members of the subject category are or are not included in the predicate category. For example, the statement ‘all scientists are people’ is a classification statement, in which ‘scientists’ is the subject and ‘people’ is the predicate. The four types of classification statement are:
- All S are P: The entire subject class lies within the predicate class. Every member of the subject class is also a member of the predicate class.
- No S are P: The entire subject class is excluded from, or outside, the predicate class. No member of the subject class is a member of the predicate class.
- Some S are P: At least one member of the subject class lies within, and is a member of, the predicate class.
- Some S are not P: At least one member of the subject class lies outside, and is not a member of, the predicate class.
Note that ‘some’ means at least one; it does not mean ‘less than all’. Thus it is possible for both statements ‘All S are P’ and ‘Some S are P’ to be true for the same S and P; if so, the former statement is more powerful. Similarly, both statements ‘Some S are P’ and ‘Some S are not P’ may be true for the same S and P.
The statements ‘All S are P’ and ‘No S are P’ are sometimes referred to as universal statements because they apply to every member of a class. In contrast, the statements ‘Some S are P’ and ‘Some S are not P’ apply not to every member but instead to a particular subset; thus they are referred to as particular statements.
Deductive Aids: Venn Diagrams and Substitution
The four classification statements can be illustrated diagrammatically as shown in Figure 17.
| All S are P | No S are P | Some S are P | Some S are not P |
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| Figure 17. Classification statements, expressed as Venn diagrams. | |||
John Venn, a 19th-century logician, invented this technique of representing the relationship between classes. Each class is represented by a circle; in this case there are only the two classes S or P. Potential members of the class are within the circle and individuals not belonging to the class are outside the circle. The overlap zone, lying within both circles, represents potential members of both classes. Hatching indicates that a zone contains no members (mathematics texts often use exactly the opposite convention). An X indicates that a zone contains at least one (‘some’) member. Zones that contain neither hatching nor an X may or may not contain members. In the next section, we will observe the substantial power of Venn diagrams for enhancing visualization of deductive statements or arguments. For now, it suffices to understand the Venn representations above of the four classification statements:
- All S are P: The zone of S that is not also P is empty (hatched), and the only possible locations of S are in the zone that overlaps P. Ergo, all S are P.
- No S are P: The zone of S that overlaps P, i.e. that is also P, is empty.
- Some S are P: The X indicates that at least one member lies within the zone that represents members of both S and P. The remaining members of S or P may or may not lie within this zone.
