Interlude:  things aren't always easy  

      The definitions previously given are not complex and use terms with which you already familiar, e.g., "true", "false", "possible".  (Here we are only giving an informal exposition of some parts of elementary logic and thus cut a few corners and rely, for example, on your intuitions about what is possible.  If you take a course in logic, you'll find that these terms and concepts can be explained in a more formal and precise manner.  You'll find one step in this direction discussed under "Truth Tables" in the section "Glances Ahead".)

      In this section, we will examine argument forms with different combinations of truth values to illustrate the point that, except in the case of an argument with true premises and a false conclusion (an invalid argument as previously defined), the truth values of the premises and the conclusions do not determine the validity of the argument.  
 
To see why things aren't always easy, consider the following cases:

         Type 1: All premises are true.
                       Conclusion is true.
Looks to be valid, right? Doesn’t it just fit the definition of validity? Here is an example:
                  Ex. All mammals have fur.
                        Lassie is a mammal.
                        Therefore, all mammals give birth to live young.
       The problem here is obvious when we examine this example.  It is not enough to assert simply that all premises are true and the conclusion is true; we must assert that if the premises are true, then the conclusion must be true.  Although in the Lassie argument, as a matter of fact, all the premises are true and the conclusion is true (let's suppose), the conclusion could still be false.  In other words, I can imagine, without contradiction, worlds in which the first two premises are true and the conclusion false.  Therefore, this argument is actually invalid.
       So are all cases of Type 1 invalid arguments?  No.  Consider
                 Ex. All mammals have fur.
                        Lassie is a mammal.
                        Therefore, Lassie has fur.
This is a case of Type 1 and is a valid argument.  (Satisfy yourself of this).  So Type 1 cases will include both valid and invalid arguments.  More generally, the validity of an argument cannot be determined simply from examination of the actual truth or falsity of the premises and conclusion.

         Type 2: One or more premises are false (all could be false).
                       Conclusion is false.
                  Ex. All men are immortal.
                        Socrates is a man.
                        Therefore, Socrates is immortal.
      The first premise and the conclusion are both false, so one might assume the argument to be invalid; however, note that the definition states that “If all of the premises are true (as in, if the premises are taken to be true), then the conclusion must be true.”  In this case, if one assumes that premises are true, then the conclusion must hold. So the Socrates argument is valid.  Let us look at another example argument of the same form (with actually true premises) to solidify this point:
                   Ex. All men are mortal.
                         Socrates is a man.
                         Therefore, Socrates is mortal
Does this mean that all cases of Type 2 are valid? No. Consider
                   Ex. All men are immortal.
                         Socrates is a man.
                         Therefore, Socrates is Swedish.
Satisfy yourself that this is invalid.
So Type 2 cases will include both valid and invalid arguments.

          Type 3: One or more premises are false (all could be false).
                        Conclusion is true.
                  Ex. The sky is blue and the grass is purple.
                        Therefore, the sky is blue.
       The premise is false, but the conclusion is true.  This argument is valid because, if we take the premise to be true, then the conclusion must follow.  The premise is a conjunction and so, if we assume the conjunction is true (which entails having two true conjuncts), then the conclusion must be true because it is one of the conjuncts.  Also, note that this is an example in which a valid argument fails to preserve truth value.
Does this mean that all cases of Type 3 are valid? No.  Consider
                  Ex. The sky is chartreuse and the grass is ivory.
                        Therefore, the sky is blue.
Satisfy yourself that this is invalid.
So Type 3 cases will include both valid and invalid arguments.  

VI. Conditions