From 46b67e2911488da5ccd6e4f20ac6b0547ff33acb Mon Sep 17 00:00:00 2001
From: grr <grr@lo2.org>
Date: Wed, 22 May 2024 21:38:48 -0400
Subject: some fixes in mathematical studies section

---
 essays/post_formalism_memories.tex | 219 ++++++++++++++++++++++---------------
 1 file changed, 128 insertions(+), 91 deletions(-)

(limited to 'essays/post_formalism_memories.tex')

diff --git a/essays/post_formalism_memories.tex b/essays/post_formalism_memories.tex
index 9b77855..847afe1 100644
--- a/essays/post_formalism_memories.tex
+++ b/essays/post_formalism_memories.tex
@@ -1,11 +1,24 @@
 \newcommand{\midheading}[1]{
 	{ \vskip 1em \centering \large \textsc{#1} \par \vskip 1em }}
 
+
+
+
+
+
+
+\newcommand{\sysname}[1]{"\textsc{#1}"}
+
+
+
+
+
+
 \chapter{Post-Formalism in Constructed Memories}
 \section{Post-Formalist Mathematics}
 
 Over the last hundred years, a philosophy of pure mathematics has 
-grown up which I prefer to call "formalism." As Willard Quine says in the 
+grown up which I prefer to call \enquote{formalism.} As Willard Quine says in the 
 fourth section of his essay "Carnap and Logical Truth,' formalism was 
 inspired by a series of developments which began with non-Euclidian 
 geometry. Quine himself is opposed to formalism, but the formalists have 
@@ -15,40 +28,40 @@ presentation of the formalist position can be found in Rudolph Carnap's
 as a heuristic aid, I will relate my study to these two standard books. (It will 
 heip if the reader is thoroughly familiar with them.) it is not important 
 whether Carnap, or Quine, or formalism---or my interpretation of them---is 
-"correct," for this essay is neither history nor philosophy. I am using history 
+\enquote{correct,} for this essay is neither history nor philosophy. I am using history 
 as a bridge, to give the reader access to some extreme mathematical 
 innovations. 
 
 The formalist position goes as follows. Pure mathematics is the 
 manipulation of the meaningless and arbitrary, but typographically 
-well-defined ink-shapes on paper 'w,' 'x,' 'y,' 'z,' '{}',' '(,' '),' '$\downarrow$,' and '$\in$.' 
+well-defined ink-shapes on paper `$w$,' `$x$,' `$y$,' `$z$,' `${}'$,' `$($,' `$)$,' `$\downarrow$,' and `$\in$.' 
 These shapes are manipulated according to arbitrary but well-detined 
 mechanical rules. Actually, the rules mimic the structure of primitive 
 systems such as Euclid's geometry. There are formation rules, mechanical 
-definitions of which concatenations of shapes are "sentences." One sentence 
-is '$((x) (x\in x) \downarrow (x) (x\in x))$.' There are transformation rules, rules for the 
+definitions of which concatenations of shapes are \enquote{\term{sentences}.} One sentence 
+is `$((x) (x\in x) \downarrow (x) (x\in x))$.' There are transformation rules, rules for the 
 mechanical derivation of sentences from other sentences. The best known 
 trasformation rule is the rule that $\psi$ may be concluded from $\varphi$ and 
 $\ulcorner \varphi \supset \psi \urcorner$; 
-where '$\supset$' is the truth-functional conditional. For later convenience, I will 
-say that $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$ are "impliors," 
-and that $\psi$ is the "implicand." 
-Some sentences are designated as "axioms." A "proof" is a series of 
+where `$\supset$' is the truth-functional conditional. For later convenience, I will 
+say that $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$ are \enquote{\term{impliors},} 
+and that $\psi$ is the \enquote{\term{implicand}.} 
+Some sentences are designated as \enquote{\term{axioms}.} A \enquote{\term{proof}} is a series of 
 sentences such that each is an axiom or an implicand of preceding sentences. 
-The last sentence in a proof is a "theorem." 
+The last sentence in a proof is a \enquote{\term{theorem}.} 
 
