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.
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:
The person in charge is Steve.
Therefore John wishes to speak to Steve.
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.
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.
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.
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."
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..."
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.
The tricky part is that you can start with false premises, proceed via valid inference, and reach a true conclusion. For example:
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.
| Premise | Conclusion | Inference |
|---|---|---|
| A | B | A => B |
| false | false | true |
| false | true | true |
| true | false | false |
| true | true | true |
If an argument is valid, and in addition it started from true premises, then it is called a sound argument. A sound argument must arrive at a true conclusion.
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
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.
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".
(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.
The Nizkor Project has another excellent list of logical fallacies; so do the reference works mentioned above.
Sadly, many of the examples below have been taken directly from Usenet,
though some have been rephrased for the sake of clarity.
"We should not speak ill of our friends"and
"We should not speak ill of our friends"Be particularly wary of this fallacy on the net, where it's easy to misread the emphasis of what's written.
The Ad Hoc fallacy is to give an after-the-fact explanation which doesn't apply to other situations. Often this ad hoc explanation will be dressed up to look like an argument. For example, if we assume that God treats all people equally, then the following is an ad hoc explanation:
"I was healed from cancer.""Praise the Lord, then. He is your healer."
"So, will He heal others who have cancer?"
"Er... The ways of God are mysterious."
"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."This is the converse of Denial of the Antecedent.
"Premise: Belief in God fills a much-needed gap."
"There's abundant proof that God exists and is still performing miracles today. Just last week I read about a girl who was dying of cancer. Her whole family went to church and prayed for her, and she was cured."It's quite valid to use personal experience to illustrate a point; but such anecdotes don't actually prove anything to anyone. Your friend may say he met Elvis in the supermarket, but those who haven't had the same experience will require more than your friend's anecdotal evidence to convince them.
Anecdotal evidence can seem very compelling, especially if the audience wants to believe it. This is part of the explanation for urban legends; stories which are verifiably false have been known to circulate as anecdotes for years.
"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."
"... Thus there is ample proof of the truth of the Bible. All those who refuse to accept that truth will burn in Hell.""... In any case, I know your phone number and I know where you live. Have I mentioned I am licensed to carry concealed weapons?"
"Microsoft software is undoubtedly superior; why else would Bill Gates have got so rich?"
The first is the abusive form. If you refuse to accept a statement, and justify your refusal by criticizing the person who made the statement, then you are guilty of abusive argumentum ad hominem. For example:
"You claim that atheists can be moral -- yet I happen to know that you abandoned your wife and children."This is a fallacy because the truth of an assertion doesn't depend on the virtues of the person asserting it. A less blatant argumentum ad hominem is to reject a proposition based on the fact that it was also asserted by some other easily criticized person. For example:
"Therefore we should close down the church? Hitler and Stalin would have agreed with you."A second form of argumentum ad hominem is to try and persuade someone to accept a statement you make, by referring to that person's particular circumstances. For example:
"Therefore it is perfectly acceptable to kill animals for food. I hope you won't argue otherwise, given that you're quite happy to wear leather shoes."This is known as circumstantial argumentum ad hominem. The fallacy can also be used as an excuse to reject a particular conclusion. For example:
"Of course you'd argue that positive discrimination is a bad thing. You're white."This particular form of Argumentum ad Hominem, when you allege that someone is rationalizing a conclusion for selfish reasons, is also known as "poisoning the well".
It's not always invalid to refer to the circumstances of an individual who is making a claim. If someone is a known perjurer or liar, that fact will reduce their credibility as a witness. It won't, however, prove that their testimony is false in this case. It also won't alter the soundness of any logical arguments they may make.
(Note that this isn't the same as assuming something is false until it has been proved true. In law, for example, you're generally assumed innocent until proven guilty.)
Here are a couple of examples:
"Of course the Bible is true. Nobody can prove otherwise."In scientific investigation, if it is known that an event would produce certain evidence of its having occurred, the absence of such evidence can validly be used to infer that the event didn't occur. It does not prove it with certainty, however."Of course telepathy and other psychic phenomena do not exist. Nobody has shown any proof that they are real."
