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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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|-

| &nbsp;

| &nbsp;

| [[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;

| [[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>

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|-

| &nbsp;

| &nbsp;

| [[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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|- 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&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.

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