Truth Tables: The left hand columns

Truth-tables give us a systmatic way of determining when the given sentences are true and when they are false. The basic idea is that we list out all of the different ways the world could be by listing out all the different possible combinations of truth-values for the atomic sentences which appear.

It will prove useful to develop a standard way of listing out these possibilities. (it will prove especially useful for me when I'm grading your work, but I think it will also help you be able to deal with truth tables more efficiently.) Here's a wuick explanation of our standard way of setting up truth tables.

After we've listed the different atomic sentence letters in the left hand column:

First, we determine how many lines (or rows) we will need. Given n different atomic sentences, we need 2n lines. So, if there are 3 different atomic sentences (as in 3.1E #2 (l)) we need 23 = 8 lines.

Second, we fill in the column under the rightmost of the sentence letters following the pattern, TFTF.... for the appropriate number of lines that we need. If we need 8 lines, we write (vertically, of course) TFTFTFTF (as we did in 3.1E #2 (l) under the "J").

Next, we fill in the column under the sentence letter immediately to the left of the previous one the following the pattern, TTFFTTFF for the appropriate number of lines that we need. If we need 8 lines, so we write TTFFTTFF (as we did in 3.1E #2 (l) under the "F").

Next, fill in the rest of the sentence letters. If there is a third sentence letter, we need to fill in its column, following the pattern TTTTFFFFTTTTFFFF... (for the appropriate number of lines). If there is a fourth sentence letter, we follow the pattern TTTTTTTTFFFFFFFF... (for the appropriate number of lines). And so on.

Notice what's happening with each new letter, we alternate between T's and F's half as often as we did with the previous letter (another way of putting it is that the string of consecutive T's is twice as long as it was with the previous letter.) That is, we start by following the pattern TFTFTF... where there is 1 T followed by 1 F. Then, for the next letter we write TTFFTTFF... where there are 2 consecutive T's followed by 2 consecutive F's. If there's a next letter we write TTTTFFFFTTTTFFFF... where there are 4 consecutive T's followed by 4 F's. If there was another sentence letter we would have 8 consecutive T's followed by 8 F's. And so on.

Hopefully you get the point. (It actually turns out to be fairly simply once you start working through specific examples, but it is hard to give a quick abstract description.)

If you are still having trouble, you might want to cunsult the books explanation of this which appears on pp. 69&70. And, as always, I'm happy to answer any questions you have. Just stop by my office hours or send me an email.

Click here to return to the explanation of 3.1E #2 (l).