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Revision as of 12:01, 12 August 2009
, 12:01, 12 August 2009→Formal development: redo TeX markup
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The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math>
The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math>
−{| align="center" border="0" cellpadding="10" cellspacing="0"
+{| align="center" cellpadding="10"
| [[Image:PERS_Figure_01.jpg|500px]] || (1)
| [[Image:PERS_Figure_01.jpg|500px]] || (1)
|-
|-
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|}
|}
−One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (EN). Under EN, the axioms read as follows:
+One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (<math>\operatorname{En}</math>). Under <math>\operatorname{En},</math> the axioms read as follows:
−{| align="center" border="0" cellpadding="10"
+{| align="center" cellpadding="10"
|
|
−<math>\begin{array}{ccccc}
+<math>\begin{matrix}
−I_1 & : &
+I_1
−\operatorname{true}\ \operatorname{or}\ \operatorname{true} & = &
+& : &
−\operatorname{true} \\
+\operatorname{true} ~\operatorname{or}~ \operatorname{true}
−I_2 & : &
+& = &
−\operatorname{not}\ \operatorname{true}\ & = &
+\operatorname{true}
−\operatorname{false} \\
+\\
−J_1 & : &
+I_2
−a\ \operatorname{or}\ \operatorname{not}\ a & = &
+& : &
−\operatorname{true} \\
+\operatorname{not}~ \operatorname{true}
−J_2 & : &
+& = &
−(a\ \operatorname{or}\ b)\ \operatorname{and}\ (a\ \operatorname{or}\ c) & = &
+\operatorname{false}
−a\ \operatorname{or}\ (b\ \operatorname{and}\ c) \\
+\\
−\end{array}</math>
+J_1
+& : &
+a ~\operatorname{or}~ \operatorname{not}~ a
+& = &
+\operatorname{true}
+\\
+J_2
+& : &
+(a ~\operatorname{or}~ b) ~\operatorname{and}~ (a ~\operatorname{or}~ c)
+& = &
+a ~\operatorname{or}~ (b ~\operatorname{and}~ c)
+\end{matrix}</math>
|}
|}
−Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (EX). Under EX, the axioms read as follows:
+Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (<math>\operatorname{Ex}</math>). Under <math>\operatorname{Ex},</math> the axioms read as follows:
−{| align="center" border="0" cellpadding="10"
+{| align="center" cellpadding="10"
|
|
−<math>\begin{array}{ccccc}
+<math>\begin{matrix}
−I_1 & : &
+I_1
−\operatorname{false}\ \operatorname{and}\ \operatorname{false} & = &
+& : &
−\operatorname{false} \\
+\operatorname{false} ~\operatorname{and}~ \operatorname{false}
−I_2 & : &
+& = &
−\operatorname{not}\ \operatorname{false} & = &
+\operatorname{false}
−\operatorname{true} \\
+\\
−J_1 & : &
+I_2
−a\ \operatorname{and}\ \operatorname{not}\ a & = &
+& : &
−\operatorname{false} \\
+\operatorname{not}~ \operatorname{false}
−J_2 & : &
+& = &
−(a\ \operatorname{and}\ b)\ \operatorname{or}\ (a\ \operatorname{and}\ c) & = &
+\operatorname{true}
−a\ \operatorname{and}\ (b\ \operatorname{or}\ c) \\
+\\
−\end{array}</math>
+J_1
+& : &
+a ~\operatorname{and}~ \operatorname{not}~ a
+& = &
+\operatorname{false}
+\\
+J_2
+& : &
+(a ~\operatorname{and}~ b) ~\operatorname{or}~ (a ~\operatorname{and}~ c)
+& = &
+a ~\operatorname{and}~ (b ~\operatorname{or}~ c)
+\end{matrix}</math>
|}
|}