-This account is ultrasimplified and non-rigorous, but it is adequate for 
+This account is ultra-simplified and non-rigorous, but it is adequate for 
 my purposes. (The reader may have noticed a terminological issue here. For 
 Quine, an implication is merely a logically true conditional. The rules which 
 are used to go from some statements to others, and to assemble proofs, are 
-rules of inference. The relevant rule of inference is the modus ponens; $\psi$ is 
+rules of inference. The relevant rule of inference is the \term{modus ponens};\editornote{i.e., "$P$ implies $Q$. $P$ is true. Therefore, $Q$ must also be true."} $\psi$ is 
 the ponential of $\varphi$ and $\ulcorner \varphi \supset \psi \urcorner$. What I 
 am doing is to use a terminology of 
 implication to talk about rules of inference and ponentials. The reason is 
 that the use of Quine's terminology would result in extremely awkward 
 formulations. What I will be doing is sufficiently transparent that it can be 
 translated into Quine's terminology if necessary. My results will be 
-unaffected.) The decisive feature of the arbitrary game called "mathematics" 
+unaffected.) The decisive feature of the arbitrary game called \enquote{mathematics} 
 is as follows. A sentence-series can be mechanically checked to determine 
 whether it is a proof. But there is no mechanical method for deciding 
 whether a sentence is a theorem. Theorems, or rather their proofs, have to be 
@@ -59,47 +72,53 @@ the ability to make inferential discoveries.
 A variety of branches of mathematics can be specialized out from the 
 basic system. Depending on the choices of axioms, systems can be 
 constructed which are internally consistent, but conflict with each other. A 
-system can be "interpreted," or given a meaning within the language of a 
+system can be \enquote{interpreted,} or given a meaning within the language of a 
 science such as physics. So interpreted, it may have scientific value, or it may 
 not. But as pure mathematics, all the systems have the same arbitrary status. 
 
-By "formalist mathematics" I will mean the present mathematical 
+By \enquote{formalist mathematics} I will mean the present mathematical 
 systems which are presented along the above lines. Actually, as many authors 
-have observed, the success of the non-Euclidian "imaginary" geometries 
+have observed, the success of the non-Euclidian \enquote{imaginary} geometries 
 made recognition of the game-like character of mathematics inevitable. 
 Formalism is potentially the greatest break with tradition in the history of 
-mathematics. In the Foreward to \booktitle{The Logical Syntax of Language}, Carnap 
+mathematics. In the \essaytitle{Foreward} to \booktitle{The Logical Syntax of Language}, Carnap 
 brilliantly points out that mathematical innovation is still hindered by the 
 widespread opinion that deviations from mathematical tradition must be 
-justified---that is, proved to be "correct" and to be a faithful rendering of 
-"the true logic." According to Carnap, we are free to choose the rules of a 
+justified---that is, proved to be \enquote{correct} and to be a faithful rendering of 
+\enquote{the true logic.} According to Carnap, we are free to choose the rules of a 
 mathematical system arbitrarily. The striving after correctness must cease, so 
-that mathematics will no longer be hindered. \said{Before us lies the boundless 
-ocean of unlimited possibilities.} In other words, Carnap, the most reputable 
+that mathematics will no longer be hindered. \enquote{\emph{Before us lies the boundless 
+ocean of unlimited possibilities.}} In other words, Carnap, the most reputable 
 of academicians, says you can do anything in mathematics. Do not worry 
 whether whether your arbitrary game corresponds to truth, tradition, or 
-reality: it is still legitimate mathematics. Despite this wonderful Principle of 
-Tolerance in mathematics, Carnap never ventured beyond the old 
+reality: it is still legitimate mathematics. Despite this wonderful \uline{Principle of 
+Tolerance} in mathematics, Carnap never ventured beyond the old 
 ink-on-paper, axiomatic-deductive structures. I, however, have taken Carnap 
-at his word. The result is my "post-formalist mathematics." I want to stress 
+at his word. The result is my \enquote{post-formalist mathematics.} I want to stress 
 that my innovations have been legitimized in advance by one of the most 
 reputable academic figures of the twentieth century. 
 
