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===Functional quantifiers===
===Functional quantifiers===
====Tables====
====Exercise 1====
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{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|}<br>
|}<br>
====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>
</blockquote>
</blockquote>
====Exercise 3====
=====Exercise 3=====
<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>
