Changes

==1.  Three Types of Reasoning==

==1.  Three Types of Reasoning==

* ''This section omitted.''

'''''This section has been omitted from the present copy.'''''

===1.1.  Types of Reasoning in Aristotle===

1.1.  Types of Reasoning in Aristotle

1.2.  Types of Reasoning in C.S. Peirce

===1.2.  Types of Reasoning in C.S. Peirce===

1.3.  Comparison of the Analyses

===1.3.  Comparison of the Analyses===

1.4.  Aristotle's "Apagogy" : Abductive Reasoning as Problem Reduction

===1.4.  Aristotle's "Apagogy" : Abductive Reasoning as Problem Reduction===

1.5.  Aristotle's "Paradigm" : Reasoning by Analogy or Example

===1.5.  Aristotle's "Paradigm" : Reasoning by Analogy or Example===

1.6.  Peirce's Formulation of Analogy

===1.6.  Peirce's Formulation of Analogy===

1.7.  Dewey's "Sign of Rain" : An Example of Inquiry

===1.7.  Dewey's "Sign of Rain" : An Example of Inquiry===

==2.  Functional Conception of Quantification Theory==

==2.  Functional Conception of Quantification Theory==

===2.1.  Higher Order Propositional Expressions===

===2.1.  Higher Order Propositional Expressions===

By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of ''higher order propositional expressions'' (HOPE's), in particular, those that stem from the propositions on 1 and 2 variables.

By way of equipping this inquiry with a bit of concrete material, I begin

with a consideration of "higher order propositional expressions" (HOPE's),

in particular, those that stem from the propositions on 1 and 2 variables.

2.1.1.  Higher Order Propositions and Logical Operators (n = 1)

====2.1.1.  Higher Order Propositions and Logical Operators (n = 1)====

A "higher order" proposition is, very roughly speaking, a proposition about propositions.

A higher order proposition is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>F : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math>

If the original order of propositions is a class of indicator functions F : X -> B, then

the next higher order of propositions consists of maps of the type m : (X -> B) -> B.

For example, consider the case where X = B.  Then there are exactly four

For example, consider the case where <math>X = B.</math> Then there are exactly four propositions <math>F : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>

propositions F : B -> B, and exactly sixteen higher order propositions  

that are based on this set, all bearing the type m : (B -> B) -> B.

Table 10 lists the sixteen higher order propositions about propositions on

Table&nbsp;10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>F : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>F_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question. The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>F_i.\!</math>

one boolean variable, organized in the following fashion:  Columns 1 & 2

form a truth table for the four F : B -> B, turned on its side from the

way that one is most likely accustomed to see truth tables, with the

row leaders in Column 1 displaying the names of the functions F_i,

for i = 1 to 4, while the entries in Column 2 give the values of

each function for the argument values that are listed in the

corresponding column head.  Column 3 displays one of the

more usual expressions for the proposition in question.

The last sixteen columns are topped by a collection of

conventional names for the higher order propositions,

also known as the "measures" m_j, for j = 0 to 15,

where the entries in the body of the Table record

the values that each m_j assigns to each F_i.

Table 10.  Higher Order Propositions (n = 1)

Table 10.  Higher Order Propositions (n = 1)

o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

|      |    |    |                                                |

|      |    |    |                                                |

o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

I am going to put off explaining Table 11, that presents a sample of

I am going to put off explaining Table&nbsp;11, that presents a sample of what I call "Interpretive Categories for Higher Order Propositions", until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit "condensed" or "degenerate" in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.

what I call "Interpretive Categories for Higher Order Propositions",

until after we get beyond the 1-dimensional case, since these lower

dimensional cases tend to be a bit "condensed" or "degenerate" in

their structures, and a lot of what is going on here will almost

automatically become clearer as soon as we get even two logical

variables into the mix.

Table 11.  Interpretive Categories for Higher Order Propositions (n = 1)

Table 11.  Interpretive Categories for Higher Order Propositions (n = 1)

o-------o----------o------------o------------o----------o----------o-----------o

o-------o----------o------------o------------o----------o----------o-----------o

</pre>

</pre>

==FL.  Note 2==

====2.1.2.  Higher Order Propositions and Logical Operators (n = 2)====

<pre>

<pre>

By way of reviewing notation and preparing to extend it to

By way of reviewing notation and preparing to extend it to

higher order universes of discourse, let us first consider

higher order universes of discourse, let us first consider