 Early in 1961, I constructed some systems which went beyond 
-formalist mathematics in two respects. 1. My sentential elements are 
-physically different from the little ink-shapes on paper used in all formalist 
+formalist mathematics in two respects. 
+\begin{enumerate}[label=\arabic*.,nosep,itemsep=0.5em]
+	\item My sentential elements are physically different from the little ink-shapes on paper used in all formalist 
 systems. My sentences are physically different from concatenations of 
 ink-shapes. My transformation rules have nothing to do with operations on 
-ink-shapes. 2. My systems do not necessarily follow the axiomatic-deductive, 
-sentence-implication-axiom-proof-theorem structure. Both of these 
-possibilities, by the way, are mentioned by Carnap in \papertitle{Languages as 
-Calculi.} A "post-formalist system," then, is a formalist system which differs 
+ink-shapes. 
+
+\item My systems do not necessarily follow the axiomatic-deductive, 
+sentence\-implication-axiom-proof-theorem structure. 
+\end{enumerate}
+
+		Both of these 
+possibilities, by the way, are mentioned by Carnap in \essaytitle{Languages as 
+Calculi.}\editornote{Also in \booktitle{The Logical Syntax of Language}.} A \enquote{post-formalist system,} then, is a formalist system which differs 
 physically from an ink-on-paper system, or which lacks the 
 axiomatic-deductive structure. 
 
 As a basis for the analysis of post-formalist systems, a list of structural 
 properties of formalist systems is desirable. Here is such a list. By 
-"implication" I will mean simple, direct implication, unless I say otherwise. 
+\enquote{implication} I will mean simple, direct implication, unless I say otherwise. 
 \begin{enumerate}
 \item A sentence can be repeated at will. 
 
@@ -118,7 +137,7 @@ mechanically whether one or two directly imply the third.
 
 \item No axiom is implied by other, different axioms. 
 
-\item The definition of "proof" is the standard definition, in terms of 
+\item The definition of \enquote{proof} is the standard definition, in terms of 
 implication, given early in this essay. 
 
 \item Given the axioms and some other sentence, it is not possible to 
@@ -128,16 +147,16 @@ Compound indirect implication is a puzzle.
 
 Now for the first post-formalist system. 
 
-\midheading{"Illusions"}
+\midheading{\sysname{Illusions}}
 
 \begin{sysrules}
-A "sentence" is the following page (with the figure on it) so long as the 
+A \enquote{sentence} is the page (page \pageref{illusions}, with figure \ref{illusions} on it) so long as the 
 apparent, perceived ratio of the length of the vertical line to that 
-of the horizontal line (the statement's "associated ratio") does not 
-change. (Two sentences are the "same" if end only if their 
+of the horizontal line (the statement's \enquote{associated ratio}) does not 
+change. (Two sentences are the \enquote{same} if end only if their 
 associated ratios are the same.) 
 
-A sentence Y is "implied by" a sentence X if and only if Y is the same as X, 
+A sentence Y is \enquote{implied by} a sentence X if and only if Y is the same as X, 
 or else Y is, of all the sentences one ever sees, the sentence having 
 the associated ratio next smaller than that of X. 
 
