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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 14:06, 9 December 2008
, 14:06, 9 December 2008→Option 1 : Less General
The relative operator takes two propositions as arguments and reports the value "true" if the first implies the second, otherwise "false".
The relative operator takes two propositions as arguments and reports the value "true" if the first implies the second, otherwise "false".
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<math>\Upsilon \langle e, f \rangle = 1 \quad \operatorname{iff} \quad e \Rightarrow f.</math>
<math>\Upsilon \langle e, f \rangle = 1 \quad \operatorname{iff} \quad e \Rightarrow f.</math>
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Expressing it another way, we may also write:
Expressing it another way, we may also write:
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<math>\Upsilon \langle e, f \rangle = 1 \quad \Leftrightarrow \quad (e (f)) = 1.</math>
<math>\Upsilon \langle e, f \rangle = 1 \quad \Leftrightarrow \quad (e (f)) = 1.</math>
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In writing this, however, it is important to notice that the 1's appearing on the left and right have different meanings. Filling in the details, we have:
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<math>\Upsilon \langle e, f \rangle = 1 \in \mathbb{B} \quad \Leftrightarrow \quad (e (f)) = 1 : \mathbb{B}^2 \to \mathbb{B}.</math>
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Finally, it is often convenient to write the first argument as a subscript, hence <math>\Upsilon_e \langle f \rangle = \Upsilon \langle e, f \rangle.</math>
====Option 2 : More General====
====Option 2 : More General====