It seems that we determine the truth or falsity of two sorts of statements:

  The most common being where some matter of essence is asserted, where the
essence in question may either have being, or not have being.  By essence
I refer to the properties of things, how things exist, or their "whatness".
By existence I refer to the fact of a things existence, or it's "thatness".

  For instance:

   "The dog has a tail"
   "I went to the store"
   "Buildings are big"
   
  In each case, the answer depends upon the existence of the essence in 
question.  For the first: whether dogs exhibit tailness, or whether tails
exist on dogs.  For the second: whether there existed a case where I was
in the store, some time ago.  For the third: whether buildings exhibit bigness,
or whether bigness exists in buildings.

  Another example deals not so much with particular essence, but with the
fact that there is essence in things at all (I call this distinction).  It
is distinction which forms the foundations of math and number theory: when
things have distinction, then they become countable.  And so, a mathematical
statement like:

   1 + 1 = 2

  Becomes the assertion "when there is a single distinguishable thing and
then another distinguishable thing, then there exists two distinguishable
things".  In both cases, however abstracted from the essence of the matter
we are (whether we are discussing the tails of dogs, or distinction in 
general), it is the existence (the realness) of it which determines the truth.

  The second case deals with the principle of existence; or consistency.  No
essence can both BE and not BE in a thing.  A car cannot both have wheels and
not have wheels.  Therefore, when someone asserts that they have a car that
has and doesn't have wheels, we know this to be false because of the limitaion
upon existence (that the essence of wheelness or nonwheelness must BE)- in 
this case .  In both cases, the substance of existence or it's presence is 
at issue: this is why Aquinas says that "truth is the same as existence".  
Further, in both cases, some matter of essence is at stake: some essence 
which may possibly exist ('like the car with and without wheels' or 'dogs 
with tails') and then determined whether such an essence CAN receive 
existence (no in the car case-- due to the nature of existence) or HAS 
received existence (yes in the dog case-- due to the act of God).