Changes

But the latter is not a theorem in anyone's philosophy, so there is really no disagreement here.

But the latter is not a theorem in anyone's philosophy, so there is really no disagreement here.

===Functional quantifiers===

==Functional Quantifiers==

The '''relative umpire operator''' <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> takes two propositions as arguments and gives the value <math>1\!</math> if and only if the first implies the second.  In symbols:

The '''relative umpire operator''' <math>\Upsilon : (\mathbb{B}^2 \to \mathbb{B})^2 \to \mathbb{B}</math> takes two propositions as arguments and gives the value <math>1\!</math> if and only if the first implies the second.  In symbols:

<br>

<br>

====Tables====

===Tables===

The auxiliary notations:

The auxiliary notations:

|}<br>

|}<br>

====Exercises====

===Exercises===

Express the following formulas in functional terms.

Express the following formulas in functional terms.

=====Exercise 1=====

====Exercise 1====

<blockquote>

<blockquote>

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>

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=====

====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>