Directory:Jon Awbrey/MathJax Problems - Revision history

Jon Awbrey: + \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

2016-11-08T19:40:37Z

+ \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

Jon Awbrey: + \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}

2016-11-08T19:38:04Z

+ \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}

Jon Awbrey: sub [ \~]

2016-11-08T19:28:26Z

sub [ \~]

Jon Awbrey: + next problem section

2016-11-08T19:00:03Z

+ next problem section

Show changes

Jon Awbrey: blank work page

2016-11-08T18:26:37Z

blank work page

Jon Awbrey: /* Variations on a theme of transitivity */ + \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

2016-11-08T18:22:13Z

Variations on a theme of transitivity: + \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

Jon Awbrey: \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} + \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

2016-11-08T18:18:31Z

\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} + \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}

Jon Awbrey: + next problem piece

2016-11-08T18:15:25Z

+ next problem piece

Jon Awbrey: /* Computation and inference as semiosis */

2016-11-08T17:35:27Z

Computation and inference as semiosis

Jon Awbrey: nope

2016-11-08T17:33:05Z

nope

@@ Line 318: / Line 318: @@
 ~ q \le r
 \\
-\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
+\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 \\
 ~ p \le q \le r

@@ Line 289: / Line 289: @@
 ~ 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
@@ Line 318: / Line 318: @@
 ~ q \le r
 \\
-=\!=\!=\!=\!=\!=\!=\!=
+\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
 \\
 ~ p \le q \le r

@@ Line 121: / Line 121: @@
+|
 <math>\begin{matrix}
-\texttt{(} p \texttt{~(} q \texttt{))}
+\texttt{(} p \texttt{ (} q \texttt{))}
 \\[4pt]
-\texttt{(} q \texttt{~(} r \texttt{))}
+\texttt{(} q \texttt{ (} r \texttt{))}
 \\[4pt]
-\texttt{(} p \texttt{~(} r \texttt{))}
+\texttt{(} p \texttt{ (} r \texttt{))}
 \\[4pt]
-\texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}
+\texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}
 \end{matrix}</math>
+|
@@ Line 148: / Line 148: @@
 | [[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;
@@ Line 154: / Line 154: @@
 | [[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;
@@ Line 160: / Line 160: @@
 | [[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;
@@ Line 166: / Line 166: @@
 | [[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>
 |}
@@ Line 386: / Line 386: @@
 |- style="height:40px; background:#f0f0ff"
 | <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 896: / Line 896: @@
 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"

← Older revision		Revision as of 18:26, 8 November 2016
Line 1:		Line 1:
−	~~==Computation and inference as semiosis==~~

