Quantifiers - University of Nebraska Omaha

MATH 2230 — FALL 2013 17
Warning: Never ever use quantifiers as abbreviations for English language
phrases “for all” or “there exists” etc. Never write anything like
“this solves the question 8x >0butwestillhavetodealwithc apple 0”
or “we know that 9 a good object, but constructing it is more complicated”.
3.2. Proofs Involving Quantifiers. What makes mathematical theorems
complicated is a sequence of alternating quantifiers, like in the
sentence in (⇤) 14 . However, even with only one quantifier around we
may get lost if the arguments are not presented properly.
Some sentences are true in every possible interpretation (model).
For instance, (9x)(x >0) ) (9x)(x >0) is always true, regardless
of our interpretation of >. All (suitable) substitutions to tautologies
of propositional calculus will have this property, but not only those.
In this part we will meet several sentences which are true regardless
of their interpretation. Sentences of this type are called logical validities,
orlogical axioms, ortautologies of the quantifier calculus. We
will give some justifications for them – these arguments could serve as
“templates” for all “common sense” proofs involving quantifiers.
So, from now on let us fix:
• the non-empty domain (universe) U of all variables,
• formulas (x), (x) withatmostonefreevariablex.
When a is an object from U, then we write (a) forthesentence
obtained from (x) byreplacingeveryfreeoccurrenceofx with a.
Similarly for .
We will use x 0 to denote some fixed objects from U —note,thisis
not a variable!
Remark 3.2.1. Many mathematical theorems have the form of an implication.
There are various strategies to prove the conditional ' ) but
most often the direct approach is best. We assume that the hypothesis
(antecedent) ' is true and we argue that the conclusion (consequent)
holds true.
Sometimes we want to argue for a biconditional statement ' , .
Then most often we prove the two implications separately (remembering
Corollary 2.2.5(9)).
Proposition 3.2.2. The sentence
(8x) (x) ) (9x) (x)