18.  USING CONSISTENCY TREES FOR QUANTIFIED EXPRESSIONS
Click on if you need more help on the use of a consistency tree.

You have already seen how we can test for validity with expressions that do not involve quantifiers.  Now we will expand to see how we can do the same when quantifiers are involved.  Earlier you saw that a complete truth table for a farly simple argument form such as (x)(Sx->Px), (Ex)~Sx |- (Ex)~Px called for sixteen lines, and we did not attempt to work with the reverse method since an easier technique would be available. 

The concept here is that, as we did with our derivations, we will instantiate a quantifed expression unless the quantifer is not itself the main connective in an expression.  What we do have to watch out for, as in the derivations, is that we respect our new name rule when working with existential quantifiers.  Let's use the same argument form that we had before.  Again, a first step is to negate the original conclusion.
(x)(Sx->Px), (Ex)~Sx |- (Ex)~Px

(x)(Sx ->Px)
(Ex)~Sx
~(Ex)~Px
(x)Px
~Sa
Pa
Sa->Pa
/    \
~Sa    Pa
x     
invalid
Any open branch indicates that the form is invalid (it is consistent to have a false conclusion with all true premises).  All closed branches tells us that the form is valid.

red flagThere is one special note.  While this technique will work for any expression with just monadic (one-place) predicates, neither it nor any other technique will work for some expressions involving polyadic (many-place) predicates  (something that happens because of the necessity of respecting our new name rule).



EXERCISES (ON YOUR OWN)

Symbolize and use trees to test for validity

1.  All logic students work hard.  Some individuals who work hard are successful.  Therefore, some logic students are successful.     
2.  Some logic students work hard.  All individuals who work hard are successful.  Therefore, some logic students are successful.     
3.  No students are lazy.  Anyone lazy will not do well.  Therefore, every student will do well.    
4.  No students are lazy.  Anyone not doing well is lazy.  Therefore, every student does well.    
5.  Every student is doing well.  Anyone lazy does not do well.  Therefore, no students are lazy.     

Go to the answer key.