User:Jon Awbrey/SANDBOX
Format Samples • Wiki Text
MathBB, MathBF, MathCal
A set of logical features, \(\mathcal{A} = \{ a_1, \ldots, a_n \},\) affords a basis for generating an \(n\)-dimensional universe of discourse, written \(A^\bullet = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].\) It is useful to consider a universe of discourse as a categorical object that incorporates both the set of points \(A = \langle a_1, \ldots, a_n \rangle\) and the set of propositions \(A^\uparrow = \{ f : A \to \mathbb{B} \}\) that are implicit with the ordinary picture of a venn diagram on \(n\) features. Accordingly, the universe of discourse \(A^\bullet\) may be regarded as an ordered pair \((A, A^\uparrow)\) having the type \((\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),\) and this last type designation may be abbreviated as \(\mathbb{B}^n\ +\!\to \mathbb{B},\) or even more succinctly as \([ \mathbb{B}^n ].\) For convenience, the data type of a finite set on \(n\) elements may be indicated by either one of the equivalent notations, \([n]\) or \(\mathbf{n}.\)
MathFrak
\(\begin{array}{lccccccccccc} \mathfrak{M} & = & \{ & \mathfrak{m}_1 & , & \mathfrak{m}_2 & , & \mathfrak{m}_3 & , & \mathfrak{m}_4 & \} \\ & = & \{ & \text{“ ”} & , & \text{“(”} & , & \text{“,”} & , & \text{“)”} & \} \\ & = & \{ & \mathrm{blank} & , & \mathrm{links} & , & \mathrm{comma} & , & \mathrm{right} & \} \end{array}\)
TextTT
For the initial case \(k = 0,\) the bound connective is an empty closure, an expression taking one of the forms \(\texttt{()}, \texttt{(~)}, \texttt{([[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]))}, \ldots\) with any number of spaces between the parentheses, all of which have the same denotation among propositions.
For the generic case \(k > 0,\) the bound connective takes the form \(\texttt{(} s_1 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.\)
Format Samples • Screenshots
MathJax Fail
MathML View
Logic of Relatives
Table 3. Relational Composition o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | L # X | Y | | o---------o---------o---------o---------o | M # | Y | Z | o---------o---------o---------o---------o | L o M # X | | Z | o---------o---------o---------o---------o |
| \(\mathit{1}\!\) | \(\mathit{1}\!\) | \(\mathit{1}\!\) | |
| \(L\!\) | \(X\!\) | \(Y\!\) | |
| \(M\!\) | \(Y\!\) | \(Z\!\) | |
| \(L \circ M\) | \(X\!\) | \(Z\!\) |
Table 9. Composite of Triadic and Dyadic Relations o---------o---------o---------o---------o---------o | # !1! | !1! | !1! | !1! | o=========o=========o=========o=========o=========o | G # T | U | | V | o---------o---------o---------o---------o---------o | L # | U | W | | o---------o---------o---------o---------o---------o | G o L # T | | W | V | o---------o---------o---------o---------o---------o |
| \(\mathit{1}\!\) | \(\mathit{1}\!\) | \(\mathit{1}\!\) | \(\mathit{1}\!\) | |
| \(G\!\) | \(T\!\) | \(U\!\) | \(V\!\) | |
| \(L\!\) | \(U\!\) | \(W\!\) | ||
| \(G \circ L\) | \(T\!\) | \(W\!\) | \(V\!\) |
Table 13. Another Brand of Composition o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | G # X | Y | Z | o---------o---------o---------o---------o | T # | Y | Z | o---------o---------o---------o---------o | G o T # X | | Z | o---------o---------o---------o---------o |
| \(\mathit{1}\!\) | \(\mathit{1}\!\) | \(\mathit{1}\!\) | |
| \(G\!\) | \(X\!\) | \(Y\!\) | \(Z\!\) |
| \(T\!\) | \(Y\!\) | \(Z\!\) | |
| \(G \circ T\) | \(X\!\) | \(Z\!\) |
Table 15. Conjunction Via Composition o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | L, # X | X | Y | o---------o---------o---------o---------o | S # | X | Y | o---------o---------o---------o---------o | L , S # X | | Y | o---------o---------o---------o---------o |
| \(\mathit{1}\!\) | \(\mathit{1}\!\) | \(\mathit{1}\!\) | |
