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Chapter 9: Valid and Sound Arguments

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Chapter 9: Valid and Sound Arguments

1.  Validity and Soundness

·      Validity is the most important concept in critical thinking. A valid argument is one where the conclusion follows logically from the premises. But what does it mean? Here is the official definition:

An argument is valid if and only if there is no logically possible situation in which the premises are true and the conclusion is false.

·      Whenever we have a valid argument, if the premises are all true, then the conclusion must also be true. What this implies is that if you use only valid arguments in your reasoning, as long as you start with true premises, you will never end up with a false conclusion. Here is an example of a valid argument:

Maya is 20 years old
Maya is more than 10 years old.

·      The validity of the argument can be determined without knowing whether the premise and the conclusion are actually true or not. 

·      Validity is about the logical connection between the premises and the conclusion. We might not know how old Maya actually is, but it is clear the conclusion follows logically from the premise. The simple argument above will remain valid even if Maya is just a baby, in which case the premise and the conclusion are both false. Consider this argument also:

Every birds can fly.
Every bat is a bird.

Every bat can fly.

·      The argument is valid—if the premises are true, the conclusion must be true. But in fact, both premises are false. Some birds cannot fly (the ostrich), and bats are mammals and not birds. 

·      What is interesting about this argument is that the conclusion turns out to be true. So, a valid argument can have false premises but a true conclusion. There are of course also valid arguments with false premises and false conclusions. What is not possible is to have a valid argument with true premises and a false conclusion.

·      A valid argument is one where it is logically impossible for the premises to be true and the conclusion to be false. But logically impossible does not mean "unlikely." Consider this argument: Laxmi is a one-month-old human baby, and so Laxmi cannot walk. This seems rational, but the argument is not valid because a one-month-old walking baby is not a logical impossibility. Imagine a scenario in which Laxmi is the product of a genetic experiment, and she is able to walk right after birth.

·      An argument that is not valid is invalid. This happens as long as there is at least one logically possible situation where its premises are true and the conclusion is false. Any such situation is known as an invalidating counterexample. It does not really matter whether the situation is realistic or whether it actually happens. What is important is that it is coherent and does not cause any contradiction.

·      Arguments are either valid or invalid, but we should not describe them as true or false. Because an argument is not a single statement, it is unclear what a true argument is supposed to be. Does it mean the argument has a true conclusion, or does it mean the argument is valid, or are we saying that the premises are true? It is confusing to speak of true and false arguments.

2.  Patterns of Valid Arguments

·      Valid arguments are useful because they guarantee true conclusions as long as the premises are true. But how do we know if an argument is valid? One indirect way is to see if we can come up with an invalidating counterexample. If we can, the argument is not valid. But of course, the weakness of this method is that when we fail to find a counterexample, this does not guarantee that the argument is valid. It is possible that we have not looked hard enough.

·      A more direct way of establishing validity is to demonstrate step by step how the conclusion of an argument can be derived using only logical principles. This is what formal logic is all about. But for everyday reasoning, a good understanding of some basic patterns of valid argument should also suit.

a.   Modus Ponens

·      Consider these two arguments: 

If whales are mammals, then whales are warm-blooded.
Whales are mammals.
Whales are warm-blooded.

If you love me, you should remember my birthday.
You love me.
You should remember my birthday. 

·      Both arguments are valid. It does not matter whether the premises and the conclusions are true or not. Furthermore, these arguments are similar to each other in the sense that they have the same logical structure, which can be represented by this pattern:
If P, then Q.
P. 
Q.

·      Here, the letters P and Q are sentence letters. They are used to translate or represent statements. By replacing P and Q with appropriate sentences, we can generate the original valid arguments. This shows that the arguments have a common form. Because this particular pattern of argument is quite common, it has been given a name. It is known as modus ponens. But do not confuse modus ponens with the following form of argument, known as affirming the consequent:
If P, then Q.
Q. 
P.

·      Not all arguments of this form are valid. Here are two invalid ones:

If Ram lives in Butwal, then Ram lives in Nepal.
Ram lives in Nepal.
Ram lives in Butwal. 

If Hari loves you, then Hari will buy you a bunch of roses.
Hari will buy you bunch of roses.
Hari loves you. 

·      Both arguments are invalid, and it is easy to find some invalid counterexamples. For example, Gopal might live in Pokhara and not Butwal, so the premises of the first argument are true but the conclusion is false. Similarly, it might be true that Govind will buy you some roses if he loves you. But perhaps he will buy them even if he does not love you. Maybe he hates you so much that he decides to send you some roses that have been sprayed with a lethal virus.

b.   Modus Tollens

·      Modus tollens is also a very common pattern of a valid argument:

If P, then Q.
Not - Q. 
Not - P.

If Superman is a human being, then Superman has human DNA.
It is not the case that Superman has human DNA. 
Superman is not a human being.

·      Note that "not- Q" simply means the negation of Q—for example, "it is not the case that Q." So, if Q means "Superman has human DNA," then not-Q would mean "it is not the case that Superman has human DNA," or "Superman does not have human DNA." But do distinguish modus tollens from the fallacious (wrong) pattern of argument known as denying the antecedent:

If P, then Q.
Not - P. 
Not – Q.

