Possible Worlds Test for Validity
It is, by definition, impossible for a valid argument to combine True Premises and a False Conclusion. Therefore, we can “test” a conclusion for its veracity – even after accepting the premises of an argument – and if we do not accept the conclusion declare the whole argument invalid.
The possible worlds test can either involve:
· the actual world
· an imaginary world where what matters is consistency. That is, can we imagine a world where we can envision alternatives that would make the argument invalid.
Actual World Example:
All whales are mammals.
All kangaroos are mammals.
Some kangaroos are brown.
______
Whales are brown kangaroos.
While we accept all the premises, this does not force us to accept the conclusion. We can test the conclusion against a “possible world” (in this case the actual world) and declare that is not true; therefore, the argument must be invalid.
Imaginary World Example #1:
There is an infinite amount of space for worlds to be in.
______
There are an infinite number of worlds.
In this case, we can imagine a possible scenario that makes the argument invalid. Imagine, for instance, a universe in which there are just two worlds, separated by an infinite distance. In this case, there would be an infinite amount of space, but only a finite number of worlds—two, to be exact. Thus it is possible for this argument to have a true premise and false conclusion; therefore, the argument is invalid.
In an imaginary world we are concerned only if it is logically possible for true premises to join a false conclusion, not whether it is physically possible (as we can imagine scenarios where physical laws as we know them do not apply). You make your imaginary world possible, and then you need to keep it consistent.
Imaginary World Example #2:
There are an infinite number of worlds.
Not every one of them is inhabited.
______
There are a finite number of inhabited worlds.
In this case we also resort to our imagination, as we actually only know of one inhabited world. To demonstrate the invalidity of this argument, consider this:
· Imagine a world in which every other world is inhabited (dealing with the second premise, but not refuting it). Imagine that in this universe every world has a number, and that even-number worlds are inhabited and odd-number ones are inhabited. This is a scenario in which the premises would be true, but the conclusion false (as there can be an infinite number of odd-number worlds). Therefore, according to this possible world scenario, the argument is invalid, as the premises are true but the conclusion false.