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Chapters and Appendices
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What is a Variable? |
Chapter 5, Islands and Divisions of Knowledge
Islands Without Roads BetweenImplications are like streets or roads. They may be travelled one-way or both ways. Streets (or implications) may lead nowhere. Others may lead to interesting and sometimes unexpected places. Each road may touch several others. Each of these others may touch several more. But by foot or car, from one road, there is no guarantee that all roads can be reached. Moreover, when some one-way roads are present, poor planning may imply no return route for every possible starting point. Maps make the exploration of any road system easy. All we have to do is read the map. Without a map, we have to explore the neighborhood in which we live, and hope we can find a path back. One-way streets are a danger here, unless another path back is available. Without a good map, we cannot say in advance, when we explore the streets, if we will get to an interesting or boring destination. To find out what is interesting, our only choice is to explore or to ask whether any one has made a map. We would like to learn from the experience of others, perhaps. By road, not all destinations are accessible or reachable. We may for example have roads on several islands with no boats, ferries, planes, bridges or ships to take us between them. Without boats, ferries, planes, bridges, or a very low-tide, we have no route or connection between one island and the next. Without these extra routes, the roads (or implications) of one island are not linked to the roads of another. The streets on even a single island need not all be connected to each other. For example, imagine on one island that a mischievous or artless road planner has provided one-way roads all leading from one end of the island to the other. On such a road system, a return to the starting point is not possible. We can imagine another island in which the planner, mischievous or not, has placed a mixture of one- and two-way roads. From some starting points you can leave but not return. From some parts or destinations, you cannot leave. Between other starting points and destinations, you can go back and forth. And after going back and forth several times, you may forget which place was the destination or the starting point. All the situations just described with one- and two-way streets can happen similarly in logic with one- and two-way implication rules. In other words, knowledge is linked by one- and two-way implication roads, spread over several islands. The map of this area is not complete. As we explore and forget, roads and routes new to us or our neighbors are uncovered or rediscovered.
Chapter Sections: Next Section: Rooms Without Doors Between Next Chapter: 6. Conditional and Biconditional Statements |
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