@@ -154,34 +173,49 @@ figures with a variety of real (measured) ratios and practicing
 judging these ratios; and so forth. 
 \end{sysrules}
 
-\imgw{2in}{img/illusions}
+\begin{figure}[p]
+	{\centering \includegraphics[width=4in]{img/illusions} \par}
+	\caption{The sentence for \sysname{Illusions}.}
+	\label{illusions}
+\end{figure}
 
-"IIlusions" has Properties 1, 3--5, and 7--8. Purely to clarify this fact, the 
+\sysname{IIlusions} has Properties 1, 3--5, and 7--8. Purely to clarify this fact, the 
 following sequence of integers is presented as a model of the order in which 
 associated ratios might appear in reality. (The sequence is otherwise totally 
-inadequate as a model of "Illusions.") 4 2 1; 4 2; 5 4 2 1; 4 3 1. The 
-implication structure would then be 
+inadequate as a model of \sysname{Illusions.}) $4 2 1$; $4 2$; $5 4 2 1$; $4 3 1$. The 
+implication structure would then be as shown in figure \ref{illusionstructure}.
 
-\imgw{4in}{img/illusionstructure}
+\begin{figure}
+	{\centering \includegraphics[width=4.5in]{img/illusionstructure} \par}
+	\caption{Example implication structure for \sysname{Illusions}.}
+	\label{illusionstructure}
+\end{figure}
 
-The axiom would be 4, and 5 could not appear in a proof. "IIlusions" has 
+The axiom would be 4, and 5 could not appear in a proof. \sysname{IIlusions} has 
 Property 1 on the basis that one can control the associated ratio. Turning to 
 Property 4, it is normally the case that when an implication is repeated, a 
 given occurrence of one of the sentences involved is unique to a specific 
-occurrence of the implication. In "Illusions," however, if two equal 
+occurrence of the implication. In \sysname{Illusions,} however, if two equal 
 sentences are next smaller than X, the occurrence of X does not uniquely 
-	belong to either of the two occurrences of the implication. Compare '\begin{tabular}{c c c} t & h & e \\ h &   &   \\ e &   & \end{tabular}', 
-where the occurrence of 't' is not unique to either occurrence of 'the'. 
-Subject to this explanation, "Illusions" has Property 4. "Illusions" has 
+belong to either of the two occurrences of the implication. Compare figure \ref{thestructure},
+where the occurrence of `$t$' is not unique to either occurrence of `$the$'. 
+Subject to this explanation, \sysname{Illusions} has Property 4. \sysname{Illusions} has 
 Property 8, but it goes without saying that the type of implication is not 
 modus ponens. Properties 3, 5, and 7 need no comment. As for Property 2, 
 the rule of implication refers to a property of sentences, rather than to 
-elements of sentences. The interesting feature of "IIlusions" is that it 
+elements of sentences. The interesting feature of \sysname{Illusions} is that it 
 reverses the situation defined by Properties 6 and 9. Compound indirect 
 implication is about the same as simple implication. The only difference is 
 the difference between being smaller and being next smaller. And there is 
 only one axiom (per person). 
 
+\begin{figure}
+	{\centering \begin{tabular}{c c c} t & h & e \\ h &   &   \\ e &   & \end{tabular} \par}
+	\caption{Structure with shared node.}
+	\label{thestructure}
+\end{figure}
+
+
 Simple direct implication, however, is subjective and illusive. It 
 essentially involves changing one's perceptions of an illusion. The change of 
 associated ratios is subjective, elusive, and certainly not numerically 
@@ -194,45 +228,48 @@ theorem. What is difficult is grasping the steps in between, the simple direct
 implications. If the brain contains a permanent impression of every sensation 
 it has received, then the implications objectively exist; but they may not be 
 thinkable without neurological techniques for getting at the impressions. In 
-any case, "proof" is well-defined in some sense---but proofs may not be 
-thinkable. "Illusions" is, after all, not so much shakier in this respect than 
+any case, \enquote{proof} is well-defined in some sense---but proofs may not be 
+thinkable. \sysname{Illusions} is, after all, not so much shakier in this respect than 
 even simple arithmetic, which contains undecidable sentences and 
 indefinable terms. 
 
