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apply fix
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<math>\begin{matrix}
 
<math>\begin{matrix}
\texttt{(} p \texttt{~(} q \texttt{))}
+
\texttt{(} p \texttt{ (} q \texttt{))}
 
\\[4pt]
 
\\[4pt]
\texttt{(} q \texttt{~(} r \texttt{))}
+
\texttt{(} q \texttt{ (} r \texttt{))}
 
\\[4pt]
 
\\[4pt]
\texttt{(} p \texttt{~(} r \texttt{))}
+
\texttt{(} p \texttt{ (} r \texttt{))}
 
\\[4pt]
 
\\[4pt]
\texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}
+
\texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
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| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)
 
| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)
 
|-
 
|-
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}\!</math>
+
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!</math>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
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| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)
 
| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)
 
|-
 
|-
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}\!</math>
+
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!</math>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
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| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)
 
| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)
 
|-
 
|-
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}\!</math>
+
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!</math>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)
 
| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)
 
|-
 
|-
| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}\!</math>
+
| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}\!</math>
 
|}
 
|}
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~ q \le r
 
~ q \le r
 
\\
 
\\
\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}
+
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
 
\\
 
\\
 
~ p \le r
 
~ p \le r
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~ q \le r
 
~ q \le r
 
\\
 
\\
=\!=\!=\!=\!=\!=\!=\!=
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\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 
\\
 
\\
 
~ p \le q \le r
 
~ p \le q \le r
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|- style="height:40px; background:#f0f0ff"
 
|- style="height:40px; background:#f0f0ff"
 
| <math>p \le q \le r\!</math>
 
| <math>p \le q \le r\!</math>
| <math>\texttt{(} p \texttt{~(} q \texttt{))}\!</math>
+
| <math>\texttt{(} p \texttt{ (} q \texttt{))}\!</math>
| <math>\texttt{(} p \texttt{~(} r \texttt{))}\!</math>
+
| <math>\texttt{(} p \texttt{ (} r \texttt{))}\!</math>
| <math>\texttt{(} q \texttt{~(} r \texttt{))}\!</math>
+
| <math>\texttt{(} q \texttt{ (} r \texttt{))}\!</math>
 
|}
 
|}
   Line 2,431: Line 2,431:  
In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.
 
In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.
   −
At first sight, the relationships seem easy enough to write out.  Figure&nbsp;75 shows how the various logical expressions are related to each other:  The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion &mdash; much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about &mdash; and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference.  From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication.
+
At first sight, the relationships seem easy enough to write out.  Figure&nbsp;75 shows how the various logical expressions are related to each other:  The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion &mdash; much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about &mdash; and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference.  From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication.
    
{| align="center" border="0" cellpadding="10"
 
{| align="center" border="0" cellpadding="10"
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