12,089
edits
Changes
Directory talk:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 02:24, 5 July 2013
, 02:24, 5 July 2013Created page with '==Work Area== ===Functional Quantifiers=== The '''umpire measure''' of type <math>{\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}}\!</math> links the constant proposit…'
==Work Area==
===Functional Quantifiers===
The '''umpire measure''' of type <math>{\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}}\!</math> links the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}\!</math> to a value of <math>1\!</math> and every other proposition to a value of <math>0.\!</math> Expressed in symbolic form:
{| align="center" cellpadding="8"
| <math>\Upsilon (f) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.\!</math>
|}
The '''umpire operator''' of type <math>{\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}}\!</math> links pairs of propositions in which the first implies the second to a value of <math>1\!</math> and every other pair to a value of <math>0.\!</math> Expressed in symbolic form:
{| align="center" cellpadding="8"
| <math>\Upsilon (e, f) = 1 \quad \Leftrightarrow \quad e \Rightarrow f.\!</math>
|}
===Functional Quantifiers===
The '''umpire measure''' of type <math>{\Upsilon : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}}\!</math> links the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}\!</math> to a value of <math>1\!</math> and every other proposition to a value of <math>0.\!</math> Expressed in symbolic form:
{| align="center" cellpadding="8"
| <math>\Upsilon (f) = 1_\mathbb{B} \quad \Leftrightarrow \quad u = 1_{\mathbb{B}^2 \to \mathbb{B}}.\!</math>
|}
The '''umpire operator''' of type <math>{\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}}\!</math> links pairs of propositions in which the first implies the second to a value of <math>1\!</math> and every other pair to a value of <math>0.\!</math> Expressed in symbolic form:
{| align="center" cellpadding="8"
| <math>\Upsilon (e, f) = 1 \quad \Leftrightarrow \quad e \Rightarrow f.\!</math>
|}