Intension vs. Extension

In 1892 Gottlob Frege, whose work focused on the mathematization and formalization of language, pointed out a very important difference between Sinn, i.e., sense, and one sense of Bedeutung, i.e., reference (Gottlob Frege, 1892). He used these words in speaking about texts, sentences, phrases, and words. In 1947 Rudolf Carnap helped the English speaking world to understand what Frege meant by these German words (Carnap, 1947). “Sense” he called the intensional meaning — the meaning that inheres as a constant value in the expression itself. “Reference” he called the extensional meaning — the meaning that the real world contributes to the mental concept being conveyed by the text. The distinction is important in coming to an understanding of the best way to formalize the kinds of meaning expressed and conveyed when we use language.

What about “intention”?  
Before delving deeper into the distinction between intensional and extensional meaning, it would be good to clarify what is meant by intentional meaning, when spelled with a t. This latter word has to do with the will to act or behave in a certain way. This kind of meaning depends critically on a person’s disposition toward past actions and desire toward future actions. In contrast, when we spell the identical sounding word with an s, the intention is to have a word with a meaning opposite to extension. One strategy of modern linguistic philosophers, seeking to avoid the inevitatble confusion that is bound to arise, is to use internal for relations involving the former term and external for those of the latter.

Concepts of mathematics.  
Many successful attempts to formalize concepts are found in mathematics. When we want to say that one value is the same as another, the most straight-forward way to express such a fact in arithmetic is: 1 = 1, 2 = 2, etc. In algebra we are able to assign letters to classes of numbers and express the fact that a constant equals itself: a = a, b = b, etc. In English we might say “George is George.” Such a statement is just as uninformative in English as it is in arithmetic and algebra. These kinds of statements are tautological. They tell us more about the nature of equals (=) and “is” than they do of what we are actually talking about.

In mathematics it is usually quite easy to distinguish between the value assigned to the constant expression, i.e. the number, and the constant expression itself, i.e., the letter of the alphabet that stands for that value. Some letters have been adopted to stand for one and only one value. Thus we have pi (π) to stand for the number representing the ratio of a circle’s circumference to its radius. This is handy, since it is not possible to translate such a ratio to any finite set of whole numbers and fractions thereof.

Frege’s puzzle.  
Examples where the difference between sense and reference is not always appreciated abound in mathematics. Suppose a scientific investigator uses some letter, say, “ʄ”, to refer to some particular quantity. An instance of “Frege’s Puzzle” is the fact that when the mathematician says “ʄ = ʄ” the statement is tautological, just like “π = π”, but when one of them says “ʄ = π” the statement is actually quite informative. The informational content has nothing to do with the fact that both letters happen to be referring to the same thing in the real world. The thing that is identical is not the thing being referred to, but the nature of the referring. The information being conveyed is that the “reference” of the letters is identical, and that the components of their “sense” must be adjusted in the mind. It is science that has made sense of the reference. When I said “1 = 1”, or “a = a”, we could make no sense out of it, aside from the meaning of equals, the “=”. But the meaning of equals did not reside in the mark, but in the nature of the symbols being made to equal. It is when the symbols on either side are different that there is a useful interpretation to the statement using it.

Frege’s own statement of his puzzle involved an actual event in the history of science. The Romans had a name for the Morning Star, “Hesperus” and another one for the Evening Star, “Phosphorus.” It was a significant discovery of empirical science when they found that Hesperus was none other than Phosphorus, simply viewed at different times of the day. What Frege was pointing out is that names have both a sense and a reference, so that different terms with the same reference do not have the same sense. Part of the sense of “Hesperus” was that it was seen in the morning, while part of the sense of “Phosphorus” was that it was viewed in the evening. They were still referring to the same object in the real world.

Extensions of historical people.  
Another example comes from the statement: “Jesus was a Jewish itinerant preacher” (Hermann Helbig, 2006, p. 307). The word “Jesus” has several aspects of meaning and rich cultural connotations with it. Here we are not talking about the different manifestations of the same entity, not different people with the same name. First, there is the extensional meaning, the reference to the individual male person who lived in the first few years of our era. It could be imagined that this person did not actually live. Second, there is the intensional meaning, the sense of his persona, his role in history, culture, and religion throughout the ages. For someone thinking of Jesus in this way we can say (metaphorically) He cannot die and will live forever. For language users focusing on the sense of this special name, the original sentence sounds like: “Jesus was (nothing but) a Jewish itinerant preacher.”

Everyday extensions.  
The backbone of language is made up of instances of words each having an identical sense but a different reference depending on the context of its use. This relationship is indispensable to human communication. I refer to my spouse as “my wife,” but you refer to your spouse possibly with those same words but possibly as “my husband.” In this case we realize that it is the relationship that is the sense of these words, not the reference. At some point in the scale from sense to reference we chose the words that the conventions of language allowed or obliged us to because of our perception of their reference in the real world. And then we change terms when the sense must be different, even though the extensional meaning, who it refers to, may be the same. My children refer to my wife as their mother. Part of the extensional meaning of these particular kinship terms is 1) their reference to a female human being, and 2) how they imply a particular relationship to the different people who make use of the reference. In the same way my boss may be your employee, my garden may be your junk heap. A personal name like “George” carries the reference to a particular individual with it when used naturally. Unlike the name “Jesus” this name has little or no sense out of context — no intensional meaning. It is a mere label to be used as needed for a chosen individual. If we are pressed for its sense out of context, we conjure up a prototypical George, we think of the people in our memory and in history, say George Washington. Oscar Wilde made use of this fact to humerous effect in his play The Importance of Being Earnest. In that play one character makes the (non-existent) intensional meaning of Earnest the precondition of her affection toward the one so named, while another makes an imagined extensional meaning her precondition to love another. Both objects of affection are willing to be rechristened to overcome their lovers’ imagined obstacles. Attention to the distinction between intensional meaning and extensional meaning is essential to a fruitful study of names.