−	Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks. Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.
−
−	By way of a very preliminary orientation, let us consider the distinction between ''information reducing inferences'' and ''information preserving inferences''. It is prudent to make make our first acquaintance with this distinction in the medium of some concrete and simple examples.
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	~~\| width="1%" \| <big>•</big>~~
−	~~\| colspan="3" \| '''Example 1. Modus Ponens'''~~
−	\|-
−	~~\|  ~~
−	~~\| width="1%" \|  ~~
−	~~\| colspan="2" \| ''Information Reducing Inference''~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\| width="1%" \|  ~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ p \Rightarrow q~~
−	\\
−	~~~ p~~
−	\\
−	~~\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}~~
−	\\
−	~~~ q~~
−	~~\end{array}</math>~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\| colspan="2" \| ''Information Preserving Inference''~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\|  ~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ p \Rightarrow q~~
−	\\
−	~~~ p~~
−	\\
−	~~\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}~~
−	\\
−	~~~ p ~ q~~
−	~~\end{array}</math>~~
−	\|}
−
−	~~Let us examine these two types of inference in a little more detail. A ''rule of inference'' is stated in the followed form:~~
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ \textit{Expression 1}~~
−	\\
−	~~~ \textit{Expression 2}~~
−	\\
−	~~\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}~~
−	\\
−	~~~ \textit{Expression 3}~~
−	~~\end{array}</math>~~
−	\|}
−
−	The expressions above the line are called ''premisses'' and the expression below the line is called a ''conclusion''. If the rule of inference is simple enough, the ''proof-theoretic turnstile symbol'' <math>{}^{\backprime\backprime} \vdash {}^{\prime\prime}\!</math> may be used to write the rule on a single line, as follows:
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	\|
−	~~<math>\textit{Premiss 1}, \textit{Premiss 2} ~\vdash~ \textit{Conclusion}.\!</math>~~
−	\|}
−
−	~~Either way, one reads such a rule of inference in the following manner:~~
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	\|
−	~~From   <math>{\textit{Expression 1}}\!</math>   and   <math>{\textit{Expression 2}}\!</math>   infer   <math>{\textit{Expression 3}}.\!</math>~~
−	\|}
−
−	Looking to Example 1, the rule of inference known as ''modus ponens'' says the following: From the premiss <math>p \Rightarrow q\!</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>
−
−	Modus ponens is an ''illative'' or ''implicational'' rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of <math>q\!</math> alone one cannot infer with any degree of certainty that <math>p \Rightarrow q\!</math> and <math>p\!</math> are the reasons why <math>q\!</math> happens to be true.
−
−	Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>
−
−	~~==Variations on a theme of transitivity==~~
−
−	The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived. But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.
−
−	In discussing this Example, it is useful to observe that the implication relation indicated by the propositional form <math>x \Rightarrow y\!</math> is equivalent to an order relation <math>x \le y\!</math> on the boolean values <math>0, 1 \in \mathbb{B},\!</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	~~\| width="1%" \| <big>•</big>~~
−	~~\| colspan="3" \| '''Example 2. Transitivity'''~~
−	\|-
−	~~\|  ~~
−	~~\| width="1%" \|  ~~
−	~~\| colspan="2" \| ''Information Reducing Inference''~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\| width="1%" \|  ~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ p \le q~~
−	\\
−	~~~ q \le r~~
−	\\
−	~~\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}~~
−	\\
−	~~~ p \le r~~
−	~~\end{array}</math>~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\| colspan="2" \| ''Information Preserving Inference''~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\|  ~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ p \le q~~
−	\\
−	~~~ q \le r~~
−	\\
−	~~\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}~~
−	\\
−	~~~ p \le q \le r~~
−	~~\end{array}</math>~~
−	\|}
−
−	In stating the information-preserving analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a concatenated syntax. Thus, <math>p \le q \le r\!</math> means <math>p \le q ~\mathrm{and}~ q \le r.\!</math> The claim that this 3-adic order relation holds among the three propositions <math>p, q, r\!</math> is a stronger claim — conveys more information — than the claim that the 2-adic relation <math>p \le r\!</math> holds between the two propositions <math>p\!</math> and <math>r.\!</math>

@@ Line 122: / Line 122: @@
 ~ q \le r
 \\
-\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
+\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 \\
 ~ p \le q \le r

← Older revision		Revision as of 18:15, 8 November 2016
Line 80:		Line 80:

	Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>		Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>
		+
		+	==Variations on a theme of transitivity==
		+
		+	The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived. But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.
		+
		+	In discussing this Example, it is useful to observe that the implication relation indicated by the propositional form <math>x \Rightarrow y\!</math> is equivalent to an order relation <math>x \le y\!</math> on the boolean values <math>0, 1 \in \mathbb{B},\!</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>
		+
		+	{\| align="center" cellpadding="8" width="90%"
		+	\| width="1%" \| <big>•</big>
		+	\| colspan="3" \| '''Example 2. Transitivity'''
		+	\|-
		+	\|
		+	\| width="1%" \|
		+	\| colspan="2" \| ''Information Reducing Inference''
		+	\|-
		+	\|
		+	\|
		+	\| width="1%" \|
		+	\|
		+	<math>\begin{array}{l}
		+	~ p \le q
		+	\\
		+	~ q \le r
		+	\\
		+	\overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)}
		+	\\
		+	~ p \le r
		+	\end{array}</math>
		+	\|-
		+	\|
		+	\|
		+	\| colspan="2" \| ''Information Preserving Inference''
		+	\|-
		+	\|
		+	\|
		+	\|
		+	\|
		+	<math>\begin{array}{l}
		+	~ p \le q
		+	\\
		+	~ q \le r
		+	\\
		+	=\!=\!=\!=\!=\!=\!=\!=
		+	\\
		+	~ p \le q \le r
		+	\end{array}</math>
		+	\|}
		+
		+	In stating the information-preserving analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a concatenated syntax. Thus, <math>p \le q \le r\!</math> means <math>p \le q ~\mathrm{and}~ q \le r.\!</math> The claim that this 3-adic order relation holds among the three propositions <math>p, q, r\!</math> is a stronger claim — conveys more information — than the claim that the 2-adic relation <math>p \le r\!</math> holds between the two propositions <math>p\!</math> and <math>r.\!</math>

