Consider the following sentence: "A glacier is a river of ice."
This is metaphorical. In some sense, a glacier isn't actually a river. A "literal" river has flowing liquid water, not ice.
Let a river(1) be defined to be an archetypal flowing-water river. A glacier isn't a river(1). Rather, a glacier shares some structure in common with a river(1). We may define river(2) to mean some broader category, of things that flow like a river(1), such as:
- A glacier
- A flowing of earth matter in a landslide
- A flowing of chemicals down an incline in a factory
and so on.
A river(2) is a concept by metaphorically extending river(1). It is, in fact, easier to explain the concept of river(2) by first clearly delineating what a river(1) is. A child will have trouble grasping the metaphorical language of "a glacier is a river of ice" until understanding what a river(1) is, such that the notion of flow that generates river(2) can be pointed to with concrete examples.
Formally, we could think of metaphorical extensions in terms of generative probabilistic models: river(2) is formed by taking some generator behind river(1) (namely, the generator of flowing substance) and applying it elsewhere. But, such formalization isn't necessary to get the idea intuitively. See also the picture theory of language; language draws pictures in others' minds, and those pictures are formed generatively/recursively out of different structures; see also generative grammar, the idea that sentences are formed out of lawful recursive structures.
Is ice a form of water?
Consider the sentence: "Ice is a form of water."
What does that mean? Suppose that, by definition, ice is frozen water. Then, the sentence is tautological.
However, the sentence may be new information to a child. What's going on?
Suppose the child has seen liquid water, which we will call water(1). The child has also seen ice, in the form of ice cubes; call the ice of ice cubes ice(1). It is new information to this child that ice(1) is a form of water(1). Concretely, you can get ice(1) by reducing the temperature of water(1) sufficiently and waiting.
At some point, water(1) is metaphorically extended into water(2) to include ice, liquid water, and water vapor. Tautologically, ice(1) is water(2). It is not strange for someone to say "The tank contains water, and some of it is frozen." However, the water(1) concept is still sometimes used, as in the sentence "Water is a liquid."
The water/ice example is, in many ways, much like the river/glacier example (and not just because both are about liquid/solid water): water(1) is metaphorically extended into water(2).
(An etymology question: why do we say that ice is a form of water, not that water is a form of ice? A philosophy question: what would be the difference between the two?)
While I've focused on extensions, other conceptual/metaphorical refinements are also possible; note that the preformal concept of temperature (temperature(1)) which means approximately "things that feel hot, make things melt/boil, and heat up nearby things" is refined into the physics definition of temperature(2) as "average kinetic energy per molecule".
A special case of metaphorical extension is a figure-ground inversion. Consider the following statements:
- All is nature (naturalism).
- All is material (materialism).
- All is physical (physicalism).
- All is mental (idealism).
- All is God (pantheism).
- All is one (monism).
- All is meaningless (nihilism).
Let's examine naturalism first. A child has a preformal concept of nature (nature(1)) from concrete acquaintance with trees, forests, wild animals, rocks, oceans, etc. Nature(1) doesn't include plastic, computers, thoughts, etc.
According to naturalism, all is nature. But, clearly, not all is nature(1). Trivially, plastic isn't nature(1).
However, it is possible to see an important sense in which all is nature(2) (things produces by the same causal laws that produce nature(1)). After all, even humans are animals (note, this is also an extension!), and the activities of humans, including the production of artifacts such as plastic, are the activities of animals, which happen according to the causal laws of the universe.
Naturalism is a kind of figure-ground inversion. We start with nature(1), initially constituting a particular part of reality (trees, rocks, etc). Then, nature(1) is metaphorically extended into nature(2), a "universal generator" than encompasses all of reality, such that even plastic is a form of nature(2). What starts as figure in the ground, becomes the ground in which all figures exist.
And, even after performing this extension, the nature(1) concept remains useful, for delineating what it delineates. While (according to naturalism) all is nature(2), some nature(2) is natural(1), while other nature(2) is unnatural(1). In fact, the nature(2) concept is mostly only useful for pointing at the way in which everything can be generated by metaphorically extending nature(1); after this extension happens, nature(2) is simply the totality, and does not need to be delineated from anything else.
Similarly, materialism extends material(1) (wood, brick, stone, water, etc) to material(2) (things that have substance and occupy space) such that material(2) encompasses all of reality. Notably, materialism is, in a sense, compatible with naturalism, in that perhaps all of reality can be formed out of the nature(2) generator, and all of reality can be formed out of the material(2) generator.
The other cases are left as exercises for the reader.
(thanks to Cassandra McClure for coming up with the terminology of figure-ground inversions applied to concepts)