1.4 Entailment = Validity
Later that day you see the Sergeant talking at the water cooler with Murphy, another detective.
"I think all criminals are felons," the Sergeant says.
"Hey, I had a feeling Quinn was guilty," Murphy replies, "but why do you think that?"
"Because all criminals have broken the law," says the Sergeant, "and all felons have broken the law. So all criminals are felons."
Your job is to evaluate the Sergeant’s argument:
1. All criminals have broken the law.
2. All felons have broken the law.
3. All criminals are are felons.
If you understand why that argument is bad, then you understand the fundamental concept of logic: in logic we want to know when some information guarantees that something is true.
That concept is called deductive logical entailment, or just entailment for short.
Here’s the definition we’ll use a lot throughout this textbook:
A set of premises entails a conclusion just in case this condition is met:
Whenever the premises are true, then the conclusion is also true.
In other words: If those premises are true, the conclusion must be true.
Previously we said that we will develop a model or tool, called a logical system, in order to study reasoning. Now we can see one of the key aspects of reasoning we want our tool to model: logical entailment. We want our logical system to show when some information does or doesn’t entail a conclusion.
You can test whether you really understand a concept by seeing if you can put it in slightly different words.
Select all the sentences below that expresses entailment.
The concept of entailment is so common and important that there are many ways of expressing it in English. Make these sentences all say the same thing as: the premises entail the conclusion.
Saying an argument is valid might sound different, since it refers to the argument rather than the premises, but it’s still another way of saying the same thing: a valid argument is one whose premises entail the conclusion.
It's worth noting that the word "valid" has other meanings in English. For example, you might say someone has a "valid point", which means they are expressing a legitimate concern or issue. In this book, though, we won't use the word in any of those other ways. We'll just use it in the specific sense as another way of talking about entailment.
Now let's put your knowledge into action. Drag the sentences into an argument so that it's valid.