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4 Categorical Propositions

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1 4 Categorical Propositions
4.3 Venn Diagrams and the Modern Square of Opposition

2 Venn Diagrams We can represent the meaning of our 4 categorical propositions using overlapping circles: S P

3 Venn Diagrams S At least one member exists P S P X S P X S P
All S are P = empty X = No S are P Some S are P Some S are not P At least one member exists S P S P X S P X

4 Boolean Interpretation
No members of S are outside of P (No commitment to existence of Ss or Ps) No members of S are inside of P At least one S exists, and that S is a P At least one S exists, and that S is not a P S P S P S P X S P X

5 Venn Diagrams lead to Square of Opposition
Compare the A and O propositions. One says nothing exists in S where it is not P, the other just the opposite. A and O propositions, then, are contradictory. S P A S P E Compare the E and the I propositions. One says nothing exists within the overlap of S and P, the other the opposite. E and I are contradictory then as well. S P X I S P X O

6 Modern Square of Opposition
Logically Undetermined A E Contra dictory Logically Undetermined Logically Undetermined Contra dictory I O Logically Undetermined Supposing one proposition is true (or false), you can tell the truth value of its opposite, but nothing beyond that.

7 Testing Immediate Inferences
Logically Undetermined A E Contra dictory Logically Undetermined Logically Undetermined Contra dictory I O Logically Undetermined Suppose that ‘Some Salads are not Priced to sell’ is true. What can we know about “All Salads are Priced to sell”? We can know from the square that it is false.

8 Testing Immediate Inferences
Logically Undetermined A E Contra dictory Logically Undetermined Logically Undetermined Contra dictory I O Logically Undetermined Suppose it is false that No Salads are Priced to sell. What can we know about “Some Salads are Priced to sell”? We can know from the square that it is true.

9 TEST: Some S are not P. Therefore, it is false that All S are P. S P X Some S are not P S P X It is false that All S are P This is the opposite of this S P

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