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Revision as of 19:28, 8 November 2016
, 19:28, 8 November 2016sub [ \~]
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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)
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| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}\!</math>
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!</math>
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| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)
| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)
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| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}\!</math>
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!</math>
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| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)
| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)
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| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}\!</math>
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!</math>
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| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)
| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)
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| <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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|- 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>
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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 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 — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — 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 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 — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — 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.
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