 In \booktitle{The Logical Syntax of Language}, Carnap distinguishes pure syntax 
 and descriptive syntax; and says that pure syntax should be independent of 
 notation, and that every system should be isomorphic to some ink-on-paper 
-system. In so doing, Carnap violates his ov'n Principle of Tolerance. Consider 
+system. In so doing, Carnap violates his own \uline{Principle of Tolerance}. Consider 
 the following trivial formalist system. 
 
-\midheading{"Order"}
+\midheading{\enquote{Order}}
 
 \begin{sysrules}
-A "sentence" is a member of a finite set of integers. 
+A \enquote{sentence} is a member of a finite set of integers. 
 
-Sentence Y is "implied by" sentence X if and only if Y=X, or else of all the 
+Sentence Y is \enquote{implied by} sentence X if and only if Y=X, or else of all the 
 sentences, Y is the one next smaller than X. 
 
 Take as the axiom the largest sentence. 
 \end{sysrules}
 
-Is the pure syntax of "\textsc{Illusions}" insomorphic to "\textsc{Order}"? The preceding 
-paragraph proved that it is not. The implication structure of "Order" is 
+Is the pure syntax of \sysname{Illusions} isomorphic to \sysname{Order}? The preceding 
+paragraph proved that it is not. The implication structure of \sysname{Order} is 
 mechanical to the point of idiocy, while the implication structure of 
-"Illusions" is, as I pointed out, elusive. The figure 
-
-\imgw{4in}{img/orderstructure}
-
+\sysname{Illusions} is, as I pointed out, elusive. Figure \ref{orderstruc}
 where loops indicate multiple occurances of the same sentence, could 
-adequately represent a proof in "Order," but could not remotely represent 
-one in "Illusions." The essence of "Illusions" is that it is coupled to the 
+adequately represent a proof in \enquote{Order,} but could not remotely represent 
+one in \sysname{Illusions.} The essence of \sysname{Illusions} is that it is coupled to the 
 reader's subjectivity. For an ink-on-paper system even to be comparable to 
-"IIlusions," the subjectivity would have to be moved out of the reader and 
+\sysname{IIlusions,} the subjectivity would have to be moved out of the reader and 
 onto the paper. This is utterly impossible. 
 
+\begin{figure}
+	{\centering \includegraphics[width=4.5in]{img/orderstructure} \par}
+	\caption{Implication structure of \sysname{Order}.}
+	\label{orderstruc}
+\end{figure}
+
 Here is the next system. 
 
-\midheading{"Innperseqs"}
+\midheading{\sysname{Innperseqs}}
 
 \begin{sysrules}
 Explanation: Consider the rainbow halo which appears to surround a small 
@@ -245,17 +282,17 @@ happens on one contracting radius of the halo, and specifically
 what happens on the segment of that radius lying in the vague 
 outer ring: the outer segment. 
 
-A "sentence" (or halopoint) is the changing halo color at a fixed point, in 
+A \enquote{sentence} (or halopoint) is the changing halo color at a fixed point, in 
 space, in the halo; until the halo contracts past the point. 
 
-Several sentences "imply" another sentence if and only if, at some instant, 
+Several sentences \enquote{imply} another sentence if and only if, at some instant, 
 the several sentences are on an outer segment, and the other 
 sentence is the inner endpoint of that outer segment. 
 
-An "axiom" is a sentence which is in the initial vague outer ring (before it 
+An \enquote{axiom} is a sentence which is in the initial vague outer ring (before it 
 contracts), and which is not an inner endpoint. 
 
