Logic lesson 5

Logic 5

One useful method of understanding statements and whether they make sense is to construct something called a Venn diagram. Here is an example.

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 circle.

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 picture.

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.

via GIPHY

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 evaluate:

(1) If Professor Oldeagle is a complainer then he is an enemy of the state.
and
(2) If Professor Oldeagle is an enemy of the state then he is a complainer.

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 bill.

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 penny.)

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.