ONT Re: Toward A Functional Conception Of Quantificational Logic
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Note 126
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chris, david, seth, ...
due to popular demand for more detailed documentation with
regard to my wild claims about quantification theory, i am
going to revive this old project paper for what it's worth
on the current market. it may be slow going, as i have to
recover this from my old mac print dump, and rewrite a bit.
| Document History:
|
| Subject: Inquiry & Analogy
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 3.20
| Created: 01 Jan 1995
| Revised: 22 Dec 2001
| Faculty: F. Mili & M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Division 2 (A Functional Conception of Quantification Theory)
| Excerpt: Subdivision 2.1 (Higher Order Propositional Expressions)
| Excerpt: Section 2.1.1 (Higher Order Propositions & Logical Operators)
Inquiry & Analogy
Abstract
This report discusses C.S. Peirce's treatment of analogy,
placing it in relation to his overall theory of inquiry.
The first order of business is to introduce the three
fundamental types of reasoning that Peirce adopted
from classical logic. In Peirce's analysis both
inquiry and analogy are complex programs of
reasoning which develop through stages of
these three types, although normally in
different orders.
Outline
1. Three Types of Reasoning
1.1 Types of Reasoning in Aristotle
1.2 Types of Reasoning in C.S. Peirce
1.3 Comparison of the Analyses
1.4 Aristotle's "Apagogy": Abductive Reasoning as Problem Reduction
1.5 Aristotle's "Paradigm": Reasoning by Analogy or Example
1.6 Peirce's Formulation of Analogy
1.7 Dewey's "Sign of Rain": An Example of Inquiry
2. A Functional Conception of Quantification Theory
Up till now quantification theory has been based on the assumption of
individual variables ranging over universal collections of perfectly
determined elements. Merely to write down quantified notations like
"(For All)_(x in X) F(x)" and "(For Some)_(x in X) F(x)" institutes
a subscription to notions of this order, as the membership relations
indited in their indices act to bear witness. Reflected on pragmatic
insights and constructive principles, however, these ideas begin to
appear as problematic hypotheses whose warrants to be admitted are
not beyond question, as projects of exhaustive determination that
overreach the powers of finite information and control to manage.
Therefore, it is worth considering how we might shift the medium
of quantification theory closer to familiar ground, toward the
particular predicates themselves that represent our continuing
acquaintance with phenomena.
2.1 Higher Order Propositional Expressions
By way of equipping the 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 & Logical Operators (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
| \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |
| F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 |
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
| | | | |
| F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | | |
| F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | | |
| F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | | |
| F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| | | | |
o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
Table 11. Interpretive Categories for Higher Propositions (n = 1)
o-------o----------o------------o------------o----------o----------o-----------o
|Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information|
o-------o----------o------------o------------o----------o----------o-----------o
| m_0 | nothing | | | | | |
| | happens | | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_1 | | | nothing | | | |
| | | just false | exists | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_2 | | | | | | |
| | | just not x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_3 | | | nothing | | | |
| | | | is x | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_4 | | | | | | |
| | | just x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_5 | | | everything | F is | | |
| | | | is x | linear | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_6 | | | | | F is not | F is |
| | | | | | uniform | informed |
o-------o----------o------------o------------o----------o----------o-----------o
| m_7 | | not | | | | |
| | | just true | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_8 | | | | | | |
| | | just true | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_9 | | | | | F is | F is not |
| | | | | | uniform | informed |
o-------o----------o------------o------------o----------o----------o-----------o
| m_10 | | | something | F is not | | |
| | | | is not x | linear | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_11 | | not | | | | |
| | | just x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_12 | | | something | | | |
| | | | is x | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_13 | | not | | | | |
| | | just not x | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_14 | | not | something | | | |
| | | just false | exists | | | |
o-------o----------o------------o------------o----------o----------o-----------o
| m_15 | anything | | | | | |
| | happens | | | | | |
o-------o----------o------------o------------o----------o----------o-----------o
have to break here --
will explain later --
jon awbrey
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