-An "innperseq" is a sequence of sequences of sentences on one radius 
+An \enquote{innperseq} is a sequence of sequences of sentences on one radius 
 satisfying the following conditions. 1. The members of the first 
 sequence are axioms, 2. For each of the other sequences, the first 
 member is implied by the non-first members of the preceding 
@@ -270,7 +307,7 @@ last sequence has one member.
 {\centering
 \begin{minipage}{1.6in}\imgw{1.3in}{img/innperseqs}\end{minipage}
 	\begin{minipage}{2.25in}
-"Sentences" at 
+\enquote{Sentences} at 
 
 	\begin{tabular}{ c r l }
 		\bimg{time1} & $time_1$: & $a_1 a_2 a_3 a_4 a_5 a_6 a_7 b$ \\
@@ -302,7 +339,7 @@ last sequence has one member.
 		& & $d,e \rightarrow\ g$ \\
 	\end{tabular}
 
-"Axioms" $a_1 a_2 a_3 a_4 a_5 a_6 a_7$
+\enquote{Axioms} $a_1 a_2 a_3 a_4 a_5 a_6 a_7$
 
 
 Innperseq \\
@@ -317,18 +354,18 @@ $(g)$
 
 In the diagram, different positions of the vague outer 
 ring at different times are suggested by different shadings. The 
-outer segment moves "down the page." The figure is by no means 
+outer segment moves \enquote{down the page.} The figure is by no means 
 an innperseq, but is supposed to help explain the definition. 
-In "Innperseqs," a conventional proof would be redundant unless all 
+In \sysname{Innperseqs,} a conventional proof would be redundant unless all 
 the statements were on the same radius. And even if the weakest axiom were 
 chosen (the initial outer endpoint), this axiom would imply the initial inner 
 endpoint, and from there the theorem could be reached immediately. In 
-other words, to use the standard definition of "proof" in "Innperseqs" 
+other words, to use the standard definition of \enquote{proof} in \sysname{Innperseqs} 
 would result in an uninteresting derivation structure. Thus, a more 
-interesting derivation structure is defined, the "innperseq." The interest of 
-an "innperseq" is to be as elaborate as the many restrictions in its definition 
-will allow. Proofs are either disregarded in "Innperseqs"; or else they are 
-identified with innpersegs, and lack Property 8. "Innperseqs" makes the 
+interesting derivation structure is defined, the \enquote{\term{innperseq.}} The interest of 
+an \enquote{\term{innperseq}} is to be as elaborate as the many restrictions in its definition 
+will allow. Proofs are either disregarded in \sysname{Innperseqs}; or else they are 
+identified with innperseqs, and lack Property 8. \sysname{Innperseqs} makes the 
 break with the proof-theorem structure of formalist mathematics. 
 
 Turning to simple implication, an implicand can have many impliors; 
@@ -344,11 +381,11 @@ one is implied by the others. But when are you given sentences? If one can
 think the sentences, then relating them is easy---but it is difficult to think the 
 sentences in the first place, even though they objectively exist. The diagram 
 suggests what to look for, but the actual thinking, the actual sentences are 
-another matter. As for Property 9, when "theorems" are identified with last 
+another matter. As for Property 9, when \enquote{theorems} are identified with last 
 members of innperseqs, I hesitate to say whether a derivation of a given 
 sentence can be constructed mechanically. If a sentence is nearer the center 
 than the axioms are, an innperseq can be constructed for it. Or can it? The 
-answer is contingent. "Innperseqs" is indeterminate because of the difficulty 
+answer is contingent. \enquote{Innperseqs} is indeterminate because of the difficulty 
 of thinking the sentences, a difficulty which is defined into the system. It is 
 the mathematician's capabilities at a particular instant which delimit the 
 indeterminacies. Precisely because of the difficulty of thinking sentences, I 
@@ -357,20 +394,20 @@ will give several subvariants of the system.
 \midheading{Indeterminacy}
 
 \begin{sysrules}
-A "totally determinate innperseq" is an innperseq in which one thinks all the 
+A \enquote{totally determinate innperseq} is an innperseq in which one thinks all the 
 sentences. 
 
-An "implior-indeterminate innperseq" is an innperseq in which one thinks 
+An \enquote{implior-indeterminate innperseq} is an innperseq in which one thinks 
 only each implicand and the outer segment it terminates. 
 
