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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 21:34, 9 December 2008
, 21:34, 9 December 2008→Option 1 : Less General: conclude section
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>[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.
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>
====Option 2 : More General====
====Option 2 : More General====