T4.1: 4 of 4

We have been reviewing basic symbolizations from chapter two. Part of the point here is that so much of English can be properly symbolized from the within SL.

Back in chapter two, though, we saw that there were certain connectives, non-truth functional ones, which could not be symbolized in SL. Thus, there are limitations to the expressive power of SL.

However, it's worth pointing out that SL can express any truth function. What this means can best be seen by way of example.

Think about the following table for a "mystery" sentence-form '???':

 P Q R ??? T T T T T T F F T F T T T F F F F T T F F T F F F F T F F F F F

The unknown sentence-form must be true only on the first and third rows. So, what is it? That is, what sentence-form will have exactly this truth table? Before reading on, think for a moment about what it might be.

In fact, there are lots of SL forms which have exactly the truth table above – they express the truth function which is true just when P and R are both true.

The first form that may come to mind is P&R. Notice that this is true in exactly rows one and three.

But others are possible. Think just a moment about a longer way of symbolizing the same thing.

To do this, first notice that (P&Q)&R is true on exactly the first row of our table. Right?

Second, what sentence-forms are true on exactly the third row? One that is is (P&~Q)&R. (You can tell, because this just states the partial truth value assignment that defines row three.)

Then just put these two together to form a sentence which says "I'm true on row one or row three":

[(P&Q)&R]v[(P&~Q)&R]

This, like P&R, is a sentence-form constructed to be true on just the first and third row. Now, make sure you notice how we constructed it: each bracketed expression represents one of the true rows. The main connective is a wedge. Now, this construction procedure will work for any sentence. You may want to try it for the following final problem...

Give a sentence-form with the following table:

 P Q R ??? T T T T T T F F T F T F T F F F F T T F F T F F F F T F F F F T

Enter your answer in the following field and press TAB or the "Enter" button. (You must type in the sentences 'P', 'Q', and 'R' instead of variables 'P','Q' and 'R'.)