We can also use the basic concept of a truth table to determine the validity of predicate expressions. Here we are looking for a falsifying interpretation, as when we ask whether saying that every student is ambitious and then that some who are ambitious are successful is enough for us to conclude that some students are successful.
First, we will assign this code:
..Fx: x is a student
..Gx: x is ambitious
..Hx: x is successful
Next, we will assume a very small world of two individuals, a and b. Here, as with our earlier truth tables, we want to see if it is possible to set each premise true and the conclusion false. Again, we are not proving that the original conclusion really is false but only that it does not have to be true even if the premises are true. Below we see that it is possible and thus that the argument is invalid.
Fa Fb Ga Gb Ha Hb || FGD | GHE | FHE
...0...0....1...0....1.....1........1.........1........0

Now let's look at another argument: Some students are ambitious, and everyone ambitious succeeds, so some students succeed.
Fa Fb Ga Gb Ha Hb || FGE | GHD | FHE
...1...0....1...0....1.....0........1.........1...X..0
What we now see is that having to have one individual with characteristics F and G is enough to show that same individual with the characteristic H. This then makes it impossible not to have an example of someone with both the characteristics F and H. The original argument is shown to be valid.

Working with interpretations is more difficult than working with truth tables. It is important, for instance, to allow enough individuals into our very small world so that we do not accidentally think we are talking about the same individuals when we say, for instance, some are rich and some are famous.

[Also it has to be noted that when we are working with arguments that make use of multiple quantifiers and two-place predicates there is a special problem that makes it impossible to determine invalidity by any mechanical means. This is because of the need for a new name rule to avoid thinking we are talking of the same individuals when in fact they might be different.]

Use the program Interpretations to experiment with these falsifying interpretations (but note that you will need to link separate premises with the postfixed signal A in order to have just a single premise string).

Go on to the use of consistency trees.

Go back to the starting page.