All About Arguments

How to Win

By Dr. Kim Johnson with The Lukeion Project

Of all the topics and ideas we cover in logic, the argument is certainly the star. Students may dream of outarguing their opponents in the courtroom, or winning an important political debate, or even proving their parents wrong about curfew, homework, or allowance.  Unfortunately, because of the nature of logical arguments, showing that someone has reached the wrong conclusion is difficult.

What makes an argument?

When logicians talk about “argument,” they don’t mean the typical immature argument between siblings (“Did too!” -- “Did not!”). The technical definition of a logical argument is “a set of premises which imply a conclusion.” That means that each argument has at least two statements, one of which is a conclusion.

Seems easy enough, right? Unfortunately, even figuring out which is the premise, and which is the conclusion, can be difficult. Consider the argument in the Declaration of Independence. Is the conclusion, “All men are created equal”? Or is it that King George had established tyranny? Or is it that it was necessary for “one people to dissolve the political bands which have connected them with another”? To discuss an argument, you need to know what is being argued.

To add insult to injury, sometimes parts of the argument are left unstated including even the conclusion. Arguments with unstated propositions are called enthymemes, from the Greek for “to keep in mind.”  Advertisements and politics are some of the worst offenders: “Open a Coke, Open Happiness.” Why?  Because Coke (according to the advertising company) is happiness. “My opponent hates cats. You would never vote for someone who hates cats.” Conclusion: Vote for me!

When faced with an enthymeme, your duty is to interpret the argument in the most favorable light for your opponent. The first reason for this is simple: treat others the way you would like to be treated.   The other reason is practical.  Responding to a command to “Surrender!” by saying, “You wish to surrender?  Very well!” is silly and soon no one will want to discuss anything with you anymore.

What makes a good argument?

Once you have found an argument, we can analyze its validity. In Lively Logician 1, we use the principles of categorical logic developed by Aristotle to analyze syllogisms, a specific form of argument with 3 terms, 2 premises and a conclusion. The logician’s favorite syllogism is as follows:

All men are mortal.

Socrates is a man.

Therefore Socrates is mortal.

In Lively Logician 2, we branch out into truth tables and use the laws of inference and replacement to judge whether an argument is valid. If each of the steps in the argument is made with valid principles, then the entire argument is valid.

Validity means that if the premises are true, then the conclusion must be true. The tricky thing is that validity refers only to the form of the argument, and not to the truth of the conclusion. Here is a valid argument (of the same form as above) with a conclusion that is obviously false:

All jellybeans are made of sugar.

All dogs are jellybeans.

Therefore all dogs are made of sugar.

What’s the difference between the argument with the false conclusion and the original one above? In the second argument, the minor premise (“All dogs are jellybeans”) is clearly false---so the conclusion is not guaranteed to be true.

Not only can valid arguments have false conclusions, but having a true conclusion and true premises doesn’t mean that the argument is valid. Here’s another syllogism:

All canines are mammals.

All dogs are mammals.

Therefore all dogs are canines.

The conclusion and premises are all true---but the argument is bad. If you’re trying to prove that “All dogs are canines,” you might want to get a different argument.  

Saying an argument is valid is a bit like showing that the structure of a house is solid and will stand. It makes no claims about the decorations on the house or the painting job or the light bulbs. Having a valid argument just means that if the premises are true, the conclusion is necessarily true as well. As we’ve seen, the implication doesn’t go the other way. A sound argument is much better.

Sound arguments start with true premises. This is not a requirement for valid arguments.  The argument in a sound argument is also valid. Thus, the conclusion of a sound argument is guaranteed to be true. If your opponent’s argument is sound, there’s not much you can do to counter it.

How can you prove someone is wrong?

This brings us to the goal: Show that your opponent (or parent, or friend) is wrong. Unfortunately, formal logic can only go so far with this goal. We can find fallacies in their arguments: ad hominem attacks, or circular reasoning, for example. Logic doesn’t have many tools to show that someone’s conclusion is false. They might need a better argument, but their conclusion could still be true.

The best way to vanquish your opponent is to create your own, sound argument including true premises and a valid argument plus a conclusion that is the opposite of what your opponent has stated. The best way to win arguments with friends and parents is not to prove them wrong, but to make your own, better argument.