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, 12:54, 9 December 2008
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| ===Functional quantifiers=== | | ===Functional quantifiers=== |
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− | '''Exercises.''' Express the following in functional terms:
| + | ====Tables==== |
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− | ====Exercise 1==== | |
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− | <blockquote>
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− | <math>(\forall x \in X)(p(x) \Rightarrow q(x))</math>
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− | </blockquote>
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− | | |
− | <blockquote>
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− | <math>\prod_{x \in X} (p_x (q_x)) = 1</math>
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− | </blockquote>
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− | This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</math> already covered here:
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− | : [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]]
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− | | |
− | Need to think a little more about the proposition <math>p \Rightarrow q</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>
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| {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | | {| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
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| |}<br> | | |}<br> |
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− | ====Exercise 2==== | + | ====Exercises==== |
| + | |
| + | Express the following formulas in functional terms. |
| + | |
| + | =====Exercise 1===== |
| + | |
| + | <blockquote> |
| + | <math>(\forall x \in X)(p(x) \Rightarrow q(x))</math> |
| + | </blockquote> |
| + | |
| + | <blockquote> |
| + | <math>\prod_{x \in X} (p_x (q_x)) = 1</math> |
| + | </blockquote> |
| + | |
| + | This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</math> already covered here: |
| + | |
| + | : [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]] |
| + | |
| + | Need to think a little more about the proposition <math>p \Rightarrow q</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math> |
| + | |
| + | =====Exercise 2===== |
| | | |
| <blockquote> | | <blockquote> |
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| </blockquote> | | </blockquote> |
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− | ====Exercise 3==== | + | =====Exercise 3===== |
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| <blockquote> | | <blockquote> |
| <math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math> | | <math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math> |
| </blockquote> | | </blockquote> |