There's a lot of debate on the net. Unfortunately, much of it is of very low quality. The aim of this document is to explain the basics of logical reasoning, and hopefully improve the overall quality of debate.
The Concise Oxford English Dictionary defines logic as "the science of reasoning, proof, thinking, or inference". Logic will let you analyze an argument or a piece of reasoning, and work out whether it is likely to be correct or not. You don't need to know logic to argue, of course; but if you know even a little, you'll find it easier to spot invalid arguments.
There are many kinds of logic, such as fuzzy logic and constructive logic; they have different rules, and different strengths and weaknesses. This document discusses simple Boolean logic, because it's commonplace and relatively easy to understand. When people talk about something being 'logical', they usually mean the type of logic described here.
What Logic Isn't
It's worth mentioning a couple of things which logic is not.
Firstly, logical reasoning is not an absolute law which governs the universe. Many times in the past, people have concluded that because something is logically impossible (given the science of the day), it must be impossible, period. It was also believed at one time that Euclidean geometry was a universal law; it is, after all, logically consistent. Again, we now know that the rules of Euclidean geometry are not universal.
Secondly, logic is not a set of rules which govern human behavior. Humans may have logically conflicting goals. For example:
John wishes to speak to whoever is in charge.
The person in charge is Steve.
Therefore John wishes to speak to Steve.
Unfortunately, John may have a conflicting goal of avoiding Steve, meaning that the reasoned answer may be inapplicable to real life.
This document only explains how to use logic; you must decide whether logic is the right tool for the job. There are other ways to communicate, discuss and debate.
Arguments
An argument is, to quote the Monty Python sketch, "a connected series of statements to establish a definite proposition".
Many types of argument exist; we will discuss the deductive argument. Deductive arguments are generally viewed as the most precise and the most persuasive; they provide conclusive proof of their conclusion, and are either valid or invalid.
Deductive arguments have three stages: premises, inference, and conclusion. However, before we can consider those stages in detail, we must discuss the building blocks of a deductive argument: propositions.
Propositions
A proposition is a statement which is either true or false. The proposition is the meaning of the statement, not the precise arrangement of words used to convey that meaning.
For example, "There exists an even prime number greater than two" is a proposition. (A false one, in this case.) "An even prime number greater than two exists" is the same proposition, re-worded.
Unfortunately, it's very easy to unintentionally change the meaning of a statement by rephrasing it. It's generally safer to consider the wording of a proposition as significant.
It's possible to use formal linguistics to analyze and re-phrase a statement without changing its meaning; but how to do so is outside the scope of this document.
Premises
A deductive argument always requires a number of core assumptions. These are called premises, and are the assumptions the argument is built on; or to look at it another way, the reasons for accepting the argument. Premises are only premises in the context of a particular argument; they might be conclusions in other arguments, for example.
You should always state the premises of the argument explicitly; this is the principle of audiatur et altera pars. Failing to state your assumptions is often viewed as suspicious, and will likely reduce the acceptance of your argument.
The premises of an argument are often introduced with words such as "Assume...", "Since...", "Obviously..." and "Because...". It's a good idea to get your opponent to agree with the premises of your argument before proceeding any further.
The word "obviously" is also often viewed with suspicion. It occasionally gets used to persuade people to accept false statements, rather than admit that they don't understand why something is 'obvious'. So don't be afraid to question statements which people tell you are 'obvious' — when you've heard the explanation you can always say something like "You're right, now that I think about it that way, it is obvious."
Inference
Once the premises have been agreed, the argument proceeds via a step-by-step process called inference.
In inference, you start with one or more propositions which have been accepted; you then use those propositions to arrive at a new proposition. If the inference is valid, that proposition should also be accepted. You can use the new proposition for inference later on.
So initially, you can only infer things from the premises of the argument. But as the argument proceeds, the number of statements available for inference increases.
There are various kinds of valid inference - and also some invalid kinds, which we'll look at later in this document. Inference steps are often identified by phrases like "therefore..." or "...implies that..."
Conclusion
Hopefully you will arrive at a proposition which is the conclusion of the argument - the result you are trying to prove. The conclusion is the result of the final step of inference. It's only a conclusion in the context of a particular argument; it could be a premise or assumption in another argument.
The conclusion is said to be affirmed on the basis of the premises, and the inference from them. This is a subtle point which deserves further explanation.
Implication In Detail
Clearly you can build a valid argument from true premises, and arrive at a true conclusion. You can also build a valid argument from false premises, and arrive at a false conclusion.
The tricky part is that you can start with false premises, proceed via valid inference, and reach a true conclusion. For example:
Conclusion: Therefore sea otters live in the ocean
There's one thing you can't do, though: start from true premises, proceed via valid deductive inference, and reach a false conclusion.
We can summarize these results as a "truth table" for implication. The symbol "=>" denotes implication; "A" is the premise, "B" the conclusion. "T" and "F" represent true and false respectively.
Here's an example of an argument which is valid, and which may or may not be sound:
Premise: Every event has a cause
Premise: The universe has a beginning
Premise: All beginnings involve an event
Inference: This implies that the beginning of the universe involved an event
Inference: Therefore the beginning of the universe had a cause
Conclusion: The universe had a cause
The proposition in line 4 is inferred from lines 2 and 3. Line 1 is then used, with the proposition derived in line 4, to infer a new proposition in line 5. The result of the inference in line 5 is then re-stated (in slightly simplified form) as the conclusion.
