Chapter 7, Tutorial 3
PD completed: ^I and strategy

To think about our last rule of inference, let's take up where we left off at the end of tutorial 2.

Think about categorical syllogisms. Here's one:

All tigers are mammals.
All mammals are animals.
So, all tigers are animals.

Not very interesting...but it is clearly valid. In symbols:

(^x)(Tx>Mx)
(^x)(Mx>Ax)
(^x)(Tx>Ax)

Now let's think about doing a derivation of the conclusion. Look at the main connective of your two premises, in both cases it is the universal quantifier. So use ^I:

 

Once you eliminate the ^ and justify lines 3 and 4, you can use HS.

But what to do next? You know only that if a is a tiger, then a is an animal!

 

 

Here's a hint. 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!)

Whenever a name is arbitrary in this way, we pulled it out of the blue (so to speak), it could stand for 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 5 we understand in this way: "If an arbitrary object is a tiger, then it's an animal ". As long as 'a' is arbitrary, then, the step to line 6 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:

^I
input:

output:
P(a)

(^x)P

Provided 'a' is arbitrary in this sense:

  •  'a' does not occur in any premise or in the assumption of a subderivation still in progress (unterminated).
  •  'a' does not occur in P.

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, 'Px' for "x is president", and 'Mx' for "x is President.)

Premise 1 Mg&Pg                 
1 ^I 2

(^x)(Mx&Px)

MISTAKE!

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.

Derivation 1 Derivation 2
Premise 1 Bc&(^x)(Lx&Jx)
1 &E 2 (^x)(Lx&Jx)
2 ^E 3 Lc&Jc
1 &E 4

Lc

2 ^I 5 (^x)Lx
Premise 1 Bc&(^x)(Lx&Jx)
1 &E 2 (^x)(Lx&Jx)
2 ^E 3 Lb&Jb
1 &E 4

Lb

2 ^I 5 (^x)Lx