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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)
|-
|-
|}
|}
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>
|}
|}