Spotting arguments
Spotting an argument is harder than spotting premises or a conclusion. Lots of people shower their writing with assertions, without ever producing anything you might reasonably call an argument.
Sometimes arguments don't follow the pattern described above. For example, people may state their conclusions first, and then justify them afterwards. This is valid, but it can be a little confusing.
To make the situation worse, some statements look like arguments but aren't. For example:
"If the Bible is accurate, Jesus must either have been insane, an evil liar, or the Son of God."
That's not an argument; it's a conditional statement. It doesn't state the premises necessary to support its conclusion, and even if you add those assertions it suffers from a number of other flaws which are described in more detail in the Atheist Arguments document.
An argument is also not the same as an explanation. Suppose that you are trying to argue that Albert Einstein believed in God, and say:
"Einstein made his famous statement 'God does not play dice' because of his belief in God."
That may look like a relevant argument, but it's not; it's an explanation of Einstein's statement. To see this, remember that a statement of the form "X because Y" can be re-phrased as an equivalent statement, of the form "Y therefore X". Doing so gives us:
"Einstein believed in God, therefore he made his famous statement 'God does not play dice'."
Now it's clear that the statement, which looked like an argument, is actually assuming the result which it is supposed to be proving, in order to explain the Einstein quote.
Furthermore, Einstein did not believe in a personal God concerned with human affairs — again, see the Atheist Arguments document.
Further Reading
We've outlined the structure of a sound deductive argument, from premises to conclusion. But ultimately, the conclusion of a valid logical argument is only as compelling as the premises you started from. Logic in itself doesn't solve the problem of verifying the basic assertions which support arguments; for that, we need some other tool.
The dominant means of verifying basic assertions is scientific enquiry. However, the philosophy of science and the scientific method are huge topics which are quite beyond the scope of this document.
For a more comprehensive introduction to logic, try Flew's "Thinking Straight", listed in the Atheist Media document. A much more detailed book is Copi's "Introduction to Logic". The Electronic Resources document also lists LOGIC-L, a LISTSERV mailing list devoted to discussing the teaching of elementary logic.
Fallacies
There are a number of common pitfalls to avoid when constructing a deductive argument; they're known as fallacies. In everyday English, we refer to many kinds of mistaken beliefs as fallacies; but in logic, the term has a more specific meaning: a fallacy is a technical flaw which makes an argument unsound or invalid.
(Note that you can criticize more than just the soundness of an argument. Arguments are almost always presented with some specific purpose in mind — and the intent of the argument may also be worthy of criticism.)
Arguments which contain fallacies are described as fallacious. They often appear valid and convincing; sometimes only close inspection reveals the logical flaw.
Below is a list of some common fallacies, and also some rhetorical devices often used in debate. The list isn't intended to be exhaustive; the hope is that if you learn to recognize some of the more common fallacies, you'll be able to avoid being fooled by them.
"If the universe had been created by a supernatural being, we would see order and organization everywhere. And we do see order, not randomness — so it's clear that the universe had a creator."
"For thousands of years Christians have believed in Jesus Christ. Christianity must be true, to have persisted so long even in the face of persecution."
"The vast majority of people in this country believe that capital punishment has a noticeable deterrent effect. To suggest that it doesn't in the face of so much evidence is ridiculous."
"All I'm saying is that thousands of people believe in pyramid power, so there must be something to it."
This line of argument isn't always completely bogus; for example, it may be relevant to refer to a widely-regarded authority in a particular field, if you're discussing that subject. For example, we can distinguish quite clearly between:
"Hawking has concluded that black holes give off radiation"
and
"Penrose has concluded that it is impossible to build an intelligent computer"
Hawking is a physicist, and so we can reasonably expect his opinions on black hole radiation to be informed. Penrose is a mathematician, so it is questionable whether he is well-qualified to speak on the subject of machine intelligence.
The question presupposes a definite answer to another question which has not even been asked. This trick is often used by lawyers in cross-examination, when they ask questions like:
"Where did you hide the money you stole?"
Similarly, politicians often ask loaded questions such as:
"How long will this EU interference in our affairs be allowed to continue?"
or
"Does the Chancellor plan two more years of ruinous privatization?"
Another form of this fallacy is to ask for an explanation of something which is untrue or not yet established.
This fallacy is similar to Post Hoc Ergo Propter Hoc. The fallacy is to assert that because two events occur together, they must be causally related. It's a fallacy because it ignores other factors that may be the cause(s) of the events.
"Literacy rates have steadily declined since the advent of television. Clearly television viewing impedes learning."
Note that this fallacy is different from Non Causa Pro Causa. That has the form "A implies B, A is false, therefore B is false", where A does not in fact imply B at all. Here, the problem isn't that the implication is invalid; rather it's that the falseness of A doesn't allow us to deduce anything about B.
"If the God of the Bible appeared to me, personally, that would certainly prove that Christianity was true. But God has never appeared to me, so the Bible must be a work of fiction."
Published: 06/20/1999