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Section 1.1: Explaining the Possibility of The Impossible |
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Often in philosophy what we are trying to do is define as best we can a concept, and to try to understand in what contexts that concept will apply. This brings us to the question: how do we define a concept? To "define" a concept is to, To define X correctly is to specify an intension which captures all X things and only X things in the extension of X. What does this mean? First, let's look at the vocabulary. Intension = a meaning or test to determine what objects apply to a concept So "red" is an intention because it gives you instructions about which objects apply to the concept in question (namely, at least all the red objects). If the intension of concept X is "circular and metallic" then you know that all of the objects to which concept X applies are circular and metallic, and that's it. Extension = the set of objects that meet the test laid out in the intension. So if the intension of concept X is "has an engine" then in the extension of X you will have all running boats, cars, lawnmowers, remote control vehicles, planes, etc., since all of these objects meet the description "has an engine". Now recall that to define X correctly is to specify an intension which captures all X things and only X things in the extension of X. Example: define "car" Let's say we define the concept of car as having the intension "having an engine". One we've made this proposed definition or intension, we should check to see what objects fit it (which objects will be in the extension of that concept). What we will discover soon is that in the extension of "car" will be all cars (let's assume that broken cars are not cars) all lawn mowers and all boats (other things too). Are we willing to call boats and lawn mowers cars? If not, then the definition needs some fixing up to rule out the objects we don't want. However, notice that the definition did mention a feature that all cars must have. Without an engine, it cannot be a car. This means that the definition has captured a necessary condition for "being a car". We know something is a necessary condition when: If N is a necessary condition for P, then If there is no N, there is no P. In other words, if "having an engine" is a necessary condition for "being a car", then if there is no engine present, there is no car present. This is good, but we need something more specific, since it is also a necessary condition for "being a boat" that you have an engine. We need sufficient conditions for being a car. A sufficient condition for X is a property that -- once we know it is present -- we know with certainty that an X is present. Knowing the presence of that property is sufficient to know that an X is present. So: If S is a sufficient condition for P, then If there is no P, there is no S. Let's think in terms of "being a triangle". Notice that "having three enclosed interior angles 90, 60, and 30" is not a necessary condition for "being a triangle" (there are right triangles, isoceles triangles, etc), but it is a sufficient condition for "being a triangle". If we know that some thing (call it X) "has three enclosed interior angles 90, 60, 30", then we know that X is a triangle (nothing else meets that description). So if we know that there is no triangle present (no P), then we know that the property of "having three enclosed angles 90, 60, 30" (S) can't be present, because if it were, there would be a triangle present (P). Now take the two concepts of "necessary" and "sufficient" condition and put them together. You have given a perfect definition of a concept when you have provided an intension (a definition) listing both the necessary and sufficient conditions for that concept. |
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