One useful method of understanding statements and whether they make
sense is to construct something called a Venn diagram. Here is an
The rectangle represents some universe of objects. This idea can be
very general. It doesn't have to apply to the Universe that we live in.
All that is needed is a "universe" that will include all of the things
that are part of the problem we are dealing with.
In this case the "universe" is roughly equivalent to what people call
"the world." The problem might be something involved with different
kinds of living creatures. We will only draw in those living creatures
that we need to deal with. So we could draw this diagram first and then
perhaps some people would argue that Shetland ponies are ponies, and
ponies are different from horse, and probably other people would argue
for including ponies within the circle for horses, but in a smaller
The way things work, humans often
discover narrow categories of things first, and only later discover
that things in the narrower categories are grouped together under
larger categories. For instance, humans have known about birds, fish,
mammals, reptiles, etc. The earliest lumping together may have been
when people started talking about "birds and beasts." Humans originally
didn't know about the dinosaurs, so they were left entirely out of the
More recently, humans have started noticing that mammals and birds are
more similar to each other than are, e.g., birds and snakes.
Functionally, at least, they are similar in being warm-blooded. Being
warm-blooded means that an organism can function in much colder weather
than can the other kinds of animals.
Let's take a rest from formal Venn studies for a while. Here is a
story. The names and places have been changed to protect the innocent.
In the Malasinus Republic, the President for Life learned thatProfessor
Oldeagle had become very critical of the government. President Pavo
said, "People who constantly complain are enemies of the state.
Professor Oldeagle must be silenced quickly and without failure!"
As the Venn diagram shows, lots of people just complain about things
all the time. They don't have anything to do with politics, much less
are they enemies of the state. Some enemies of the state don't
complain. They stay in the shadows as much as they can. What do you
think? What if people said, "All people who study formal logic or even
just read a lot are cowardly intellectuals who ought to get beaten up?"
A football coach at a major college said to a visiting professor that
all the college's non-sports teachers were
putrid intellectuals and just plain punks. The putrid professor
demonstrated gansekinage (scroll down to see it)
on the football coach. Sometimes the
region where two circles of a Venn diagram intersect doesn't contain
all the humans in the universe.
Please note the big question marks on each of the above Venn diagrams.
Professor Oldeagle belonged in the complainers circle but not the part
of it that was overlapped by the "enemies of the state" circle, and
Professor Putrid would more charitably be put in the general
"academic types" category and not among "the punks." It would have
clarified just what he was if the red arrow had pointed into the part
of "academic types" overlapped by the "Aikido player."
People can misuse language and ideas sometimes. If you knew Professor
Oldeagle you might draw the above Venn diagram and then think about
complainers. You probably know several people who complain all of the
time but have no interest in anything political, military, or anything
else that might make them enemies of the state.
Now back to formal logic for a moment. Here are two statements to
(1) If Professor Oldeagle is a complainer then he is an enemy of the
(2) If Professor Oldeagle is an enemy of the state then he is a
Before we do anything else with these statements we need to evaluate
the truth of each of them. We can't do that job with logic. If I say
that the temperature outside is right at the freezing point this very
moment, the only way to nail that issue is to go outside with a
thermometer. If it says 0° Centigrade or 32° Fahrenheit, then the
statement is true. If the thermomenter reads anything else, then I was
wrong. A miss is as good as a mile in this kind of case.
Statement 1 is false if
investigation shows that Professor Oldeagle is a complainer and it also
shows that he is not an enemy of the state. When we find out the true
facts and learn that somebody has made a statement that is false, then
what are the results that follow along with that discovery? We now know
that whoever made the statement was either a deliberate liar or else
that he or she was badly informed about the facts and ought not to have
made reckless accusations.
This is analogous to the example, "If I give you a penny, then I will
give you a hundred dollar bill," which is wrong only if I do give you a
penny, but I break my promise and don't give you the hundred dollar
Statement 2, " If Professor Oldeagle is an enemy of the state then he is a
complainer," is false if
investigation shows that Professor Oldeagle is an enemy of the state
but it also shows that he does notmake
it his business to complain (True statement → False statement)
Since we have established by investigation that he is not an enemy of
the state and that he is a complainer (False statement → True statement
), statement 2 will be found true. (That is like the case wherein I
just give you a hundred dollars, despite the fact that you gave me no
Those two results ought to tell us
that people who are opposed to a government in a significant way may
well try not to do anything to help the government find them out.
People who are not enemies of the state may complain, and people who
are enemies of the state may intentionally avoid complaining and other
things that would draw attention. In this case, Professor Oldeagle is
not an enemy of the state, and he also is not a complainer. That is not
a surprising finding because spies and other such subversive types of
people often do not complain or otherwise make trouble because
they want to look like the majority of ordinary citizens, but there is
no necessary connection between not complaining and being an enemy of
the state either.
One mistake people often make is to turn a true statement of the form A
→ B into B → A. That does not work. It may well be true that, e.g., "If
Bertie is a bad guy then Bertie will destroy houses", but it doesn't
follow that "If Bertie destroys houses, then Bertie is a bad guy."
Maybe Bertie is one of those guys whose job it is to demolish old and
unwanted buildings to make room for new buildings.
A demolition expert is usually a highly paid "good guy."
One of the big problems with the arguments we can make by using formal
logic is that so far we have no way of using logic symbols to
distinguish statements such as, "Some complainers are enemies of the
state," which is true, from, "All complainers are enemies of the
state," which is definitely not true.
What is the big take-home discovery here? It is that people can say
anything and they often can make it look good, but to know the truth
about what is really going on you have to do the investigations.