Valid and Invalid Arguments Content from the guide to life, the universe and everything

Valid and Invalid Arguments

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A man saying, 'if not x, then why?' as his computer replies, 'Can we do something easier? This is hurting my diodes'.

Undoubtedly you've had an argument with someone at some point in your life - parents, spouse, law enforcement...

Would you believe there is a method to arguing? This entry explains some of the theory behind arguments, and it may help you the next time you have a disagreement with your sister over whose turn it is to use the computer.

Introduction to Arguments

In logic1, an argument consists of a set of statements. The first statements are called the 'premises', while the last statement is called the 'conclusion'. The conclusion is usually the statement that you want somebody to accept, which is why you're arguing in the first place.

Here's an example of an argument:

All birds have wings.
A cuckoo is a bird.
Therefore, a cuckoo has wings.

As you can see, this argument has premises ('All birds have wings' and 'A cuckoo is a bird'), and a conclusion ('A cuckoo has wings'). In this particular argument, the premises force the conclusion. Anyone who believes the premises must also believe the conclusion. This is therefore called a 'valid' argument, or it could be said that the conclusion follows 'validly' from the premises.

In the above example, both premises are true, so the conclusion must be true as well. By 'true', we mean that they are statements that accurately reflect reality. (What reality actually is, and how it is possible to reflect it in statements, are questions we will leave for the philosophers.)

It is possible for an argument to be valid, yet for the premises and conclusion to be false. For example:

All Japanese people keep pet aardvarks.
Tony Blair is Japanese.
Therefore, Tony Blair keeps pet aardvarks.

Both the premises and the conclusion are demonstrably false, and yet the argument is valid. Anyone who believed both premises would also have to believe the conclusion. This shows that validity is a feature of the form of the argument, and has nothing to do with its content.

When an argument is valid, and the premises are true, then the conclusion must also be true. Then the argument is called sound, or cogent, and whoever you are arguing with is forced to agree that you are right, or else to resort to violence.

Valid Arguments

Since validity has to do with the form of an argument, it is possible to identify valid forms, and some of these have been studied by logicians, and have been given names in Latin.

Valid Argument Form Number One - Modus Ponens

Modus Ponens ('proposing mode') is the most common form of valid argument. The cuckoo argument and the Tony Blair argument above are both examples of Modus Ponens. A generalized Modus Ponens argument looks like this:

All A are B.
x is A
Therefore, x is B.

This is what Modus Ponens looks like with certain kinds of statements, namely those involving quantifiers. (A quantifier is a word like 'all', 'some' or 'none'.) Modus Ponens arguments can also be constructed with conditional statements, also called 'if/then' statements:

If a Barghest yowls, then Lascadua sells out.
A Barghest yowls.
Therefore, Lascadua sells out.

We might not know anything about Barghests or about Lascadua, but we must admit that the argument is perfectly valid, and that anyone who believes the premises must also believe the conclusion.

Valid Argument Form Number Two - Modus Tollens

All h2g2 Researchers are debonair.
Idi Amin is not debonair.
Therefore, Idi Amin is not an h2g2 Researcher.

... because if Dr Amin were an h2g2 Researcher, he would be debonair, by the first premise, but according to the second premise, he is not debonair, so he cannot be an h2g2 Researcher! This form is called Modus Tollens, (removing mode), and it is the second type of valid argument. It is more commonly called the 'Law of Indirect Reasoning'.

Modus Tollens also works perfectly well with conditional statements:

If Marvin is cheerful, then it is a cold day on Venus.
It is not a cold day on Venus.
Therefore, Marvin is not cheerful.

Invalid Arguments

Now, take a look at a different argument:

Some vertebrates are warm-blooded.
Frogs are vertebrates.
Therefore, frogs are warm-blooded.

This argument is invalid. Both of the premises are true, but the conclusion is false. In a valid argument, the conclusion is never false when the premises are true. This particular argument is invalid because Modus Ponens does not work with the quantifier 'some', only with the quantifier 'all'.

