the concept “An apple is a fruit” has a fairly easy rendition compared to “Fruits consist of apples and other things” . In combination with the fact that the linguistic variable (the word to be defined) occurs on the left, this might be taken as evidence of a holistic point of view (i.e. one which does not build concepts up from smaller things, but rather by dividing larger things). On the other hand, we might simply be observing a tendency to put the (to be determined) linguistic variable on the left hand side in conjunction with an Subject-Verb-Object language (i.e. one which structures its subjects, verbs, and objects from left to right).