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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 04:04, 4 July 2008
, 04:04, 4 July 2008→Review and Transition
It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>
It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
|+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''
|+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
! Other Notations
! Other Notations
|-
|-
| " "
|
| True.
| <math>\operatorname{True}</math>
| 1
| <math>1\!</math>
|-
|-
| ( )
| <math>(~)</math>
| False.
| <math>\operatorname{False}</math>
| 0
| <math>0\!</math>
|-
|-
| A
| <math>A\!</math>
| A.
| <math>A\!</math>
| A
| <math>A\!</math>
|-
|-
| (A)
| <math>(A)\!</math>
| Not A.
| <math>\operatorname{Not}\ A</math>
| A’ <br> ~A <br> ¬A
|
<math>\begin{matrix}
~A' \\
\tilde A \\
\lnot A \\
\end{matrix}</math>
|-
|-
| A B C
| <math>A\ B\ C</math>
| A and B and C.
| <math>A\ \operatorname{and}\ B\ \operatorname{and}\ C</math>
| A ∧ B ∧ C
| <math>A \land B \land C</math>
|-
|-
| ((A)(B)(C))
| <math>((A)(B)(C))\!</math>
| A or B or C.
| <math>A\ \operatorname{or}\ B\ \operatorname{or}\ C</math>
| A ∨ B ∨ C
| <math>A \lor B \lor C</math>
|-
|-
| (A (B))
| <math>(A (B))\!</math>
| A implies B. <br> If A then B.
|
| A ⇒ B
<math>\begin{matrix}
A\ \operatorname{implies}\ B \\
\operatorname{If}\ A\ \operatorname{then}\ B \\
\end{matrix}</math>
| <math>A \Rightarrow B\!</math>
|-
|-
| (A, B)
| <math>(A, B)\!</math>
| A not equal to B. <br> A exclusive-or B.
|
| A ≠ B <br> A + B
<math>\begin{matrix}
A\ \operatorname{not~equal~to}\ B \\
A\ \operatorname{exclusive~or}\ B \\
\end{matrix}</math>
|
<math>\begin{matrix}
A \neq B \\
A + B \\
\end{matrix}</math>
|-
|-
| ((A, B))
| <math>((A, B))\!</math>
| A is equal to B. <br> A if & only if B.
|
| A = B <br> A ⇔ B
<math>\begin{matrix}
A\ \operatorname{is~equal~to}\ B \\
A\ \operatorname{if~and~only~if}\ B \\
\end{matrix}</math>
|
<math>\begin{matrix}
A = B \\
A \Leftrightarrow B \\
\end{matrix}</math>
|-
|-
| (A, B, C)
| <math>(A, B, C)\!</math>
|
|
<math>\begin{matrix}
A B’C ∨<br>
\operatorname{Just~one~of} \\
A B C’
A, B, C \\
\operatorname{is~false}. \\
\end{matrix}</math>
|
<math>\begin{matrix}
A'B~C~ & \lor \\
A~B'C~ & \lor \\
A~B~C' & \\
\end{matrix}</math>
|-
|-
| ((A),(B),(C))
| <math>((A),(B),(C))\!</math>
|
|
A B’C’ ∨<br>
<math>\begin{matrix}
\operatorname{Just~one~of} \\
A, B, C \\
\operatorname{is~true}. \\
& \\
\operatorname{Partition~all} \\
\operatorname{into}\ A, B, C. \\
\end{matrix}</math>
|
<math>\begin{matrix}
A~B'C' & \lor \\
A'B~C' & \lor \\
A'B'C~ & \\
\end{matrix}</math>
|-
|-
|
|
A + B + C<br> <br>
<math>\begin{matrix}
A B C ∨<br>
((A, B), C) \\
A B’C’ ∨<br>
& \\
(A, (B, C)) \\
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{Oddly~many~of} \\
A, B, C \\
\operatorname{are~true}. \\
\end{matrix}</math>
|
<p><math>A + B + C\!</math></p>
<br>
<p><math>\begin{matrix}
A~B~C~ & \lor \\
A~B'C' & \lor \\
A'B~C' & \lor \\
A'B'C~ & \\
\end{matrix}</math></p>
|-
|-
| (Q, (A),(B),(C))
| <math>(Q, (A),(B),(C))\!</math>
| Partition Q <br> into A, B, C.<br>
|
Genus Q comprises <br> species A, B, C.
<math>\begin{matrix}
\operatorname{Partition}\ Q \\
\operatorname{into}\ A, B, C. \\
& \\
\operatorname{Genus}\ Q\ \operatorname{comprises} \\
\operatorname{species}\ A, B, C. \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
Q A B’C’ ∨<br>
Q'A'B'C' & \lor \\
Q A’B C’ ∨<br>
Q~A~B'C' & \lor \\
Q~A'B~C' & \lor \\
Q~A'B'C~ & \\
\end{matrix}</math>
|}
|}
'''Note.''' The usage that one often sees, of a plus sign "+" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:
'''Note.''' The usage that one often sees, of a plus sign "+" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do: