15 Definitions (Ad)

Introduction

In order to be able to formulate Peirce's rules precisely, we first need to give some definitions. The definitions will be accompanied by explanations and examples.

Definitions

Negative context

A negative context is a concept that has the negation-sign in front of it:

[Proposition:
    [Person: JohnTheBaptist]- - - - -[Christ: #]
 ]
"John the Baptist is not the Christ"

Outermost context

Every graph must -- explicitly or implicitly -- be part of a knowledge base, and every knowledge base has an outermost context. The outermost context is a concept of type Assertion whose referent is a collection of conceptual graphs.

Evenly enclosed

A concept or conceptual graph is said to be evenly enclosed if, going outwards to the outermost context, we encounter an even number of negative contexts on our way out.

For example:

[Proposition:
   [Proposition:
      [Men: ]->(Attr)->[Mortal]
 ]
"It is not the case that it is not the case that all men are mortal"
"All men are mortal"

here, the innermost graph,

[Men: ]->(Attr)->[Mortal]

is evenly enclosed because, going outwards towards the outermost context, we encounter an even number of negative contexts (in this case, 2).

Here is another example:

[Proposition:
   [Situation:
      [Person: John]<-(Agnt)<-[Act: Say]-
         ->(Thme)->[Proposition:
                            [Person: #I]- - - -[Swindler]
                         ]
   ]
 ]
"It is not the case that there is the following situation:
   John said, 'I am not a swindler"
"John never said, 'I am not a swindler"

Here, the innermost graph:

[Person: #I]- - - -[Swindler]

is evenly enclosed because we encounter an even number (two) negative contexts on our way outwards to the outermost context. Notice that there is an extra context, "Situation" in between the outer negative context ("Proposition") and the inner negative context (also "Proposition"). Thus the predicate even applies to negative contexts, not just any contexts.

Oddly enclosed

A concept or conceptual graph is said to be oddly enclosed if, going outwards to the outermost context, we encounter an odd number of negative contexts.

For example:

[Situation:
  [Person: JohnTheBaptist]<-(Agnt)<-[Act: Say]-
     ->(Thme)->[Proposition:
                   [Person: #I]- - - -[Christ: #]
                ]  
]
"There is this situation: John the baptist says, 'I am not the Christ'".

Here, the innermost graph:

[Person: #I]- - - -[Christ: #]

is oddly enclosed because, on our way outwards to the outermost context, we meet an odd (one) number of negative contexts.

Again, it is precisely the number of negative contexts we meet on our way that counts, not the number of contexts.

Domination

If a context y occurs nested in a context x, then x is said to dominate y. This is true for any level of nesting that can obtain between y and x.

For example:

[x:
  [y:
     [z]
  ]
]

y dominates z, while x dominates both y and z.

Double negation

A double negation is two nested negative contexts of type Proposition, where one is directly nested inside the other:

[Proposition:
   [Proposition:
          // Content
   ]
 ]

If we draw a double negation around any conceptual graph, the effect is to leave the graph's meaning unchanged. This is because ((p)) is the same as p. (To see this, try to say, "It is not the case that I am not hungry" and derive the meaning, "I am hungry".)

Summary

A concept c is said to be a negative context if it has the negation sign () in front of it.

A concept c is evenly enclosed if, going outwards from c to the outermost context, we encounter an even number of negative contexts. Likewise, a concept c is oddly enclosed if, going outwards from c to the outermost context, we encounter an odd number of negative contexts.

A concept x is said to dominate another concept y if y occurs inside the referent of x. This is regardless of whether y occurs directly in x's referent or nested inside x's referent.

A double negation is two negative contexts of type Proposition where one is nested directly inside the other.

Next

With these definitions, we are ready to formulate Peirce's rules.


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