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Revision as of 20:32, 13 December 2008
, 20:32, 13 December 2008change of notation : [u, v] → [i, j] & [x, y] → [u, v]
====Higher Order Propositions and Logical Operators (''n'' = 2)====
====Higher Order Propositions and Logical Operators (''n'' = 2)====
By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [\mathcal{X}] = [x_1, x_2] = [x, y],</math> based on two logical features or boolean variables <math>x\!</math> and <math>y.\!</math>
By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [\mathcal{X}] = [x_1, x_2] = [u, v],</math> based on two logical features or boolean variables <math>u\!</math> and <math>v.\!</math>
The universe of discourse <math>X^\circ\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.
The universe of discourse <math>X^\circ\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.
: The points of <math>X^\circ</math> form the space:
: The points of <math>X^\circ</math> form the space:
:: <math>X = \langle \mathcal{X} \rangle = \langle x, y \rangle = \{ (x, y) \} \cong \mathbb{B}^2.</math>
:: <math>X = \langle \mathcal{X} \rangle = \langle u, v \rangle = \{ (u, v) \} \cong \mathbb{B}^2.</math>
: Each point in <math>X\!</math> may be described by means of a ''singular proposition'', that is, a proposition that describes it uniquely. This form of representation leads to the following enumeration of points:
: Each point in <math>X\!</math> may be described by means of a ''singular proposition'', that is, a proposition that describes it uniquely. This form of representation leads to the following enumeration of points:
:: <math>X = \{\ (\!|x|\!)(\!|y|\!),\ (\!|x|\!) y,\ x (\!|y|\!),\ x y\ \} \cong \mathbb{B}^2.</math>
:: <math>X = \{\ (\!|u|\!)(\!|v|\!),\ (\!|u|\!) v,\ u (\!|v|\!),\ u v\ \} \cong \mathbb{B}^2.</math>
: Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>x\!</math> and <math>y\!</math> take on that point. This form of representation leads to the following enumeration of points:
: Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point. This form of representation leads to the following enumeration of points:
:: <math>X = \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} \cong \mathbb{B}^2.</math>
:: <math>X = \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} \cong \mathbb{B}^2.</math>
As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.
As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.
The next higher order universe of discourse that is built on <math>X^\circ</math> is <math>X^{\circ 2} = [X^\circ] = [[x, y]],</math> which may be developed in the following way. The propositions of <math>X^\circ</math> become the points of <math>X^{\circ 2},</math> and the mappings of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> become the propositions of <math>X^{\circ 2}.</math> In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form <math>w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.</math>
The next higher order universe of discourse that is built on <math>X^\circ</math> is <math>X^{\circ 2} = [X^\circ] = [[u, v]],</math> which may be developed in the following way. The propositions of <math>X^\circ</math> become the points of <math>X^{\circ 2},</math> and the mappings of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> become the propositions of <math>X^{\circ 2}.</math> In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form <math>w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.</math>
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>[x, y]\!</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.
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 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 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.
<br>
<br>
The umpire measure is defined at the level of truth functions, but can also be understood in terms of its implied judgments at the syntactic level. Interpreted this way, <math>\Upsilon_1\!</math> recognizes theorems of the propositional calculus over <math>[x, y],\!</math> giving a score of "1" to tautologies and a score of "0" to everything else, regarding all contingent statements as no better than falsehoods.
The umpire measure is defined at the level of truth functions, but can also be understood in terms of its implied judgments at the syntactic level. Interpreted this way, <math>\Upsilon_1\!</math> recognizes theorems of the propositional calculus over <math>[u, v],\!</math> giving a score of "1" to tautologies and a score of "0" to everything else, regarding all contingent statements as no better than falsehoods.
One remark in passing for those who might prefer an alternative definition. If we had originally taken <math>\Upsilon\!</math> to mean the absolute measure, then the relative vesrion could have been defined as <math>\Upsilon_e f = \Upsilon (e (f)).\!</math>
One remark in passing for those who might prefer an alternative definition. If we had originally taken <math>\Upsilon\!</math> to mean the absolute measure, then the relative vesrion could have been defined as <math>\Upsilon_e f = \Upsilon (e (f)).\!</math>
| <math>f_{14}</math>
| <math>f_{14}</math>
| 1110
| 1110
| <math>((u)(y))\!</math>
| <math>((u)(v))\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the ''existential interpretation''. As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives. Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions. In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic. At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above. At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.
Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the ''existential interpretation''. As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives. Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions. In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic. At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above. At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.
