Chapter Four, Tutorial One

Truth Tables

In this tutorial and the next, we will show how to write a **truth table**
for a sentence of SL. This will allow us to fully analyze the meaning of such
a sentence.

Later in this chapter, we will see how to apply truth tables to test, for instance, whether a sentence is logically true or an argument is valid.

To write truth tables for an *arbitrary* sentence of SL, you will have to know
the tables which define the connectives. Here is one truth table definition as a reminder.

P | Q | P&Q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

Remember that P and Q are *variables* ranging over all SL sentences. Thus, the table applies to *any* sentence of our symbolic language.
But this may be a bit hard to fathom at first. Our box and oval are better:

& | ||

T | T | T |

T | F | F |

F | T | F |

F | F | F |

and now move on.

P | Q | P&Q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

We can't fit our big placeholders in everywhere though. Below is one table for all five connectives; the P and Q are just like the box and oval.

P | Q | P&Q | PvQ | P>Q | P=Q | ~P |

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

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

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

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

We can substitute in for P and Q with any sentences of SL. Then the connective showing in the table would be the "main" connective.

The truth value (true or false) of the whole sentence is the value the table gives. For example if P and Q are both long sentences and both are FALSE (say) then read across the bottom row:

P | Q | P&Q | PvQ | P>Q | P=Q | ~P |

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

then we see from the bottom row (where both P and Q are false that P&Q and PvQ are both FALSE (notice the 'F' under both). But the other sentences you could form by adding one further connective are TRUE.

When this idea of applying our truth table definitions of the connectives makes sense, move on...

Here's a very easy application of our table definition of the connectives. Suppose for definiteness that 'B' and 'C' continue to mean that Bob will attend law school and that Carola will attend law school and assume that both are TRUE. What can we say about '~B>C'?

Click here to see the demonstration applying the truth table definitions.

MouseOver
the sign to
** pause** the demonstration. Click it to pass on.

First, notice that 'B' and 'C' are both assigned true: the 'T' underneath each indicates this.