@@ Line 22: / Line 22: @@
 ~ p
 \\
-\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
+\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
 \\
 ~ q
@@ Line 40: / Line 40: @@
 ~ p
 \\
-\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
+\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 \\
 ~ p ~ q

← Older revision		Revision as of 17:33, 8 November 2016
Line 1:		Line 1:
−	==Computation and inference as semiosis ~~: Version 1==~~	+	==Computation and inference as semiosis==
−
−	Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks. Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.
−
−	By way of a very preliminary orientation, let us consider the distinction between ''information reducing inferences'' and ''information preserving inferences''. It is prudent to make make our first acquaintance with this distinction in the medium of some concrete and simple examples.
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	~~\| width="1%" \| <big>•</big>~~
−	~~\| colspan="3" \| '''Example 1. Modus Ponens'''~~
−	\|-
−	~~\|  ~~
−	~~\| width="1%" \|  ~~
−	~~\| colspan="2" \| ''Information Reducing Inference''~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\| width="1%" \|  ~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ p \Rightarrow q~~
−	\\
−	~~~ p~~
−	\\
−	~~\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}~~
−	\\
−	~~~ q~~
−	~~\end{array}</math>~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\| colspan="2" \| ''Information Preserving Inference''~~
−	\|-
−	~~\|  ~~
−	~~\|  ~~
−	~~\|  ~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ p \Rightarrow q~~
−	\\
−	~~~ p~~
−	\\
−	~~\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}~~
−	\\
−	~~~ p ~ q~~
−	~~\end{array}</math>~~
−	\|}
−
−	~~Let us examine these two types of inference in a little more detail. A ''rule of inference'' is stated in the followed form:~~
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	\|
−	~~<math>\begin{array}{l}~~
−	~~~ \textit{Expression 1}~~
−	\\
−	~~~ \textit{Expression 2}~~
−	\\
−	~~\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}~~
−	\\
−	~~~ \textit{Expression 3}~~
−	~~\end{array}</math>~~
−	\|}
−
−	The expressions above the line are called ''premisses'' and the expression below the line is called a ''conclusion''. If the rule of inference is simple enough, the ''proof-theoretic turnstile symbol'' <math>{}^{\backprime\backprime} \vdash {}^{\prime\prime}\!</math> may be used to write the rule on a single line, as follows:
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	\|
−	~~<math>\textit{Premiss 1}, \textit{Premiss 2} ~\vdash~ \textit{Conclusion}.\!</math>~~
−	\|}
−
−	~~Either way, one reads such a rule of inference in the following manner:~~
−
−	~~{\| align="center" cellpadding="8" width="90%"~~
−	\|
−	~~From   <math>{\textit{Expression 1}}\!</math>   and   <math>{\textit{Expression 2}}\!</math>   infer   <math>{\textit{Expression 3}}.\!</math>~~
−	\|}
−
−	Looking to Example 1, the rule of inference known as ''modus ponens'' says the following: From the premiss <math>p \Rightarrow q\!</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>
−
−	Modus ponens is an ''illative'' or ''implicational'' rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of <math>q\!</math> alone one cannot infer with any degree of certainty that <math>p \Rightarrow q\!</math> and <math>p\!</math> are the reasons why <math>q\!</math> happens to be true.
−
−	Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>
−
−	~~==Computation and inference as semiosis : Version 2~~==

	Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks. Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.		Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks. Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.