Chapter Four, Tutorial Two

Full Truth Tables

In the last tutorial, we determined the truth values of SL sentences *given* or *supposing *the truth values for their atomic components.

...we do not know who Bob and Carola are. For example, we can't say that Bob is not a law student ('~B').

What we *can* say is that '~Bv~C' is false *if* 'B'
and 'C' are true. That is, it is *possible* that both will be law students,
and if that possibility holds, then '~Bv~C' is false.

B | C | ~ | B | v | ~ | C | |

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

A truth table is our way of going through *all such possibilities* at
once:

B | C | |

row one: | T | T |

row two: | T | F |

row three: | F | T |

row four: | F | F |

Instead of doing one row, i.e., one possibility, we can go through all possibilities at once when constructing a truth table. That way we don't have to assume we know which of 'B' and 'C' are true.

The following demonstration applies our truth table definitions and the reasoning of tutorial one, to all possibilities. This is a "full" truth table (i.e., a table going through all possibilities).

Click here to see the demonstration applying the truth table definitions.MouseOver the MouseOver
the
sign to ** pause** the demo.

First, notice that we have four rows in this table. One for each way in which Bob and Carola is or is not to be a law student.

As
usual, we take the truth values assigned on the left and move them under the
complex sentence's atomic components.

Then
we work on the negated atomic sentences: each negation is assigned the *opposite*
truth value from its atomic component.

Finally,
under the main connective, 'v', we write the truth value of the whole sentence.
(Each value here is determined by the values of the immediate components,
here the two negations.)

Notice
that the procedure here is just like what we did in tutorial one, except
now we have four rows instead of one. But each of the rows works just like
those of the one row tables.

Click here to replay.

When you are finished with this more complex table, move on to the next page.