If Einstein is a biologist, then Einstein is a scientist.
But Einstein is not a biologist.

Einstein is not a scientist.

c.    Disjunctive Syllogism

·      Both patterns are valid:

P or Q.
Not - P. 
Q.

P or Q.
Not - Q. 
P.

·      Either we should break up, or we should get married.
We should not get married. 
We should break up.

d.   Hypothetical Syllogism

·      Another pattern of a valid argument:

If P or Q.
If Q then R. 
If P then R.

·      If God created the universe, then everything is perfect.
If everything is perfect, then there is no evil. 
If God created the universe, then there is no evil.

e.    Constructive Dilemma 

·      A pattern of a valid argument with three premises:

P or Q.
If P then R.
If Q then S. 
R or S.

Either the president is lying, or he is telling the truth.
If the president is lying, then he is wicked.
If the president is telling the truth, then he is mad. 
Either the president is wicked, or he is mad.

·      When R is the same as S, this is an equally valid pattern:

P or Q.
If P then R.
If Q then R. 
R.

·      Either our actions are random, or our actions are determined.
If our actions are random, we do not have free will.
If our actions are determined, we do not have free will. 
We do not have free will.

f.     Destructive Dilemma

If P then Q.
If R then S.
Either P or R. 
Therefore, either Q or S.

If the sun shines tomorrow, [then] we will go to the beach.
If it rains tomorrow, [then] we will go to the museum. 
Tomorrow we will either not go to the museum or not go to the beach [or both]. 
Therefore it will either not rain or the sun will not shine [or both].

 

g.   Reductio ad Absurdum 

·      Reductio ad absurdum is Latin for "reduced to absurdity." It is a method for showing that a certain statement S is false:

a)    First assume that S is true.

b)   From the assumption that S is true, show that it leads to a contradiction, or a claim that is false or absurd.

c)    Conclude that S must be false.

·      Suppose someone claims that a human being's right to life is absolute and so we should never kill or destroy human life. 

·      But is this acceptable? If it is, then it follows that when you are being attacked, it will be wrong for you to kill your attacker if this is the only way to prevent yourself from being harmed. But surely this is not correct. Most people would agree that in some situations when your life is threatened you can respond by deadly force, and this is recognized by our legal system. Since the original assumption leads to an absurd conclusion, this demands that the right to life is not absolute. 

h.   Combining Patterns to form more Complex Arguments:

·      The patterns of valid arguments can be combined together to form a complex argument. This valid argument involves three applications of modus tollens:
If P then Q.
If Q then R.
If R then S.
Not – S.
Not – P.

For example, 
If Ram is coming to Kathmandu, then he will come by bus.
If Ram is coming by bus, then he will arrive tomorrow.
If Ram will arrive tomorrow, then he will have a chances to stay with us for a week.
Ram is not going to stay with us for a week.
So, Ram is not coming to Kathmandu. 

3.  Arguments Involving Generalizations

·      A generalization, or a general statement, is a statement that talks about the properties of a certain class of objects. We shall be concerned only with the following three main kinds of generalizations:

Type

Example

 

Universal

Every F is G; all Fs are Gs.

Every great idea is ridiculed in the beginning.

Feminists hate men.

Asians are good at math.

For example, to show that "all politicians are corrupt" is false, all you need to find is one single politician who is not corrupt.

All men are mortal.
Socrates is a man.
Socrates is mortal.

Existential

Some F is G; at least one F is G. 

Some dinosaur is warm-blooded.

All men are mortal.
Socrates is a man.
Something is mortal. 

All whales are mammals.
Some whales are carnivorous.  
All carnivorous organisms eat other animals. 
Therefore, some mammals eat other animals.

Statistical

Statements that say that certain proportion of Fs are Gs.

1 out of 3 children in the U.S. is born out of wedlock.

19 of the 21 hijackers on September 11 were Saudi Arabians.

 

a.  Patterns of Valid Argument

Here are few valid argument patterns involving every, with some examples:

Every F is G.

X is F.

X is G. 

Every F is G.

X is not G.

X is not G.

Every F is G.

Every G is H.

Every F is H.

Every dog is hairy.

Harry is a dog.

Harry is hairy.

Every dog is hairy.

Harry is not hairy.

Harry is not a dog.

Every terrier is a dog.

Every dog is an animal.

Every terrier is an animal.

But notice that all the arguments below are not valid:

Every F is G.

X is G.

X is F.

Every F is G.

X is not F.

X is not G.

Every Fs is Gs.

Most Gs are Hs.

Most Fs are Hs.

Every dog is hairy.

Harry is hairy.

Harry is a dog.

Every dog is hairy.

Harry is not dog.

Harry is not hairy.

Most birds are creatures that can fly.

Most creatures that can fly are insects.

So, most birds are insects.

*Terrier: Hunting /tracking

4.   Soundness

·     We know that if the premises are true, the conclusion is also true. But validity does not tell us whether the premises or the conclusion are actually true. 

·     If an argument is valid, and all the premises are true, then it is called a sound argument. So, the conclusion of a sound argument must be true.

·     We should try out best to provide sound arguments to support an opinion. The conclusion of the argument will be true, and anyone who disagree would have to show that at least one premise is false, or the argument is invalid, or both. 

END OF THE PART

 

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