| \(L,\!\) | \(X\!\) | \(X\!\) | \(Y\!\) |
| \(S\!\) | \(X\!\) | \(Y\!\) | |
| \(L,\!S\) | \(X\!\) | \(Y\!\) |
Table 18. Relational Composition P o Q o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | P # X | Y | | o---------o---------o---------o---------o | Q # | Y | Z | o---------o---------o---------o---------o | P o Q # X | | Z | o---------o---------o---------o---------o |
| \(\mathit{1}\!\) | \(\mathit{1}\!\) | \(\mathit{1}\!\) | |
| \(P\!\) | \(X\!\) | \(Y\!\) | |
| \(Q\!\) | \(Y\!\) | \(Z\!\) | |
| \(P \circ Q\) | \(X\!\) | \(Z\!\) |
Table 20. Arrow: J(L(u, v)) = K(Ju, Jv) o---------o---------o---------o---------o | # J | J | J | o=========o=========o=========o=========o | K # X | X | X | o---------o---------o---------o---------o | L # Y | Y | Y | o---------o---------o---------o---------o |
| \(J\!\) | \(J\!\) | \(J\!\) | |
| \(K\!\) | \(X\!\) | \(X\!\) | \(X\!\) |
| \(L\!\) | \(Y\!\) | \(Y\!\) | \(Y\!\) |
Grammar Stuff
| ||||||
| ||||||
|
| ||||||||||
| ||||||||||
| ||||||||||
|
| ||||||||||
| ||||||||||
| ||||||||||
|
Table Stuff
| \(F\!\) | \(F\!\) | \(F()\!\) | \(F\!\) |
| \(\underline{0}\) | \(F_0^{(0)}\!\) | \(\underline{0}\) | \((~)\) |
| \(\underline{1}\) | \(F_1^{(0)}\!\) | \(\underline{1}\) | \(((~))\) |
| \(F\!\) | \(F\!\) | \(F(x)\!\) | \(F\!\) | |
| \(F(\underline{1})\) | \(F(\underline{0})\) | |||
| \(F_0^{(1)}\!\) | \(F_{00}^{(1)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \((~)\) |
| \(F_1^{(1)}\!\) | \(F_{01}^{(1)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \((x)\!\) |
| \(F_2^{(1)}\!\) | \(F_{10}^{(1)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(x\!\) |
| \(F_3^{(1)}\!\) | \(F_{11}^{(1)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(((~))\) |
| \(F\!\) | \(F\!\) | \(F(x, y)\!\) | \(F\!\) | |||
| \(F(\underline{1}, \underline{1})\) | \(F(\underline{1}, \underline{0})\) | \(F(\underline{0}, \underline{1})\) | \(F(\underline{0}, \underline{0})\) | |||
| \(F_{0}^{(2)}\!\) | \(F_{0000}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \((~)\) |
| \(F_{1}^{(2)}\!\) | \(F_{0001}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \((x)(y)\!\) |
| \(F_{2}^{(2)}\!\) | \(F_{0010}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \((x) y\!\) |
| \(F_{3}^{(2)}\!\) | \(F_{0011}^{(2)}\!\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \((x)\!\) |
| \(F_{4}^{(2)}\!\) | \(F_{0100}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(x (y)\!\) |
| \(F_{5}^{(2)}\!\) | \(F_{0101}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \((y)\!\) |
| \(F_{6}^{(2)}\!\) | \(F_{0110}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \((x, y)\!\) |
| \(F_{7}^{(2)}\!\) | \(F_{0111}^{(2)}\!\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \((x y)\!\) |
| \(F_{8}^{(2)}\!\) | \(F_{1000}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{0}\) | \(x y\!\) |
| \(F_{9}^{(2)}\!\) | \(F_{1001}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(\underline{1}\) | \(((x, y))\!\) |
| \(F_{10}^{(2)}\!\) | \(F_{1010}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{0}\) | \(y\!\) |
| \(F_{11}^{(2)}\!\) | \(F_{1011}^{(2)}\!\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \(\underline{1}\) | \((x (y))\!\) |
| \(F_{12}^{(2)}\!\) | \(F_{1100}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{0}\) | \(x\!\) |
| \(F_{13}^{(2)}\!\) | \(F_{1101}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \(\underline{1}\) | \(((x)y)\!\) |
| \(F_{14}^{(2)}\!\) | \(F_{1110}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{0}\) | \(((x)(y))\!\) |
| \(F_{15}^{(2)}\!\) | \(F_{1111}^{(2)}\!\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \(\underline{1}\) | \(((~))\) |
| fi‹x, y› |
|
|
|
fj‹u, v› | ||||||
|
|
|
| A |
|
|
|
B | ||||||
|
|
|
|
|
| ||||||
|
|
|
|
|
|