Quantifiers and Implications

One of the most important things I learned this week was about the differences between universal quantifiers and existential quantifiers.  If a statement has a universal quantifier (all), in order for it to be true it must have no falsifying example; there must be no instance where the statement is false.  Therefore, to prove the statement false, you need only one example where it is not true.  If instead you have a statement with an existential quantifier (some), to prove the statement True there must be one example where the statement holds.  To prove this statement False, there must be no instance where it is True.  When dealing with an empty string, it will prove a statement with a universal quantifier True, and one with an existential quantifier False.

Implications are interesting, and sometimes confusing to construct from English sentences.  An implication is made up of an antecedent and a consequent, in the direction of:

if antecedent then consequent

      In English, however, it is sometimes said: 

 consequent if antecedent 

     or it has confusing modifying words stuck in, such as unless, which is handled as an if not statement.  I personally find this a bit confusing as you add other modifiers, and so there may be a bit of extra practice done on this subject.

Personally, I am a little excited about this new outlook on implications; I can see future arguments with my siblings becoming more interesting when I can say that their point doesn't falsify mine, when in that context it probably should.