Basic Logical Concepts

 

What is an argument?
An argument is a group of statements in which some of the statements (the premises) are claimed to provide support for another (the conclusion).  All arguments have just one conclusion, but may have several premises (at least one premise is required).

Statements are sentences which have a truth value, which is to say, a sentence that is either true or false.   "Watch your step!", for example, is a sentence, but it is not a statement because it is neither true nor false.

Arguments may be said to be valid or invalid, sound or unsound.  They are not properly said to be true or false.
Statements may be said to be true or false.  They are not properly said to be valid or invalid, sound or unsound.


Validity

Put most simply, if an argument is valid, the truth of the premises insures the truth of the conclusion.  Otherwise put, if an argument is valid, there will be no case in which all the premises are true and the conclusion is false.  The following argument, for example, is valid:  

All B are C
All A are B
All A are C    

Think about the argument above.  Could you substitute terms in for A, B, and C (substituting consistently, of course) such that the two premises come out true but the conclusion false?   Experiment for a moment to see if you can do this.  Eventually, you will be convinced that you can't.  (And no, you cannot subsitute so as to have the same word but with two different meanings - e.g. "plant" - that's a particular kind of error called "equivocation" that we'll discuss later.)

An informal way to gauge the validity of an argument is to ask yourself the following question:  IF the premises are true, MUST the conclusion also be true?  Note that in any given case, the premises may or may not be true - but pretend for a moment that they ARE true, and ask yourself if their truth also insures the truth of the conclusion.  If they do, then the argument is valid.

It will have occurred to you, then, that it is possible to have a valid argument in which some or all of the premises are false.  It is even possible to have a valid argument with a false conclusion.  The one case that is NOT possible in a valid argument is to have all true premises and a false conclusion.

Consider the following instances of the valid argument form presented above:
 

#1:    All bats are camels       (false)      
        All aardvarks are bats      (false)   
        All aardvarks are camels    (false) 
 

#2:     All maples are dogs    (false)      
         All poodles are maples     (false) 
         All poodles are dogs        (true)  
 

#3:    All vertebrates are felines    (false)
        All cats are vertebrates   (true)
        All cats are felines             (true)
 

#4:   All bears are mammals    (true)
        All animals are bears      (false)
        All animals are mammals   (false)
 

#5:    All trees are plants   (true)
        All oaks are trees     (true)
        All oaks are plants   (true)
 

Again, all five of the arguments above are valid.  That's because, in each case, IF the premises were true, the conclusion would have to be true.
 

Now, what happens if the argument is invalid?  Then you aren't guaranteed anything about the truth - or falsity - of the conclusion.  Sometimes, invalid arguments have conclusions which just happen to be true, but not because their truth was supported by the premises.  With invalid arguments, the premises may be true or false, and the conclusion may be true or false: all combinations are possible.  Here, for example, is an invalid argument:

Some B are C
All A are B
All A are C

In this argument, the truth of the premises does not guarantee the truth of the conclusion.  In other words, it will be possible to find instances of this argument form in which the premises are all true and the conclusion is false.  Here is such a case:

   
Some Christians are Baptists     (true)
All Catholics are Christians.      (true)
All Catholics are Baptists.         (false)
 

It might have occurred to you by now that if you have an argument in which all the premises are true and the conclusion is false, then that argument must be invalid (remember - there is no such thing as a valid argument with all true premises and false conclusion).

But again, remember that all combinations are possible in an invalid argument.  For example, here is an invalid argument with a true conclusion.  (It has the same form as the argument immediately above.)
 

Some canines are dogs.  (true)
All poodles are canines.  (true)
All poodles are dogs.   (true)
 
 

Soundness
As we've just seen, just because an argument is valid doesn't necessarily mean that the conclusion is true.  But if, in addition to knowing that the argument is valid, you also know that the premises are true, THEN you know that the conclusion must be true.  In this case the argument is said to be sound.  In other words:

An argument is sound if and only if both of the following conditions are met:
1. The argument is valid.
2. All the premises are true.

If the argument is sound, then the conclusion must be true.


Here is a quick self-test on some of the ideas presented so far:

True or False

1. You can have a valid argument with all false premises.
2. You can have a valid argument with some true premises and some false premises.
3. You can have a valid argument with all false premises and a false conclusion.
4. You can have a valid argument with all true premises and a false conclusion.
5. You can have a sound argument with a false premise.
6. You can have a sound argument with a false conclusion.
7. You can have a sound argument that is invalid.
8. You can have a valid argument that is unsound.
9.  You can have an invalid argument with all true premises and a true conclusion.
10.  You can have a false argument.
11. You can have a true argument.
12.  You can have a valid statement.

Click here for the answers.
   


 Here is a summary table of some of the ideas discussed above:



Valid Argument Example
Invalid Argument Example
True Premises
True Conclusion
All dogs are mammals  (T)
All terriers are dogs (T)
All terriers are mammals(T)

All dogs are mammals (T)
All terriers are mammals (T)
All terriers are dogs (T)

True Premises
False Conclusion

  One doesn't exist!

All dogs are mammals (T)
All cats are mammals (T)
All cats are dogs  (F)
False Premises
True Conclusion
All mammals are fish (F)
All barracuda are mammals (F)
All barracuda are fish (T)
All mammals are dogs (F)
All cats are dogs (F)
All cats are mammals (T)
False Premises
False Conclusion
All terriers are cats  (F)
All mammals are terriers (F)
All mammals are cats (F)
All dogs are cats (F)
All mammals are cats (F)
All mammals are dogs (F)
Mixed (both T and F) Premises
True Conclusion
All mammals are dogs (F)
All terriers are mammals (T)
All terriers are dogs (T)
All mammals are dogs (F)
All terriers are dogs (T)
All terriers are mammals (T)
Mixed (both T and F) Premises
False Conclusion
All terriers are dogs  (T)
All cats are terriers (F)
All cats are dogs  (F)
All terriers are dogs (T)
All mammals are dogs (F)
All mammals are terriers (F)


   Question: Which of the 12 argument examples above (highlighted in white) is/are sound?  Why?