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32 bytes added ,  12:54, 9 December 2008
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===Functional quantifiers===
 
===Functional quantifiers===
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'''Exercises.'''  Express the following in functional terms:
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====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>
      
{| 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>
   −
====Exercise 2====
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====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>
   −
====Exercise 3====
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=====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>
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