To broaden our understanding of the notion of validity, we will examine a modern version of a branch of logic first developed in ancient times— categorical logic. Categorical logic concerns immediate inferences and syllogisms that are composed of categorical propositions, so we need to begin by explaining what a categorical proposition is.
In the argument above, the first premise asserts some kind of relation- ship between squares and rectangles; the second premise asserts some kind of relationship between rectangles and things with parallel sides; finally, in virtue of these asserted relationships, the conclusion asserts a relationship between squares and things having parallel sides. Our task is to understand these relationships as clearly as possible so that we can discover the basis for the validity of this argument. Again, we shall adopt the strategy of start- ing from simple cases and then use the insights gained there for dealing with more complicated cases.
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A natural way to represent the relationships expressed by the proposi- tions in an argument is through diagrams. Suppose we draw one circle standing for all things that are squares and another circle standing for all things that are rectangles. The claim that all squares are rectangles may be represented by placing the circle representing squares completely inside the circle representing rectangles.