-A "sententially indeterminate innperseq" is an innperseq in which one thinks 
+A \enquote{sententially indeterminate innperseq} is an innperseq in which one thinks 
 only the outer segment, and its inner endpoint, as it progresses 
 inward. 
 \end{sysrules}
 
 
 Let us return to the matter of pure and descriptive syntax. The interest 
-of "Illusions" and "Innperseqs" is precisely that their abstract structure 
+of \enquote{Illusions} and \enquote{Innperseqs} is precisely that their abstract structure 
 cannot be separated from their physical and psychological character, and 
 thus that they are not isomorphic to any conventional ink-on-paper system. I 
 am trying to break through to unheard of, and hopefully significant, modes 
@@ -385,11 +422,11 @@ one. (I have not combined the two essays because their approaches are too
 different.) I will define post-formalist systems in constructed memories, 
 beginning with a system in an M*-Memory. 
 
-\midheading{"Dream Amalgams"}
+\midheading{\enquote{Dream Amalgams}}
 
 \begin{sysrules}
-A "sentence" is a possible method, an $A_{a_i}$. with respect to an M*-Memory. 
-The sentence $A_{a_p}$ "implies" the sentence $A_{a_q}$ if and only if the $a_q$th 
+A \enquote{sentence} is a possible method, an $A_{a_i}$. with respect to an M*-Memory. 
+The sentence $A_{a_p}$ \enquote{implies} the sentence $A_{a_q}$ if and only if the $a_q$th 
 M*-assertion is actually thought; and either $A_{a_q} = A_{a_p}$, or else there is 
 cross-method contact of a mental state in $A_{a_q}$ with a state in $A_{q_p}$\footnote{sic?}
 
@@ -404,13 +441,13 @@ combine totally different episodes in the dreamer's life into an
 amalgam. 
 \end{sysrules}
 
-"\textsc{Dream Amalgams}" has Properties 1-5. For the first time, sentences are 
+\enquote{\textsc{Dream Amalgams}} has Properties 1--5. For the first time, sentences are 
 structurally composite, with mental states being the relevant sentential 
 elements. Implication has an unusual character. The traditional type of 
-implication, modus ponens, is "directed," because the conditional is 
+implication, modus ponens, is \enquote{directed,} because the conditional is 
 directed. Even if $\ulcorner\varphi\supset\phi\urcorner$ is true 
 $\ulcorner\varphi\supset\phi\urcorner$ may not be. Now implication is also 
-directed in "\textsc{Dream Amalgams,}" but for a very different reason. 
+directed in \enquote{\textsc{Dream Amalgams,}} but for a very different reason. 
 Cross-method contact, unlike the conditional, has a symmetric character. 
 What prevents implication from being necessarily symmetrical is that the 
 implicand's M*-assertion actually has to be thought, while the implior's 
@@ -421,7 +458,7 @@ remembering, one is automatically aware of the cross-method contacts of
 states in $A_{a_q}$. The conditions on the axioms ensure that they will have 
 implications without losing Property 7. 
 
-As for compound implication in "\textsc{Dream Amalgams,}" the organism 
+As for compound implication in \enquote{\textsc{Dream Amalgams,}} the organism 
 with the M*-Memory can't be aware of it at all; because it can't be aware 
 that at different times it remembered different methods to be the one 
 actually used. (In fact, the organism cannot be aware that the system has 
@@ -446,7 +483,7 @@ study post-formalist modes of inference which are anything but deductive.
 To aid in understanding the next system, which involves infalls in a 
 D-Memory, here is an 
 
-{ \vskip 1.5em \centering \large \framebox[1.1\width]{"Exercise to be Read Aloud"} \par\vskip 1.5em}
+{ \vskip 1.5em \centering \large \framebox[1.1\width]{\enquote{Exercise to be Read Aloud}} \par\vskip 1.5em}
 
 (Read according to a timer, reading the first word at O' O", and prolonging 
 and spacing words so that each sentence ends at the time in parentheses after 
-- 
cgit v1.2.3

