Chapter Eight, Tutorial Two
The final two rules of PD are a bit more complicated. ^I will be our means of deriving a statement of form (^x)P. But it's hard to see what premises are sufficient to allow one to "introduce" the universal quantifier. (Even if you know that a bunch of things in the universe of discourse are P and you know of no exceptions, this does not prove that everything is P.)
The rule, %E, to allow one to draw conclusions from an assumption of form (%x)Px, is similarly difficult: A statement of this form is not specific about the object which satisfies P. This makes the formulation of this rule a little tricky.
To provide motivation for our rule of ^I, think about the following English reasoning.
If you make more than four thousand dollars, then you must file. And if you are required to file but do not, then you will be prosecuted. Thus, if you make more than four thousand dollars yet do not file, you will be prosecuted.
Yes, this is IRS thinking! And it's clear who the "you" is, anyone in the IRS's domain of discourse: US citizens and residents. So, the "you" is arbitrary. The thinking is about anyone in this universe of discourse.
Most importantly, notice that the conclusion really amounts to a universal generalization: "All who make more than four thousand dollars without filling income taxes will be prosecuted". From the IRS thinking, then, one can see how to derive a universal generalization: argue from statements about arbitrary individuals.
Let's see this in action within a derivation. To have a clear interpretation in mind, let
Mx: x makes more than four thousand dollars.
Then the IRS argument may be formalized as follows:
Let's do a derivation showing this to be valid. Begin with two applications of ^E.
We've substituted in with 'a', but of course we could have picked 'b', or 'c', or any name we might want to give for anything in the universe of discourse. (Hint: this will mean that 'a' is arbitrary!)
Now ask: what would happen if 'Ma&~Fa' is true, if some arbitrary individual both makes more than $4000 yet does not file? Consider the subderivation:
All this only goes to prove that '(Ma&~Fa)>Pa' is true -- a mere substitution instance of our goal. But wait, 'a' is arbitrary. Again, look where it comes from: lines 3 and 4. As far as this derivation is concerned, 'a' could be a name of anything. That's the idea behind ^I: we generalize from a particular instance, so long as the instance uses the arbitrary name.
So, the last two lines of our derivation will look like this:
Line 11 we understand in this way: "If an arbitrary person makes more than four thousand dollars but does not file, then he or she will be prosecuted". As long as 'a' is arbitrary, then, the step to line 12 makes sense.
As you see, this is exactly what we call ^I: the move from a substitution instance with arbitrary variable to the universally quantified sentence.
Now for the big question: How do we tell when a name is arbitrary? Our prescription is fairly simple: make sure the name does not occur in any premise or undischarged assumption and does not occur in the line derived by ^I. Here's the general formulation:
Notice the provisos! They make sure the constant instantiated in the output of the rule is arbitrary: it's not about some particular object described in premise, assumption or conclusion of the rule application.
Finally, it worth seeing how this rule should not be applied. (Let 'g' stand for George W. Bush, 'Mx' for "x is president", and 'Px' for "x is President.)
Obviously, just because George is male and president, one should not conclude that everyone is! Because 'g' occurs the premise of line 1, it follows that it does not count as arbitrary. (This is so because it fails to meet the first proviso. Can you think of an example that fails to be a proper application of the rule because it fails to meet the second proviso?)
Now, which of the following is correct? Look carefully and click on the derivation number of the correct application of ^I.