Does a reference to an extension ever really exist?  
A modern linguistic philosopher of some influence has made names out to be rigid designators in the sense that they refer to objects in the real world in a way that other words do not (Saul Kripke, 1972). However, even this serious attempt has recently been shown to fall into the same trap of attributing to intensional language some special power of designating a real extension (Wolfram Hinzen, 2007). Noam Chomsky has expressed doubts about the existence of a real relation between words and the extensions of those things (A. Noam Chomsky, 2000). Even very common words as a matter of course “refer” to contradictory extensions. Consider, for example, the obvious shifting of extension for the book in: “The book about linguistics which John wrote in his head weighs two kilos in hard cover;” and for the city of London in: “The city of London is so unpleasant and frenetic that it should be destroyed utterly and rebuilt ten miles further up the Thames.” The implication is that reference of language to an extension must be taken as a technical term arising from the needs of some particular theory of cognition. Furthermore, the idea of building up a thought-experiment to explicate such a technical term is either circular or irrelevant.

Principle of compositionality.  
One thing that Frege wanted to point out was that when we have an opinion about some fact or object, or take any other attitude toward it, we must state that attitude in terms of sense, not reference. This is the case with the study of semantics. The investigator must focus on the way the texts, sentences, phrases, words, and even parts of words, in a language work together and how their arrangements go about making sense. “Frege’s Principle” or the “Principle of Compositionality,” adopted by many semanticists today, is derived from Frege’s work on sense and reference. This principle states that expressions of the language are composed of parts, and that the analysis of the parts and the way the parts go into making the whole must be done in terms of their sense. The solution of Frege’s puzzle warns us that their reference is relative and transitory and cannot be counted on to tell us the way things really are.

Extensions in science.  
The scientist does not necessarily want to study the phenomena of nature in the same way that language has come to refer to them. A particularly apt case in point concerns the discovery that heat and work do not refer to anything tangible (Peter Atkins, 2003, pp. 110–112) Before 1798 heat was something like a liquid, which scientists called “caloric,” that could flow from one object to another. This theory arose despite the fact that heat was “subtle” (could enter any substance), and was “imponderable” (could not be weighed). In that year Benjamin Thompson (1753–1814) showed that heat could be created at will and was inexhaustible. Since it was generated by friction, it must be related to the motion of particles. Involved with extensive study using the steam engine scientists attempted to show how work, measured in ergs, could be related to heat, measured in calories. Sometime before 1875 James Joule (1818–1889) found that there was a direct equivalence between the work done (mechanical energy) and the heat generated (heat energy). In fact they are two manifestations of one and the same thing! “Heat” and “work” are really verbs, i.e., two ways of transferring energy from one object (location) to another. We speak of an object as “hot,” but the more accurate truth is that the object, if it must be “storing energy,” is doing so by the rapid vibrations of its parts. Heat is the agency of transfer and not an entity being transferred.

The “mind-body” problem.  
Since the time of René Descartes philosophers have wrestled with the philosophical foundations of the mind, whether it is material or etherial (Philosophical Essays: Meditations on First Philosophy, 1641). Though this man through reason alone placed the seat of the soul in the peneal gland, subsequent philosophers have been unable to explain how an immaterial soul could impinge on a body of flesh. Even Isaac Newton’s theory of gravitation required that physical bodies exert an etherial force across distances of many miles. Science has increasingly required more and more from their instruments of observation, and less and less from human perceptions. This tendency is particularly accute in the realms of particle physics and cosmology. Chomsky points out that the whole problem of mind-body requires a coherent notion of “body,” which appears to be increasingly difficult to formulate (A. Noam Chomsky, 2000).

Deceptive metaphors.  
Many things in our language and culture are imponderable and subtle in the above sense. Like heat and work these concepts would be candidates for investigation as manifestations or epiphenomena of things more concrete. Sometimes light is better seen as a wave and sometimes as a particle. These are two metaphors, two senses, for the same reference. Language has no word for light in the sense of science’s mathematical model. Compare this with the scientific position toward the word weight where it is often more precise to call it mass. Today there are even theories about the mind that explain it as an epiphenomenon of the brain. Maybe both ways of modeling it are appropriate in their own contexts. Similarly the spirit or soul of man is brought into this framework of explanation. But their reference to the real world is very subtle and, despite many attempts to prove otherwise, imponderable. Language is full of sensible metaphorical expressions that help us along the road to understanding. But these senses cannot be explained by science without building models which work out their references to elements of some not yet very well understood systems in the real world.