7.1ex IV
Semantics

Matching. Drag sentences from the right to the correct location in the proof box.

universe of discourse = the counting numbers: 1,2,3, etc.
Gxy: x is greater than y

Now, show that the sentence

(1) '(%w)(^z)Gzw' is FALSE.

Assume          __________

Then           _________

But          ________
So,           _________

But          ________

(for contradiction) that
'(%w)(^z)Gzw' is true.
some substitution instance of (1)
    (^z)Gzn
must also be true (where 'n' is picked to name the appropriate number).
because '(^z)Gzn' is true, all it's substitution instances must be true.
'(^z)Gzn's substitution instance
     Gnn
is true.

this last result that 'Gnn' is is true is an absurdity for no number is greater than itself!

Hence, the initial assumption is wrong and (*) is false.

 

 

 

Q.E.D.