1. A first definition of logic
Logic is the study of the structure of arguments.
To understand this definition you must first understand what the word argument means as it used in logic. The word argument as it is used in logic does not mean anything like verbal dispute. The word as it is used in logic means something technical. What exactly does it mean? Let’s take things step-by-step and find out.
2. Statements
To understand what an argument is (as the word is used in logic—a qualifier which will be implicit from now on), you must first understand what a statement is. So let’s talk about statements and then we’ll get back to arguments.
There’s not much to it: a statement is just a sentence that claims something. Here are some statements:
(A) 1 + 1 = 2
(B) The Earth is flat
(C) If it is the case that if we do not seize the initiative then we will lose it, and this will be bad for us unless we do not need it, then if we need it we had better try our best to seize it, unless we either want things to be bad for us or don’t care.
You might wonder whether all sentences are statements. Are there sentences that don’t claim anything? Sure. Here are a few:
(D) Ouch!
(E) What time is it?
(F) Come with me if you want to live.
3. Truth values
Most statements are either true or false in the exclusive sense of or—they are one or the other, but not both.
The way to say this in philosophical lingo is that most statements have a definite truth-value. There are two truth-values in standard logic: true and false. True statements have a truth-value of true, and false statements have a truth-value of false. What is the point of ever talking this way? Talking this way can make some things clearer and easier to say.
Note that a statement can have a truth-value even if we don’t know what it is. For instance, even though we don’t know whether or not there is life in other galaxies, the statement that there is life in other galaxies certainly either is true or false (and not both).
Digression 1: I said above that most statements have a definite truth-value. If you are wondering, there are paradoxical statements that have either both or neither truth-value depending on how one looks at it. If you spend some time considering which truth-value the statement “This statement is false” has, you will see the point.
Digression 2: Some logicians reserve the term statement for sentences that have definite truth-values, and use the term declarative sentence to designate the broader category of sentences that claim something. I’m not doing that in this article.
4. Arguments
Now that we have discussed what statements are, you should be in a position to understand what an argument is. Here you go:
In logic, an argument is a set of statements, consisting of one or more premises and one conclusion, where the premises are intended jointly to support the conclusion.
That’s really all there is to it, although in practice people generally also allow into arguments an indicator (like the word therefore) that tells us which statement is the conclusion.
Here is an example of an argument:
All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
In this case, the premises are “All men are mortal” and “Socrates is a man.” The conclusion is “Socrates is mortal.” We know which is which because of the word therefore: it tells us that the statements preceding it are intended to jointly entail the truth of the statement following it.
Often, arguments are presented more formally and the premises and conclusion may be labeled for easier reference. The above argument, for instance, might be rendered:
(P1) All men are mortal.
(P2) Socrates is a man.
Therefore,
(C) Socrates is mortal.
4.1. More examples
Here are a few other examples of arguments:
We can have an idea only of that which we directly experience. The only things that we directly experience are the contents of our own minds. If matter exists, then it is not one of the contents of our minds. Therefore, we cannot even have an idea of matter.
Either we have free will or we do not. If the laws of physics are true, then we do not have free will. If we are morally responsible for our actions, then we must have free will. Hence, either the laws of physics are false or we are not morally responsible for our actions.
Bad things happen to good people as much as to bad people. The best explanation for this is that reality at its most fundamental is indifferent to justice. So, reality at its most fundamental probably is indifferent to justice.
4.2. The conclusion doesn’t necessarily come last.
Sometimes the premises and conclusion of an argument are presented informally out of order but that’s not a problem as long as we know which is which. For instance, whether you say
It’s cloudy outside. And if it’s cloudy outside, then it’s going to rain. Therefore, it’s going to rain.
or
It’s going to rain. You see, if it’s cloudy outside, then it’s going to rain. And it is indeed cloudy outside.
you are making the same argument, namely the following:
(P1) If it is cloudy outside, then it is going to rain.
(P2) It is cloudy outside.
Therefore,
(C) It is going to rain.
5. Revisiting the definition of logic
5.1. An analogy
In the first section, I defined logic as the study of the structure of arguments. Let me elaborate on that a little bit. Let’s start with an extended analogy:
Suppose you are trying to solve an arithmetic problem with a calculator. You want to be sure of at least two things: (1) that you press the keys you are supposed to press, (2) that your calculator works properly, in the sense that if you press the keys you are supposed to press, then your calculator is guaranteed to give the correct solution to your problem. If you either press the wrong keys or your calculator does not work properly, then you might still get the correct answer, but only by pure chance, and that’s not what you want.
Now, we can imagine two different professions, one of which specializes in checking whether you have pressed the right keys, and the other of which specializes in checking whether your calculator works properly. A person in the key-checking profession doesn’t really care whether your calculator works properly: checking that is not his job. Likewise, a person in the calculator-checking profession doesn’t really care whether you have pressed the right keys: checking that is not his job. All the calculator-checking person cares about is whether your calculator functions in such a way that if you had pressed the keys you were supposed to press—whether or not you actually did—then your calculator would have been guaranteed to give the right answer.
5.2. The parallel in logic
Here is the parallel in logic to the above analogy:
If you are trying to argue for something, you want to be sure of at least two things: first, that the premises of your argument are true, and second, that your argument is structured properly, in the sense that if your premises are true, then they do entail the truth (or probable truth, if that is all you are after) of your conclusion. If you start out with false premises or your argument is structured incorrectly, then it will be purely a matter of chance whether your conclusion turns out to be true (or probably true), and that’s not what you want.
There are all sorts of fields that specialize in checking whether your premises are true, but only one field specializes in checking whether your argument is structured correctly: logic.
Logicians don’t worry about whether the premises of your argument are true. All they are concerned with is whether your argument is structured such that if the premises of your argument were true, then they would have entailed the truth of your conclusion (or, again, its probable truth if that is all you are after).
6. Deductive and inductive arguments
A given argument can be classified as deductive or inductive, depending on the intent of the person who has made the argument. If the person’s intent is that the truth of the premises alone should guarantee the truth of the conclusion, then the argument is a deductive argument. If the person’s intent is that the truth of the premises alone would support the truth of the conclusion without guaranteeing it, then the argument is an inductive argument.
Deductive logic studies the structure of deductive arguments. Inductive logic studies the structure of inductive arguments.
6.1. Validity and soundness
A properly structured deductive argument—an argument the truth of whose premises alone (whether or not they actually are true) would guarantee the truth of the conclusion—is called a valid argument. An improperly structured deductive argument—an argument the truth of whose premises alone would not guarantee the truth of the conclusion—is called an invalid argument
A valid argument with true premises is called a sound argument, but remember that logic examines only the structure of arguments, so logic asks only whether a given argument is valid, not whether it is sound.
There is no universally accepted technical term for properly structured inductive arguments or for properly structured inductive arguments that also have true premises, though some texts use the words strong and cogent for these, respectively. The key things that you need to understand about analyzing arguments are merely the following: (1) all it takes for an argument to be a bad argument is for one of the premises to be false or for the argument to be improperly structured; (2) whether or not an argument is properly structured has nothing to do with whether or not the premises or the conclusion actually are true.