Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory - Revision history

Jon Awbrey: table captions

2013-07-05T02:22:24Z

table captions

Jon Awbrey: update

2013-07-05T01:48:43Z

update

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Jon Awbrey: /* Higher Order Propositional Expressions */ $for $$

2009-10-28T17:26:46Z

Higher Order Propositional Expressions: <math> for $

Jon Awbrey: /* Higher Order Propositional Expressions */ + note on notation

2009-10-28T17:00:08Z

Higher Order Propositional Expressions: + note on notation

Jon Awbrey: /* Extending the Existential Interpretation to Quantificational Logic */ convert graphic to PNG

2009-10-25T01:01:00Z

Extending the Existential Interpretation to Quantificational Logic: convert graphic to PNG

Jon Awbrey: /* Application of Higher Order Propositions to Quantification Theory */ italix

2009-10-05T16:08:19Z

Application of Higher Order Propositions to Quantification Theory: italix

Jon Awbrey: /* Application of Higher Order Propositions to Quantification Theory */ wrong wrappers

2009-10-05T16:04:32Z

Application of Higher Order Propositions to Quantification Theory: wrong wrappers

Jon Awbrey: /* Application of Higher Order Propositions to Quantification Theory */ de-awk

2009-10-05T16:01:01Z

Application of Higher Order Propositions to Quantification Theory: de-awk

Jon Awbrey: /* Higher Order Propositions and Logical Operators (''n'' = 2) */ typo

2009-09-30T23:58:06Z

Higher Order Propositions and Logical Operators (''n'' = 2): typo

Jon Awbrey: /* Readings */

2009-05-06T17:00:22Z

Readings

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 {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%"
-|+ '''Table 1.  Higher Order Propositions (''n'' = 1)'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | align="right" | <math>x:\!</math>
@@ Line 106: / Line 107: @@
 {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 2.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | Measure
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 {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%"
-|+ '''Table 3.  Higher Order Propositions (''n'' = 2)'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | align="right" | <math>u\!:</math><br><math>v\!:</math>
@@ Line 570: / Line 573: @@
 {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 4.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\!</math>'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | align="right" | <math>u\colon\!</math><br><math>v\colon\!</math>
@@ Line 920: / Line 924: @@
 {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 5.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\!</math>'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | align="right" | <math>u\colon\!</math><br><math>v\colon\!</math>
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 {| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 7.  Syllogistic Premisses as Higher Order Indicator Functions'''
+|+ style="height:30px" |
+|
 <math>\begin{array}{clcl}
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 {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 8.  Simple Qualifiers of Propositions (Version 1)'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | align="right" | <math>u\colon\!</math><br><math>v\colon\!</math>
@@ Line 1,614: / Line 1,621: @@
 {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 9.  Simple Qualifiers of Propositions (Version 2)'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | align="right" | <math>u\colon\!</math><br><math>v\colon\!</math>
@@ Line 1,824: / Line 1,832: @@
 {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
-|+ '''Table 10.  Relation of Quantifiers to Higher Order Propositions'''
+|+ style="height:30px" |
 |- style="background:ghostwhite"
 | <math>\text{Mnemonic}\!</math>

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 By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of ''higher order propositional expressions'', in particular, those that stem from the propositions on 1 and 2 variables.
-'''Note on notation.'''  The discussion that follows uses [[minimal negation operations]], expressed as bracketed tuples of the form $\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},$ and logical conjunctions, expressed as concatenated tuples of the form $e_1 ~\ldots~ e_k$, as the sole expression-forming operations of a calculus for [[boolean-valued functions]] or "propositions".  The expressions of this calculus parse into data structures whose underlying graphs are called ''cacti'' by graph theorists.  Hence the name ''[[cactus language]]'' for this dialect of propositional calculus.
+'''Note on notation.'''  The discussion that follows uses [[minimal negation operations]], expressed as bracketed tuples of the form <math>\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},</math> and logical conjunctions, expressed as concatenated tuples of the form <math>e_1 ~\ldots~ e_k,</math> as the sole expression-forming operations of a calculus for [[boolean-valued functions]] or "propositions".  The expressions of this calculus parse into data structures whose underlying graphs are called ''cacti'' by graph theorists.  Hence the name ''[[cactus language]]'' for this dialect of propositional calculus.
 ====Higher Order Propositions and Logical Operators (''n'' = 1)====

← Older revision		Revision as of 17:00, 28 October 2009
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	By way of equipping this inquiry with a bit of concrete material, I begin with a consideration of ''higher order propositional expressions'', 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'', in particular, those that stem from the propositions on 1 and 2 variables.
		+
		+	'''Note on notation.''' The discussion that follows uses [[minimal negation operations]], expressed as bracketed tuples of the form $\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},$ and logical conjunctions, expressed as concatenated tuples of the form $e_1 ~\ldots~ e_k$, as the sole expression-forming operations of a calculus for [[boolean-valued functions]] or "propositions". The expressions of this calculus parse into data structures whose underlying graphs are called ''cacti'' by graph theorists. Hence the name ''[[cactus language]]'' for this dialect of propositional calculus.

	====Higher Order Propositions and Logical Operators (''n'' = 1)====		====Higher Order Propositions and Logical Operators (''n'' = 1)====

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 Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].\!</math>  Figure&nbsp;6 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>
-<pre>
+{| align="center" cellpadding="10" style="text-align:center"
-o-----------------------------------------------------------o
+| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.png]]
-|                                                           |
+|-
-|                             o                             |
+| <math>\text{Figure 6.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]</math>
-|                            / \                            |
+|}
 ===Application of Higher Order Propositions to Quantification Theory===

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 Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.
-It's hard to tell if it makes any difference from a purely formal point of view, but it serves intuition to devise a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions.  Therefore, let us declare the type of _existential-valued_ functions <math>f : \mathbb{B}^k \to \mathbb{E},</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}</math> is a pair of values that indicate whether or not anything exists in the cells of the underlying universe of discourse.  As usual, let's not be too fussy about the coding of these functions, reverting to binary codes whenever the intended interpretation is clear enough.
+It's hard to tell if it makes any difference from a purely formal point of view, but it serves intuition to devise a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions.  Therefore, let us declare the type of ''existential-valued functions'' <math>f : \mathbb{B}^k \to \mathbb{E},</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}</math> is a pair of values that indicate whether or not anything exists in the cells of the underlying universe of discourse.  As usual, let's not be too fussy about the coding of these functions, reverting to binary codes whenever the intended interpretation is clear enough.
 With these qualifications in mind we note the following correspondences between classical quantifications and higher order indicator functions:

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 To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.
-There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>  Table&nbsp;3 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero.  The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures.  In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.
+There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>  Table&nbsp;3 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 15, with blank entries in the Table being optional for values of zero.  The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures.  In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.
 {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%"

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 ==Readings==
-* Quine, W.V. (1969/1981), "On the Limits of Decision", ''Akten des XIV. Internationalen Kongresses für Philosophie, vol. 3 (1969).  Reprinted, pp. 156&ndash;163 in Quine (ed., 1981), ''Theories and Things'', Harvard University Press, Cambridge, MA.
+* Quine, W.V. (1969/1981), "On the Limits of Decision", ''Akten des XIV. Internationalen Kongresses für Philosophie'', vol. 3 (1969).  Reprinted, pp. 156&ndash;163 in Quine (ed., 1981), ''Theories and Things'', Harvard University Press, Cambridge, MA.
 ==Document History==