Symbolism And Diagram for Categorical Proposition
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This document discusses categorical propositions in classical logic and their symbolic representations. Categorical propositions make claims about whether one class is included in another. They can be symbolized using sets and logical operators. The four basic categorical propositions are A ("All S is P"), E ("No S is P"), I ("Some S is P"), and O ("Some S is not P"). Their relationships can be depicted using Venn diagrams, which visually represent class membership and contradictions between propositions.
Symbolism And Diagram for Categorical Proposition
- 3. Symbolism and Diagrams for Categorical Propositions
- 4. Categorical proposition is the base for the Classical Logic. They are called categorical propositions because they are about categories or classes. Such propositions affirm or deny that some class S is included in some other class p, completely or partially.
- 5. Boolean interpretation of categorical propositions depends heavily on the notation of an empty class, it is convenient to have a symbol to represent it. The symbol zero 0, is used for this purpose. The term S has no members, we write an equal sign between S and 0.
- 6. Thus the equation S = 0 says that there are no S’s, or that S has no members. To deny that S is empty S does have members. We symbolize that denial by drawing a slanting line through the equals sign. The inequality S ≠ 0 says that there are S’s, by denying that S is empty.
- 7. To symbolize A proposition. The A proposition, “All S is P”, says that all members of class S are also member of the class P. That is, there are no members of the class S that that are not members of P or “No S is non- P”
- 8. To symbolize E propositions. The E propositions, “No S is P”, says that no members of the class S are the members of class P. This can be rephrased by saying that the product of the two classes is empty which is symbolized by the equation SP = 0.
- 9. To symbolize I proposition. The I proposition “Some S is P”, says that at least one member of S is also a member of P. This means that the product of the classes S and P is not empty. It is symbolized by the inequality SP ≠ 0.
- 10. To symbolize O proposition. The O proposition, “Some S is not P”, obverts to the logically equivalent to I propositions, “Some S is non-P”. It is symbolized by the inequality SP ≠ 0.
- 11. The relationship between these propositions contradict each other in several ways, as can be illustrated here.
- 12. This diagram shows that : 1. Contradictories 2. Contrary 3. Sub-Contrary 4. Sub-Alternation
- 13. Contradictory means they have opposite truth values. A and O propositions are contradictory as are E and I proposition. They are opposite of each other in both Quantity and Quality therefore, have opposite truth values. When any Categorical statement is true, its partner across the diagonal is false. When false its contradictory must be true.
- 14. Example: if “all rubies are red stones” (A) is true, then “some rubies are not red stones” (O) must be false. Similarly if “no mammals are aquatic” (E) is false, then “some mammals are aquatic” (I) must be true.
- 15. A and E propositions are contrary. Propositions are contrary when they cannot both be true. Example: An A proposition e.g. “all giraffes have long necks” cannot be true at the same time as the corresponding E proposition: “no giraffes have long necks”. They are opposite in Quality only (both are universals). however that corresponding A and E proposition while contrary are not contradictory.
- 16. I and O propositions are Sub Contrary. Propositions are Sub Contrary when it is impossible for both to false. Example: “some lunches are free” is false, “some lunches are not free” must be true. However that is possible for corresponding I and O both to be true.
- 17. Example: “some nations are democratic” and “some nations are not democratic”. Again I and O propositions are sub contrary, but not contrary or contradictory
- 18. Sub alternation are same in Quality but different in Quantity. From A to I ( is A is true then, the I is true). From E to O ( if E is true then, O is true). Now for Falsity From I to A ( if the I is false, then A is false). From O to E (if O is false then, E is false).
- 19. All S are P. No S are P. Some S are P. Some S are not P.
- 20. The Venn diagrams constitute an iconic representation of the standard form categorical propositions, in which spatial inclusions and exclusions correspond to the non-spatial inclusions and exclusions of classes. They provide an exceptionally clear method of notation.