Five counter-intuitive logic lessons

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Logic teaches a number of useful lessons such as how to identify fallacies, how to avoid cognitive biases, and how to construct good arguments.  But, some of the lessons of logic are counter=intuitive.  Let's look at a few of these.

​1. An argument with true statements can be invalid.  An argument is said to be valid if the premises provide necessary support for the conclusion.  An argument can do this with either true or false statements depending on how they are formulated.  However, just because the premises of an argument are true does not mean the argument is valid.  While the argument may sound persuasive, it could be the case that the premises do not provide support for the conclusion.  A good example of this is the following argument which contains all true statements but is, nevertheless, invalid: All banks are financial institutions.  Chase is a financial institution.  Therefore, Chase is a bank.

2.  Statements can sound very different, yet mean exactly the same thing.  One of the insights you can learn from categorical logic is that statements which sound entirely different are, in fact, equivalent in meaning.  One of the purposes of studying categorical logic is to learn precisely this insight.  Another purpose is to give you the power to simplify complex statements such as this one: Some employees who are not currently on the payroll are not ineligible for workers' benefits.  Categorical logic can show that this rather unclear statement is really the same as this much simpler statement: Some of those eligible for workers' benefits are not currently on the payroll.

3.  There is a mathematical-like rigor to ordinary language.  Certain words in ordinary language such as "and," "or," "if...then," and "if and only if" function somewhat like the mathematical operators for addition, subtraction, multiplication, and division.  What this means is that you can determine whether statements are true or false without knowing everything about the statement's content.  For example, in the statement "Nixon resigned the presidency and Clinton wrote the Gettysburg Address" you can determine that this statement is false if all you know is that Clinton did not write the Gettysburg Address.  Partial information can lead you to detect when statements are false (or true).  

4.  It is possible to evaluate an argument's merits without entirely understanding its content.  This possibility exists in logic due to the previous point and the fact that we can build upon it a set of rules which allow anyone to deduce an argument's validity without reference to its content.  Just as in math where you can add numbers without worrying about what the numbers reference (2+3=5 and you don't need to know what you're adding 2 of and 3 of to deduce that) you can also infer an argument's validity without worrying about the argument's reference.  While this is one of the most difficult points to master in the study of logic it turns out to be a very powerful tool for the evaluation of everyday arguments.  

5.  Fallacies of thinking are extremely common in ordinary discourse.  With all the power of logical reasoning, it is still quite common for people to be persuaded by faulty arguments.  What's worse is that many of these fallacies are easy to recognize with only a little training in the very basic principles of logic.  Certainly one of the reasons why fallacies of thinking are so common is because they are so effective.  These fallacies in reasoning are effective in part because our brains are wired up to be persuaded in ways that are not always rational and because without some basic knowledge of logic it is easy to overlook these fallacies.