Chapter Two, Tutorial 4

Deductive Concepts

Remember how we defined valid? Think about it for a moment: an argument is valid if and only if what? (Pick one.)

Right.

An argument is valid just in case it is not possible that its conclusion be false while its premises are all true.

And our first example about Chris (remember: he'll get an 'A' or a 'B', but not an 'A' it turns out) is pretty clear in this:

AvB

__~A
__ B

If you think about it a minute, you'll see that given how we've defined the possibilities for our connectives to be true, there is no way for B to be false (given the premises); it is inescapable.

A | B | AvB | |

row one: | T | T | T |

row two: | T | F | T |

row three: | F | T | T |

row four: | F | F | F |

A | B | AvB | |

row one: | T | T | T |

row two: | T | F | T |

row three: | F | T | T |

row four: | F | F | F |

A | B | AvB | ~A | |

row one: | T | T | T |
F |

row two: | T | F | T |
F |

row three: | F | T | T |
T |

row four: | F | F | F | T |

A | B | AvB | ~A | |

row one: | T | T | T |
F |

row two: | T | F | T |
F |

row three: | F | T | T |
T |

row four: | F | F | F | T |

A | B | AvB | ~A | |

row one: | T | T | T |
F |

row two: | T | F | T |
F |

row three: | F | T | T |
T |

row four: | F | F | F | T |

What is *logically* possible is what is allowed by language.

So, this notion of possibility is relative to a given language. At this point we should be thinking about logical possibility in English or in SL. So, it's *logically* possible that Barack Obama is a plumber (or that 'Pg' be true). Our language *allows* this possibility.

Is it possible that Obama is *now* a plumber?

No, it's not possible, not in the way meant by the question. Barack Obama (now in 2010) is definitely no plumber. We *know* he's now U.S. president and not working on the pipes. In a *way* then, it's just not possible that Obama is a plumber...we *know* he's not. Just keep in mind that **this sort of possibility is different from logical possibility**. It's not *language* that excludes Obama from that practical profession! It's that he has another job. (We would want to say that he might have been a plumber, had his early life been very different. This shows what language allows.)

Next: Another example of a central concept defined in terms of logical possibility.

Remember that we defined logical equivalence or sameness of meaning *in terms of impossibility.* Two sentences are logically equivalent iff what?

- it is impossible for each sentence to be false.
- it is impossible for one sentence to be true while the other is false.
- it is impossible for one sentence of the pair to be false.

Hint: We've defined this one