ONT Re: Toward A Functional Conception Of Quantificational Logic
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Note 130
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| Document History:
|
| Subject: Inquiry & Analogy
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 3.21
| Created: 01 Jan 1995
| Revised: 24 Dec 2001
| Faculty: F. Mili & M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Section 2.1.4 (Measure for Measure)
2.1.4 Measure for Measure
An acquaintance with the functions of the umpire operator can be gained
from Tables 13 & 14, where the 2-dimensional case is worked out in full.
The auxiliary notations:
!a!_i f = !Y!(f_i, f),
!b!_i f = !Y!(f, f_i),
define two series of measures:
!a!_i, !b!_i : (%B%^2 -> %B%) -> %B%,
incidentally providing compact names for
the column headings of these two Tables.
Table 13. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
| | y | 1010 | |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
| f \ | | |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| f_0 | 0000 | () | 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 1 1 |
| | | | |
| f_5 | 0101 | (y) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (x, y) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_8 | 1000 | x y | 1 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| | | | |
| f_10 | 1010 | y | 1 1 1 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 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 14. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | x | 1100 | f |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
| | y | 1010 | |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
| f \ | | |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| f_0 | 0000 | () |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 1 1 1 1 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_5 | 0101 | (y) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (x, y) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (x y) | 1 1 |
| | | | |
| f_8 | 1000 | x y | 1 1 1 1 1 1 1 1 |
| | | | |
| f_9 | 1001 | ((x, y)) | 1 1 1 1 |
| | | | |
| f_10 | 1010 | y | 1 1 1 1 |
| | | | |
| f_11 | 1011 | (x (y)) | 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 |
| | | | |
| f_15 | 1111 | (()) | 1 |
| | | | |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Applied to a given proposition f, the qualifiers !a!_i and !b!_i tell whether
f rests "above f_i" or "below f_i", respectively, in the implication ordering.
By way of example, let us trace the effects of several such measures, namely,
those that occupy the limiting positions of the Tables.
!a!_00 f = %1% iff f_00 => f iff %0% => f, hence !a!_00 f = %1% for all f.
!a!_15 f = %1% iff f_15 => f iff %1% => f, hence !a!_15 f = %1% iff f = %1%.
!b!_00 f = %1% iff f => f_00 iff f => %0%, hence !b!_00 f = %1% iff f = %0%.
!b!_15 f = %1% iff f => f_15 iff f => %1%, hence !b!_15 f = %1% for all f.
In short, !a!_0 = !b!_15 is a totally indiscriminate measure,
one that accepts all propositions f : %B%^2 -> %B%, whereas
!a!_15 and !b!_0 are measures that appreciate the constant
propositions %1% : %B%^2 -> %B% and %0% : %B%^2 -> %B%,
respectively, above all others.
Finally, in conformity with the use of the fiber notation to
indicate sets of models, it is natural to use notations like:
[| !a!_i |] = (!a!_i)^(-1)(%1%),
[| !b!_i |] = (!b!_i)^(-1)(%1%),
[| !Y!_p |] = (!Y!_p)^(-1)(%1%),
to denote sets of propositions that satisfy the umpires in question.
Jon Awbrey
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