8.1.2

Here are some more you can practice with. First translate them into symbols, then calculate their truth values using the truth-functional definitions of the operators.

1. Obama and Hillary are Democrats if Newt is a Republican.

2. Either Obama will run or Newt is a Democrat.

3. If Clinton runs then she is at least 35 years old.

4. Obama is commander in chief if and only if he is President.

5. Being born in America is a necessary condition for being president.

6. If being born here is a necessary condition for running, then the Governator cannot run.

7. Either Hume did not invent truth tables or else if Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege; however, Kant denied that “existence” was a predicate if and only if Aristotelian logic dominated for two thousand years.

8. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege; but Kant denied that “existence” was a predicate only if Aristotelian logic dominated for two thousand years.

9. Either Hume did not invent truth tables or else Wittgenstein wrote the Tractatus, and Russell’s paradox was bad news to Frege only if Kant denied that “existence” was a predicate, given that Aristotelian logic dominated for two thousand years.

10. If it is false both that Hume invented truth tables and that Kant denied “existence” was a predicate, then given that Aristotelian logic dominated for two thousand years, Wittgenstein’s writing the Tractatus implies that Russell’s paradox was bad news to Frege.

Translations and calculations of the truth values of these statements are provided below.

1. Obama and Hillary are Democrats if Newt is a Republican.

N | > | (O | . | H) | |

T | T | T | |||

T | |||||

T |

2. Either Obama will run or Newt is a Democrat.

O | v | D |

T | F | |

T |

3. If Clinton runs then she is at least 35 years old.

C | > | T |

T | T | |

T |

4. Obama is commander in chief if and only if he is President.

O | = | P |

T | T | |

T |

5. Being born in America is a necessary condition for being president.

Requires some interpretation: an instantiation to a case: e.g., if Colbert is president then he was born in America. Let “P” mean “Colbert is President.”

P | > | B |

f | t | |

t |

6. If being born here is a necessary condition for running, then Schwartzenegger cannot run.

If Schwartzenegger’s being born here is necessary for him to run, then he cannot run.

“B” = He was born here

“R”= He can run

(R | > | B) | > | ~R |

F | F | T | ||

T | ||||

T |

7. Either Hume did not invent truth tables or else if Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege; however, Kant denied that “existence” was a predicate if and only if Aristotelian logic dominated for two thousand years.

[(~H | v | (W | > | R)] | . | (K | = | A) |

F | T | T | F | T | ||||

T | T | F | ||||||

T | ||||||||

F |

(Hume did not invent truth-tables, Wittgenstein did, and he wrote the Tractatus too. Russell’s paradox was very bad news to Frege (and not only to him!). It is to Kant that we owe the insight to not treat “existence” as a predicate, and of course Aristotlean logic did dominate Western philosophy for two thousand years –until modern symbolic logic was developed by people like Frege, Russell and Wittgenstein.)

8. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege; but Kant denied that “existence” was a predicate only if Aristotelian logic dominated for two thousand years.

[(~H | v | W) | > | R] | . | (K | > | A) |

T | T | T | F | T | ||||

T | T | |||||||

T | ||||||||

T |

9. Either Hume did not invent truth tables or else Wittgenstein wrote the Tractatus, and Russell’s paradox was bad news to Frege only if Kant denied that “existence” was a predicate, given that Aristotelian logic dominated for two thousand years.

(~H | v | W) | . | [A | > | (R | > | K)] |

T | T | T | T | F | ||||

T | F | |||||||

F | ||||||||

F |

10. If it is false both that Hume invented truth tables and that Kant denied “existence” was a predicate, then given that Aristotelian logic dominated for two thousand years, Wittgenstein’s writing the Tractatus implies that Russell’s paradox was bad news to Frege.

~(H | . | K) | > | [A | > | (W | > | R)] |

F | F | T | T | T | ||||

F | T | |||||||

T | T | |||||||

T |