Changes

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, let me limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.

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, expressed in the form <math>y \succ \{ d, f \},</math> that appear to be worth investigating.  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 \succ f \cdot d.</math>

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, expressed in the form <math>y >\!\!= \{ d, f \}</math>, that appear to be worth investigating.  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.

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

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

<li> The notion of a "generic inquiry" is ambiguous.  Its meaning in practice depends on whether this descriptive term 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 equipotential or prototypical in a yet to be specified way.  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 notion of a "generic inquiry" is ambiguous.  Its meaning in practice depends on whether this descriptive term 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 equipotential or prototypical in a yet to be specified way.  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> 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> Given the formula <math>y_0 = y \cdot y,</math> the subordination <math>y \succ \{ 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> 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>

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

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

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

<li> Inquiry into indices (for example, the <math>0</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, 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 decomposition or subordination (<math>\succ</math>).</li>

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

<li> Inquiry into containment or inclusion.  In particular, examine the claim that <math>F \subseteq M \subseteq D</math> which conditions 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 claim that <math>F \subseteq M \subseteq D</math> which conditions 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> A "problem" calls for a plan of action to resolve the difficulty that is present in it.  This difficulty is associated with a difference between observations and intentions.

<li> A "problem" calls for a plan of action to resolve the difficulty that is present in it.  This difficulty is associated with a difference between observations and intentions.

To express this diversity in a unified formula:  Both types of inquiry begin with a "delta", a compact term that admits of expansion as a debt, a difference, a difficulty, a discrepancy, a dispersion, a distribution, a doubt, a duplicity, or a duty.</li>

To express this diversity in a unified formula:  Both types of inquiry begin with a "delta", a compact term that admits of expansion as a debt, a difference, a difficulty, a discrepancy, a dispersion, a distribution, a doubt, a duplicity, or a duty.</li>

A certain arbitrariness has to be faced in the terms that one uses to talk about reasoning, to split it up into different parts and to sort it out into different types.  It is like the arbitrary choice that one makes in assigning the midpoint of an interval to the subintervals on its sides.  In setting out the forms of a nomenclature, in fitting the schemes of my terminology to the territory that it disturbs in the process of mapping, I cannot avoid making arbitrary choices, but I can aim for a strategy that is flexible enough to recognize its own alternatives and to accommodate the other options that lie within their scope.

A certain arbitrariness has to be faced in the terms that one uses to talk about reasoning, to split it up into different parts and to sort it out into different types.  It is like the arbitrary choice that one makes in assigning the midpoint of an interval to the subintervals on its sides.  In setting out the forms of a nomenclature, in fitting the schemes of my terminology to the territory that it disturbs in the process of mapping, I cannot avoid making arbitrary choices, but I can aim for a strategy that is flexible enough to recognize its own alternatives and to accommodate the other options that lie within their scope.

If I make the mark of deduction the fact that it reduces the number of terms, as it moves from the grounds to the end of an argument, then I am due to devise a name for the process that augments the number of terms, and thus prepares the grounds for any account of experience.

If I make the mark of deduction the fact that it reduces the number of terms, as it moves from the grounds to the end of an argument, then I am due to devise a name for the process that augments the number of terms, and thus prepares the grounds for any account of experience.