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With this preamble, I return to develop my own account of formalization, with special attention to the kind of step that leads from the inchoate chaos of casual discourse to a well-founded discussion of formal models.  A formalization step, of the incipient kind being considered here, has the peculiar property that one can say with some definiteness where it ends, since it leads precisely to a well-defined formal model, but not with any definiteness where it begins.  Any attempt to trace the steps of formalization backward toward their ultimate beginnings can lead to an interminable multiplicity of open-ended explorations.  In view of these circumstances, let me limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.

With this preamble, I return to develop my own account of formalization, with special attention to the kind of step that leads from the inchoate chaos of casual discourse to a well-founded discussion of formal models.  A formalization step, of the incipient kind being considered here, has the peculiar property that one can say with some definiteness where it ends, since it leads precisely to a well-defined formal model, but not with any definiteness where it begins.  Any attempt to trace the steps of formalization backward toward their ultimate beginnings can lead to an interminable multiplicity of open-ended explorations.  In view of these circumstances, I will limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.

It begins like this:  I ask whether it is possible to reason about inquiry in a way that leads to a productive end.  I pose this question as an inquiry into inquiry, and I use the formula <math>y_0 = y \cdot y</math> to express the relationship between the present inquiry, <math>y_0\!</math>, and a generic inquiry, <math>y\!</math>.  Then I propose a couple of components of inquiry, discussion and formalization, that appear to be worth investigating, expressing this proposal in the form <math>y >\!\!= \{ d, f \}</math>.  Applying these components to each other, as must be done in the present inquiry, I am led to the current discussion of formalization, <math>y_0 = y \cdot y >\!\!= f \cdot d</math>.

It begins like this:  I ask whether it is possible to reason about inquiry in a way that leads to a productive end.  I pose my question as an inquiry into inquiry, and I use the formula <math>y_0 = y \cdot y</math> to express the relationship between the present inquiry, <math>y_0\!</math>, and a generic inquiry, <math>y\!</math>.  Then I propose a couple of components of inquiry, discussion and formalization, that appear to be worth investigating, expressing this proposal in the form <math>y >\!\!= \{ d, f \}</math>.  Applying these components to each other, as must be done in the present inquiry, I am led to the current discussion of formalization, <math>y_0 = y \cdot y >\!\!= f \cdot d</math>.

There is already much to question here.  At least, so many repetitions of the same mysterious formula are bound to lead the reader to question its meaning.

There is already much to question here.  At least, so many repetitions of the same mysterious formula are bound to lead the reader to question its meaning. Some of the more obvious issues that arise are these:

<ol style="list-style-type:decimal">

<ol style="list-style-type:decimal">

<li> The term ''generic inquiry'' is ambiguous.  Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech.  In the literal case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> denotes a particular inquiry, <math>y \in Y\!</math>, one that is assumed to be prototypical in yet to be specified ways.  In the figurative case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> is simply a variable that ranges over a collection <math>Y\!</math> of nominally conceivable inquiries.</li>

<li> The term ''generic inquiry'' is ambiguous.  Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech.  In the literal case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> denotes a particular inquiry, <math>y \in Y\!</math>, one that is assumed to be prototypical in yet to be specified ways.  In the figurative case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> is simply a variable that ranges over a collection <math>Y\!</math> of nominally conceivable inquiries.</li>

<li> On first reading, the recipe <math>y_0 = y \cdot y</math> appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.</li>

<li> First encountered, the recipe <math>y_0 = y \cdot y</math> appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.</li>

<li> Given the formula <math>y_0 = y \cdot y</math>, the subordination <math>y >\!\!= \{ d, f \}</math>, and the successive containments <math>F \subseteq M \subseteq D</math>, the <math>y\!</math> that looks into <math>y\!</math> is not restricted to examining <math>y \operatorname{'s}</math> immediate subordinates, <math>d\!</math> and <math>f\!</math>, but it can investigate any feature of <math>y \operatorname{'s}</math> overall context, whether objective, syntactic, interpretive, whether definitive or incidental, and finally it can question any supporting claim of the discussion.  Moreover, the question <math>y\!</math> is not limited to the particular claims that are being made here, but applies to the abstract relations and the general notions that are invoked in making them.  Among the many kinds of inquiry that suggest themselves, there are the following possibilities:</li>

<li> Contemplating the formula <math>y_0 = y \cdot y</math> in the context of the subordination <math>y >\!\!= \{ d, f \}</math> and the successive containments <math>F \subseteq M \subseteq D</math>, the <math>y\!</math> that inquires into <math>y\!</math> is not restricted to examining <math>y \operatorname{'s}</math> immediate subordinates, <math>d\!</math> and <math>f\!</math>, but it can investigate any feature of <math>y \operatorname{'s}</math> overall context, whether objective, syntactic, interpretive, and whether definitive or incidental, and finally it can question any supporting claim of the discussion.  Moreover, the question <math>y\!</math> is not limited to the particular claims that are being made here, but applies to the abstract relations and the general concepts that are invoked in making them.  Among the many kinds of inquiry that suggest themselves, there are the following possibilities:</li>

<ol style="list-style-type:lower-alpha">

<ol style="list-style-type:lower-alpha">

<li> Inquiry into propositions about application and equality.<br>Start with the formula <math>y_0 = y \cdot y</math> itself.</li>  

<li> Inquiry into propositions about application and equality. One may well begin with the forms of application and equality that are invoked in the formula <math>y_0 = y \cdot y</math> itself.</li>

<li> Inquiry into application (&nbsp;<math>\cdot</math>&nbsp;).</li>

<li> Inquiry into application <math>(\cdot)</math>, for example, the way that the term <math>y \cdot y</math> indicates the application of <math>y\!</math> to <math>y\!</math> in the formula <math>y_0 = y \cdot y</math>.</li>

<li> Inquiry into equality (<math>=\!</math>).</li>

<li> Inquiry into equality <math>(=)\!</math>, for example, the meaning of the equal sign in <math>y_0 = y \cdot y</math>.</li>

<li> Inquiry into indices (for example, the <math>0</math> in <math>y_0\!</math>).</li>

<li> Inquiry into indices, for example, the significance of <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}</math> in <math>y_0\!</math>.</li>

<li> Inquiry into terms, namely, constants and variables.<br>What are the functions of <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}</math> in this respect?</li>

<li> Inquiry into terms, specifically, constants and variables. What are the functions of <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}</math> in this respect?</li>

<li> Inquiry into decomposition or subordination (<math>>\!\!=</math>).</li>

<li> Inquiry into decomposition or subordination, for example, as invoked by the sign <math>{}^{\backprime\backprime} >\!\!= {}^{\prime\prime}</math> in the formula <math>y >\!\!= \{ d, f \}</math>.</li>

<li> Inquiry into containment or inclusion.  In particular, examine the assumption that formalization <math>F</math>, mediation <math>M</math>, and discussion <math>D</math> are ordered as <math>F \subseteq M \subseteq D</math>, a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.</li>

<li> Inquiry into containment or inclusion.  In particular, examine the assumption that formalization <math>F</math>, mediation <math>M</math>, and discussion <math>D</math> are ordered as <math>F \subseteq M \subseteq D</math>, a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.</li>