Invalid arguments are also called fallacies. Let us look at some very common forms of fallacy:

Invalid Argument Form Number One - Affirming the Consequent

If x, then y.
Therefore, x.

This fallacy is a common logical mistake, sometimes called 'abductive reasoning' (as opposed to deductive reasoning, of which Modus Ponens is an example). It may seem at first glance to make sense. It is invalid because the conclusion does not follow from the premises, even if it happens to be a true statement. It is very easy to mix this form up with the Modus Ponens form. Here's a concrete example:

If I am with the one I love, then I am happy.
I am happy.
Therefore, I am with the one I love.

This argument doesn't work, because the one you love could be far away, and you could be happy for some other reason, perhaps having to do with food, money, or central nervous system stimulants. Here is a very well-known example of abductive reasoning:

Fire causes smoke
There is smoke
Therefore, there is fire

As tempting as it may be to accept this conclusion, we only know that fire is one cause of smoke. There may also be other causes, so the presence of smoke does not necessitate that of fire. The statements in this argument are not explicitly phrased as conditional statements, or as quantified statements, but the rules of reasoning still apply. 'Fire causes smoke' could be rephrased as 'If there is fire, then there is smoke'.

Invalid Argument Form Number Two - Denying the Antecedent

If x, then y.
Not x.
Therefore, not y.

This is similar to Affirming the Consequent, except that it takes on a negative form. Here's an example:

All dogs have four legs.
Francis the Talking Mule is not a dog
Therefore, Francis the Talking Mule does not have four legs

Even though both of the premises are true, the conclusion is false, because dogs are not the only animals that have four legs.

Ad Hominem and Authority

Ad homimen (meaning 'towards the person') arguments and arguments based on authority are very similar fallacies. Ad hominem arguments are very common in politics, and authority-based arguments are very common in religion2.

In an ad hominem argument, a statement is said to be wrong because the person making that statement is foolish, or biased, or has been wrong before. This is a fallacy because even a foolish, biased, often wrong person can make correct statements.

Mr X says that Y is true.
But Mr X also said that Z was true, and was proved wrong.
Therefore, Y is also untrue.

Citing authority is like a positive version of ad hominem. An argument based on authority is one in which a statement is said to be true, because the person who made the statement is smart, or inspired, or usually right. This is a fallacy because everybody can be wrong, sometimes.

Some claim that arguments based on human authority are fallacious, but that arguments based on divine authority are not. This claim is not a logical one, but a theological one, and therefore beyond the scope of this entry, thank God.

Circular Reasoning

One type of fallacy that is very common with longer, more complicated arguments is called the circular reasoning fallacy. Circular reasoning is when the conclusion is, itself, used as one of the premises of the argument. The conclusion then follows quite easily, but nothing has really been proven.

The classic example of circular reasoning runs something like this:

'This scripture is the inspired word of God.'
'How do you know?'
'Because it says so right here, in this scripture.'
'Why should I believe what it says there?'
'Because this scripture is the inspired word of God...'

One Last Invalid Argument

Here is another argument. The premises are both (arguably) true, the form seems valid, the argument isn't circular, and yet the conclusion seems false!

Nothing is better than eternal bliss.
A peanut butter sandwich is better than nothing.
Therefore, a peanut butter sandwich is better than eternal bliss.

The fallacy in this argument is left as an exercise, for the Researcher to find, with the recommendation that the Researcher brush up on his or her knowledge of paradoxes.


Perhaps you've learned something new about arguing from this entry, and the next time you get in an argument, you will put this to good use. Remember: true premises + valid argument = true conclusion. If that doesn't seem to work, it might be wise to back up your arguing skills with a good working knowledge of martial arts.

1Logic, the science of making sense, lives somewhere in between philosophy and mathematics. It was systematised among the ancients by Aristotle, and incorporated into modern mathematics by George Boole, as Boolean Algebra.2They say you shouldn't bring up politics or religion in polite conversation...

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