Speaking the Truth

March 27th, 2017

Speaking the Truth

The Universe contains

*Things*- For example, “I,” “London,” “roses,” “redness,” “old English books,” “a letter”

Things have

*Attributes*- For example, “large,” “red,” “old,” “which I received yesterday”
- It is the same as to say that Things have
*Properties*

From “Symbolic Logic” by Lewis Carroll

One *Thing* may have many *Attributes*, and one *Attribute* may belong to many *Things*

- Thus, the
*Thing*“a rose” may have the*Attributes*“red,” “scented,” “full-blown,” etc.; and the*Attribute*“red” may belong to the*Things*“a rose,” “a brick,” “a ribbon,” etc.

A phrase stating if a *Thing* has (or hasn’t) an *Attribute*

- “The rose is red”
- “London is a small city”

Predicates are either *TRUE* or *FALSE*. That is called the *truth value* of the predicate

Abstracting a little (to make it general), we can say that

- If \(x\) is a
*Thing*and \(A\) is an*Attribute* - then \(A(x)\) is a
*Predicate*

We can combine *simple predicates* to make complex phrases that can be either *TRUE* or *FALSE*

- “The rose is red AND London is a small city”
- “The rose is red OR London is a small city”
- “The rose is NOT red”
- “IF the rose is red THEN London is a small city”

Let \(P(x)\) and \(Q(y)\) be two predicates. The complex predicate \(P(x)\text{ AND }Q(y)\) has a truth value depending on the truth values of \(P(x)\) and \(Q(y)\). We can see it in a *truth table*

\(P(x)\) | \(Q(y)\) | \(P(x)\) AND \(Q(y)\) |
---|---|---|

TRUE | TRUE | TRUE |

TRUE | FALSE | FALSE |

FALSE | TRUE | FALSE |

FALSE | FALSE | FALSE |

\(P(x)\) | \(Q(y)\) | \(P(x)\) OR \(Q(y)\) |
---|---|---|

TRUE | TRUE | TRUE |

TRUE | FALSE | TRUE |

FALSE | TRUE | TRUE |

FALSE | FALSE | FALSE |

Notice that this is an *Inclusive OR*

\(P(x)\) | \(Q(y)\) | \(P(x)\) XOR \(Q(y)\) |
---|---|---|

TRUE | TRUE | FALSE |

TRUE | FALSE | TRUE |

FALSE | TRUE | TRUE |

FALSE | FALSE | FALSE |

Now the result if *FALSE* if both \(P(x)\) and \(Q(y)\) are *TRUE* at the same time

This rule applies to a single predicate \(P(x)\)

\(P(x)\) | NOT \(P(x)\) |
---|---|

TRUE | FALSE |

FALSE | TRUE |

We can easily combine all the previous operations

- \(P(x)\) AND \(Q(y)\) AND R(z)
- \(P(x)\) OR NOT Q(x)

We use parenthesis to avoid ambiguity. For example

- NOT \(P(x)\) AND \(Q(y)\) can be
- NOT (\(P(x)\) AND \(Q(y)\))
- (NOT \(P(x)\)) AND \(Q(y)\)

Two predicates are *equivalent* when they have the same *truth table*

\(P(x)\) | \(Q(y)\) | \(P(x)\) AND \(Q(y)\) | \(Q(y)\) AND \(P(x)\) |
---|---|---|---|

TRUE | TRUE | TRUE | TRUE |

TRUE | FALSE | FALSE | FALSE |

FALSE | TRUE | FALSE | FALSE |

FALSE | FALSE | FALSE | FALSE |

For AND and OR, the order is not important

Let’s compare the two interpretations

\(P(x)\) | \(Q(y)\) | NOT (\(P(x)\) AND \(Q(y)\)) | (NOT \(P(x)\)) AND \(Q(y)\) |
---|---|---|---|

TRUE | TRUE | FALSE | FALSE |

TRUE | FALSE | TRUE | FALSE |

FALSE | TRUE | TRUE | TRUE |

FALSE | FALSE | TRUE | FALSE |

They are different, so parenthesis are important

\(P(x)\) | \(Q(y)\) | NOT (\(P(x)\) AND \(Q(y)\)) | (NOT \(P(x)\)) OR (NOT \(Q(y)\)) |
---|---|---|---|

TRUE | TRUE | FALSE | FALSE |

TRUE | FALSE | TRUE | TRUE |

FALSE | TRUE | TRUE | TRUE |

FALSE | FALSE | TRUE | TRUE |

Negation of AND is the OR of negations

\(P(x)\) | \(Q(y)\) | NOT (\(P(x)\) OR \(Q(y)\)) |
(NOT \(P(x)\)) AND (NOT \(Q(y)\)) |
---|---|---|---|

TRUE | TRUE | FALSE | FALSE |

TRUE | FALSE | FALSE | FALSE |

FALSE | TRUE | FALSE | FALSE |

FALSE | FALSE | TRUE | TRUE |

Negation of OR is the AND of negations

When it is true that “*IF* \(P(x)\) *THEN* \(Q(y)\)”?

\(P(x)\) | \(Q(y)\) | IF \(P(x)\) THEN \(Q(y)\) |
---|---|---|

TRUE | TRUE | TRUE |

TRUE | FALSE | FALSE |

FALSE | TRUE | TRUE |

FALSE | FALSE | TRUE |

From the list of questions produced in Homework 2 (and any new questions you have)

- Which ones are
*Scientific Questions*? - Which ones are interesting to other people?

The question “why?” can have two types of answers

- What is the purpose of something
- What is the mechanisms that causes something