Chapter Seven, Tutorial One
PD: Derivations for PL Made Easy

We need to introduce new derivation rules. Why? Well, we have two quantifiers and we need rules for each. We'll need enough rules so that we can break down premises with main connective ^ or % and so that we can build up sentences with those connectives. Fortunately, these rules are mostly very easy. In fact we've already mentioned one while symbolizing.

To make this task of introducing the rules as intuitive as possible, keep this old interpretation in mind. It's from 6.1d: the symbolization exercise about cats and only about cats! The universe of discourse is just the set of all cats. Oh, and I've added "Cx"...to stand for "x is a carnivorous".

Universe of Discourse: Cats (and only cats! for this symbolization felines are our only subject)

Cx: x is carnivorous; Mx: x is a mammal, Rx: x is a reptile; Wx: x is wild.

f: Felix the Cat (pretend he's real)
t: Tony the Tiger (of course, he's real, he was on TV)

 

Universal Elimination: ^E

The idea here is simple. Suppose you have a derivation beginning like this:

Premise 1 (^x)(Mx&~Rx)
Premise 2 (^x)Cx
     
     

Because line 2 says that everything in the universe of discourse, i.e. all cats, are carnivorous, it follows that anything 'f' might name is a C. So, at line 3, we can reasonably conclude that f is Cs. So, Felix is a carnivore, if you will! But we could have written any other lower case letter in place of 'x'. We justify this by ^E:

Premise 1 (^x)(Mx&~Rx)
Premise 2 (^x)Cx
2 ^E 3 Cf
     

Make sure you see that this makes sense! Line 2 premises that everything is C -- all the cats are carnivorous -- so we conclude on line 3 that f is C -- Felix is a carnivorous. OK? (If Felix is a cat and absolutely every cat is carnivorous, then Felix is too!) We call this rule "^E".

We could apply the same thinking and the same rule, ^E, to line 1. This could result in which of the following?

  1. Mx&~Rx
  2. Ma&~Ra
  3. Mt&~Mt