<text><span class="style10">ets and Paradoxes (6 of 7)</span><span class="style7"></span><span class="style10">VENN DIAGRAMS AND VALIDITY</span><span class="style7">Venn diagrams are very useful when testing the validity of arguments. It is, however, always important that all possible combinations of the sets are represented in the diagram. If, for example, we are interested in three distinct but possibly intersecting sets, we can use three circles, making sure we include areas for the intersections of each pair and of all three. For more than three sets, the diagrams can become quite complex. The use of colors or shading can be helpful.To check that an argument is valid, we have to ensure that the information given in the conclusion to an argument is already contained in the premises. For example, suppose we wish to show whether the following is a valid argument - or not. Premise 1: All ice skaters have a good sense of balance. Premise 2: Some pianists are ice skaters. Conclusion: So some pianists have a good sense of balance.We have three sets to show on our Venn diagram: ice skaters (</span><span class="style26">S</span><span class="style7">), pianists (</span><span class="style26">P</span><span class="style7">) and those with a good sense of balance (</span><span class="style26">B</span><span class="style7">). Now, premise 1 is equivalent to saying that all members of S are also members of </span><span class="style26">B</span><span class="style7">. That is, the part of set </span><span class="style26">S</span><span class="style7"> that is outside </span><span class="style26">B</span><span class="style7"> must be empty (i.e. have no members). On the diagram this area is colored red. Premise 2 tells us that at least one pianist is an ice skater, that is, the intersection of sets </span><span class="style26">S</span><span class="style7"> and </span><span class="style26">P</span><span class="style7"> has at least one member. Part of this area (colored red) is already known to be empty, so any members must be in the area colored green. We mark this area with a tick.The conclusion that some pianists have a good sense of balance means that the intersection of </span><span class="style26">B</span><span class="style7"> and </span><span class="style26">P</span><span class="style7"> is not empty. This area is surrounded by a heavy line in the diagram. We see that the area already contains a tick, meaning it is not empty, so the conclusion must be true when the premises are; that is the argument is valid.In contrast, we can consider an invalid argument: Premise 1: All ice skaters have a good sense of balance. Premise 2: Some pianists are ice skaters. Conclusion: So all pianists have a good sense of balance.Premises 1 and 2 are the same as before, so we can consider the same Venn diagram. However, the conclusion asserts that set </span><span class="style26">P</span><span class="style7"> is empty except for its intersection with set </span><span class="style26">B</span><span class="style7"> (the heavily outlined area). The premises do not tell us whether this is true or false; the conclusion goes beyond the premises and the argument is thus shown to be invalid.</span></text>
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<text>ΓÇó THE SCIENTIFIC METHODΓÇó CORRESPONDENCE, COUNTING AND INFINITYΓÇó COMPUTERSΓÇó LOGIC</text>
