Let P and Q be statements, each of which is either true or false (but neither is both; there are weird statements that are both---"this statement is false"). "P or Q" is a compound statement, called an alternation, which is false only when both P and Q are false (and true in all other cases). Just to make it VERY official, here is a table which lists every possible combination of truth and falseness for P and Q and states whether "P or Q" is true for each possible combination. I apologize for repeating the obvious! In the tables, below, T stands for true, and F stands for false.

P | Q | P or Q |
---|---|---|

T | T | T |

T | F | T |

F | T | T |

F | F | F |

Consider the top row of this table. That row represents this situation: P is true and Q is true; in this situation "P or Q" is true. Ordinary English is confusing on this case. For example, if a father says "finish your term paper or you will be grounded for the Unit", most people would distrust a father who grounded you even though you finished your term paper! However, in the technical sciences (mathematics, engineering, physics, chemistry, etc.), there is agreement to read alternation as allowing both parts of the compound statement to be true. Technical scientists use a different symbol, "P xor Q", to mean P is true or Q is true BUT NOT BOTH P AND Q ARE TRUE. Ordinary English uses one word for both kinds of alternation; in the sciences, we had to separate them. [Remember, computers are complete idiots and must be told everything quite literally---no leaps of the imagination!]

What is the truth of "Q or P"? What difference does it make to "commute" the two statements P and Q by putting Q ahead of P? Having both P and Q false has nothing to do with which is stated first (please forgive this review of what may seem obvious). It makes no difference, which of P and Q is stated first in an alternation. "P or Q" MEANS EXACTLY THE SAME AS "Q or P"; the two compound statements are true in exactly the same situations. Again, to make it very official, we display this sameness of truth values in the table below.

P | Q | P or Q | Q or P |
---|---|---|---|

T | T | T | T |

T | F | T | T |

F | T | T | T |

F | F | F | F |

What is the truth of "P or P"? If P is true, both alterns of this alternation statement are true; by our technical understanding of alternation, the compound "or" statement is true. If P is false, both alterns of this alternation statement are false, and thus the compound "or" statement is false. Thus, "P or P" MEANS THE SAME THING AS "P". It may be redundant, even boring, but it's true. Surprisingly, it is useful as well. Sometimes one wants to remove redundancy to make shorter but equivalent statements. Sometimes one wants to introduce redundancy as part of a transformation into something that IS interesting. The following table is intended to make this observation quite official. Note that this table has only two rows because there are only two situations to consider: P is true or P is false.

P | P | P or P |
---|---|---|

T | T | T |

F | F | F |

P | Q | R | P or Q | (P or Q) or R |
---|---|---|---|---|

T | T | T | T | T |

T | T | F | T | T |

T | F | T | T | T |

T | F | F | T | T |

F | T | T | T | T |

F | T | F | T | T |

F | F | T | F | T |

F | F | F | F | F |

We now repeat this table for "P or (Q or R)":

P | Q | R | Q or R | P or (Q or R) |
---|---|---|---|---|

T | T | T | T | T |

T | T | F | T | T |

T | F | T | T | T |

T | F | F | F | T |

F | T | T | T | T |

F | T | F | T | T |

F | F | T | T | T |

F | F | F | F | F |

**Questions To Test Your Understanding.**

- How many different compound statements can be made from "(P or Q) or R" by permuting the letters P, Q, and R? How many more can be made by permuting the letters P, Q, and R and moving the parentheses to enclose the last two statements of the compound expression? Do the compound statements created by permutation and/or moving parentheses all mean the same thing? Try to list all the possibilities in an order in which each one is obtained from the previous one by one of two actions: commuting two statements in an alternation or by moving parenthesis.
- Consider four statements alternated: "(P or Q) or (R or S)". List all the possible equivalent statements obtained by moving parentheses and/or permuting the letters P, Q, R and S. How many are there? Do they all mean the same thing?
- Suppose that 99 statements are alternated in some order, with parentheses placed so that each alternation connects only two statements (possibly compound themselves). When is this compound statement true?