MyWikiBiz, Author Your Legacy — Tuesday November 26, 2024
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, 18:08, 11 December 2008
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| 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. |
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− | ===Functional quantifiers===
| + | ==Functional Quantifiers== |
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| 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: |
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| <br> | | <br> |
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− | ====Tables====
| + | ===Tables=== |
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| The auxiliary notations: | | The auxiliary notations: |
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| |}<br> | | |}<br> |
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− | ====Exercises====
| + | ===Exercises=== |
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| Express the following formulas in functional terms. | | Express the following formulas in functional terms. |
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− | =====Exercise 1=====
| + | ====Exercise 1==== |
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| <blockquote> | | <blockquote> |
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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> | | 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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− | =====Exercise 2=====
| + | ====Exercise 2==== |
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| <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> |