Wednesday, December 19, 2007

Tom's Types in Literal Diagrams and Welton Diagrams

Tom has argued for eight types of categorical propositions rather than the traditional four. In this post I want to develop his idea a bit in order to show diagrammatically what his eight types of propositions are. Since in logic everything nice is worth doing twice, I'll use two (related) methods of diagramming here: Carroll's literal diagrams and Welton diagrams. I've talked about Carroll's diagrams before; but I only recently came across Welton diagrams, when reading J. Welton's 1896 Manual of Logic. The two are related, in that you can think of Welton diagrams as being literal diagrams for lines rather than squares. Our biliteral diagram consists of the following squares:




SPS-P
-SP-S-P


I will use an X to indicate that a box is definitely empty; and an O to indicate that it is definitely full.

Welton's diagrams are similar, but they are along a line:

| S P | S-P |-S P |-S-P |

(I have rearranged Welton's cells so that the similarities to the biliteral diagrams are more obvious.) Welton proposed an unbroken line to indicate presence, a blank to indicate absence, and a broken line when the matter is undecided. I'll use Os and Xs again instead, just because it is easier.

Tom's types are then diagrammed in the following ways:

[-S-P]



X 
  

| X |   |   |   |

[-S+P]



 X
  

|   | X |   |   |

[+S-P]



  
X 

|   |   | X |   |

[+S+P]



  
 X

|   |   |   | X |


(+S+P)



O 
  

| O |   |   |   |

(+S-P)



 O
  

|   | O |   |   |

(-S+P)



  
O 

|   |   | O |   |

(-S-P)



  
 O

|   |   |   | O |

From this the character of the eight types is visible.

Tom suggests that the following holds true:

-[-S+P] = (+S-P)
-[-S-P] = (+S+P)
-[+S-P] = (-S+P)
-[+S+P] = (-S-P)

It can easily be seen from the diagrams that he's right. To negate [ ] you switch out your X's for O's; to negate ( ) you switch out your O's for X's.

More can be said on this; and I'll probably say a bit more in a future post (I don't know when it will come out) on using Welton diagrams for propositional logic.

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