Let us return to the 2-dimensional case <math>X^\circ = [x, y].</math> In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{uv} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:
Let us return to the 2-dimensional case <math>X^\circ = [u, v].</math> In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:
<center><math>\begin{array}{llllll}
<center><math>\begin{array}{llllll}
\ell_{00} f & =
\ell_{00} f & =
\ell_{(x)(y)} f & =
\ell_{(u)(v)} f & =
\alpha_1 f & =
\alpha_1 f & =
\Upsilon_{(x)(y)} f & =
\Upsilon_{(u)(v)} f & =
\Upsilon_{(x)(y)\ \Rightarrow f} & =
\Upsilon_{(u)(v)\ \Rightarrow f} & =
f\ \operatorname{likes}\ (x)(y) \\
f\ \operatorname{likes}\ (u)(v) \\
\ell_{01} f & =
\ell_{01} f & =
\ell_{(x) y} f & =
\ell_{(u) v} f & =
\alpha_2 f & =
\alpha_2 f & =
\Upsilon_{(x) y} f & =
\Upsilon_{(u) v} f & =
\Upsilon_{(x) y\ \Rightarrow f} & =
\Upsilon_{(u) v\ \Rightarrow f} & =
f\ \operatorname{likes}\ (x) y \\
f\ \operatorname{likes}\ (u) v \\
\ell_{10} f & =
\ell_{10} f & =
\ell_{x (y)} f & =
\ell_{u (v)} f & =
\alpha_4 f & =
\alpha_4 f & =
\Upsilon_{x (y)} f & =
\Upsilon_{u (v)} f & =
\Upsilon_{x (y)\ \Rightarrow f} & =
\Upsilon_{u (v)\ \Rightarrow f} & =
f\ \operatorname{likes}\ x (y) \\
f\ \operatorname{likes}\ u (v) \\
\ell_{11} f & =
\ell_{11} f & =
\ell_{x y} f & =
\ell_{u v} f & =
\alpha_8 f & =
\alpha_8 f & =
\Upsilon_{x y} f & =
\Upsilon_{u v} f & =
\Upsilon_{x y\ \Rightarrow f} & =
\Upsilon_{u v\ \Rightarrow f} & =
f\ \operatorname{likes}\ x y \\
f\ \operatorname{likes}\ u v \\
\end{array}</math></center>
\end{array}</math></center>
Intuitively, the <math>\ell_{uv}\!</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 = [x, y],</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] = [[x, y]].\!</math> Figure 6 summarizes the action of the <math>\ell_{uv}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>
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 6 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>
<pre>
<pre>
| / \ |
| / \ |
| / \ |
| / \ |
| /x y\ |
| /u v\ |
| / o---o \ |
| / o---o \ |
| o \ / o |
| o \ / o |
| / \ | / \ |
| / \ | / \ |
| / \ @ / \ |
| / \ @ / \ |
| / x y \ / x y \ |
| / u v \ / u v \ |
| o o---o o o---o o |
| o o---o o o---o o |
| / \ \ / \ / / \ |
| / \ \ / \ / / \ |
| / \ @ / \ @ / \ |
| / \ @ / \ @ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| / y \ / \ / y \ |
| / v \ / \ / v \ |
| o @ o @ o o o |
| o @ o @ o o o |
| / \ / \ / \ | / \ |
| / \ / \ / \ | / \ |
| / \ / \ / \ @ / \ |
| / \ / \ / \ @ / \ |
| / \ /x y\ / \ / \ |
| / \ /u v\ / \ / \ |
| / x y \ / o o \ / x y \ / x y \ |
| / u v \ / o o \ / u v \ / u v \ |
| o @ o \ / o o o o o o |
| o @ o \ / o o o o o o |
| |\ / \ o / \ | / \ \ / /| |
| |\ / \ o / \ | / \ \ / /| |
| | \ / \ | / \ @ / \ @ / | |
| | \ / \ | / \ @ / \ @ / | |
| | \ / \ @ / \ / \ / | |
| | \ / \ @ / \ / \ / | |
| | \ / x \ / x y \ / x \ / | |
| | \ / u \ / u v \ / u \ / | |
| | o @ o o---o o o o | |
| | o @ o o---o o o o | |
| | |\ / \ \ / / \ | /| | |
| | |\ / \ \ / / \ | /| | |
| | | \ / \ @ / \ @ / | | |
| | | \ / \ @ / \ @ / | | |
| | | \ / \ / \ / | | |
| | | \ / \ / \ / | | |
| |L_11| \ / o y \ / x o \ / |L_00| |
| |L_11| \ / o v \ / u o \ / |L_00| |
| o---------o | o | o---------o |
| o---------o | o | o---------o |
| | \ x @ / \ @ y / | |
| | \ u @ / \ @ v / | |
| | \ / \ / | |
| | \ / \ / | |
| | \ / \ / | |
| | \ / \ / | |
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 6. Higher Order Universe of Discourse [L_uv] c [[x, y]]
Figure 6. Higher Order Universe of Discourse [L_ij] c [[u, v]]
</pre>
</pre>
\mathrm{A} &
\mathrm{A} &
\mathrm{Universal~Affirmative} &
\mathrm{Universal~Affirmative} &
\mathrm{All}\ x\ \mathrm{is}\ y &
\mathrm{All}\ u\ \mathrm{is}\ v &
\mathrm{Indicator~of}\ x (y) = 0 \\
\mathrm{Indicator~of}\ u (v) = 0 \\
\mathrm{E} &
\mathrm{E} &
\mathrm{Universal~Negative} &
\mathrm{Universal~Negative} &
\mathrm{All}\ x\ \mathrm{is}\ (y) &
\mathrm{All}\ u\ \mathrm{is}\ (v) &
\mathrm{Indicator~of}\ x \cdot y = 0 \\
\mathrm{Indicator~of}\ u \cdot v = 0 \\
\mathrm{I} &
\mathrm{I} &
\mathrm{Particular~Affirmative} &
\mathrm{Particular~Affirmative} &
\mathrm{Some}\ x\ \mathrm{is}\ y &
\mathrm{Some}\ u\ \mathrm{is}\ v &
\mathrm{Indicator~of}\ x \cdot y = 1 \\
\mathrm{Indicator~of}\ u \cdot v = 1 \\
\mathrm{O} &
\mathrm{O} &
\mathrm{Particular~Negative} &
\mathrm{Particular~Negative} &
\mathrm{Some}\ x\ \mathrm{is}\ (y) &
\mathrm{Some}\ u\ \mathrm{is}\ (v) &
\mathrm{Indicator~of}\ x (y) = 1 \\
\mathrm{Indicator~of}\ u (v) = 1 \\
\end{array}</math>
\end{array}</math>
|}<br>
|}<br>
| <math>\text{Some}\ u\ \text{is}\ v</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
|
|
| <math>\text{Some}\ u\ \text{is}\ y</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
| <math>\ell_{11}\!</math>
| <math>\ell_{11}\!</math>
|}<br>
|}<br>