For example:
"A flood as described in the Bible would require an enormous volume of water to be present on the earth. The earth doesn't have a tenth as much water, even if we count that which is frozen into ice at the poles. Therefore no such flood occurred."It is, of course, possible that some unknown process occurred to remove the water. Good science would then demand a plausible testable theory to explain how it vanished.
Of course, the history of science is full of logically valid bad predictions. In 1893, the Royal Academy of Science were convinced by Sir Robert Ball that communication with the planet Mars was a physical impossibility, because it would require a flag as large as Ireland, which it would be impossible to wave.
"Monks are more likely to possess insight into the meaning of life, as they have given up the distractions of wealth."
"Take the fraction 16/64. Now, cancelling a six on top and a six on the bottom, we get that 16/64 = 1/4.""Wait a second! You can't just cancel the six!"
"Oh, so you're telling us 16/64 is not equal to 1/4, are you?"
"I did not murder my mother and father with an axe! Please don't find me guilty; I'm suffering enough through being an orphan."
On Usenet, your argument is often less likely to be heard if you repeat it over and over again, as people will tend to put you in their kill files.
"BeOS is a far better choice of operating system than OpenStep, as it has a much newer design."
"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."
"Pornography must be banned. It is violence against women.""For thousands of years people have believed in Jesus and the Bible. This belief has had a great impact on their lives. What more evidence do you need that Jesus was the Son of God? Are you trying to tell those people that they are all mistaken fools?"
"Isaac Newton was a genius and he believed in God."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.
"Either man was created, as the Bible tells us, or he evolved from inanimate chemicals by pure random chance, as scientists tell us. The latter is incredibly unlikely, so..."
"Homosexuals must not be allowed to hold government office. Hence any government official who is revealed to be a homosexual will lose his job. Therefore homosexuals will do anything to hide their secret, and will be open to blackmail. Therefore homosexuals cannot be allowed to hold government office."Note that the argument is entirely circular; the premise is the same as the conclusion. An argument like the above has actually been cited as the reason for the British Secret Services' official ban on homosexual employees. Another example is the classic:
"We know that God exists because the Bible tells us so. And we know that the Bible is true because it is the word of God."Circular arguments are surprisingly common, unfortunately. If you've already reached a particular conclusion once, it's easy to accidentally make it an assertion when explaining your reasoning to someone else.
"Have you stopped beating your wife?"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.
"The bicycle is made entirely of low mass components, and is therefore very lightweight.""A car uses less gasoline and causes less pollution than a bus. Therefore cars are less environmentally damaging than buses."
"Jim Bakker was an insincere Christian. Therefore all Christians are insincere."
"If educational standards are lowered, the quality of argument seen on the Internet worsens. So if we see the level of debate on the net get worse over the next few years, we'll know that our educational standards are still falling."This fallacy is similar to the Affirmation of the Consequent, but phrased as a conditional statement.
"Literacy rates have steadily declined since the advent of television. Clearly television viewing impedes learning."This fallacy is a special case of the more general non causa pro causa.
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."This is the converse of the fallacy of Affirmation of the Consequent.
"Christians generally dislike atheists. You are a Christian, so you must dislike atheists."This fallacy is often committed by people who try to decide moral and legal questions by mechanically applying general rules.
"You are studying at a rich college. Therefore you must be rich.""Ants can destroy a tree. Therefore this ant can destroy a tree."
"What could be more affordable than free software? But to make sure that it remains free, that users can do what they like with it, we must place a license on it to make sure that will always be freely redistributable."One way to avoid this fallacy is to choose your terminology carefully before beginning the argument, and avoid words like "free" which have many meanings.
Here's real example from an online debate about anti-cryptography legislation:
"I believe it is always wrong to oppose the law by breaking it.""Such a position is odious: it implies that you would not have supported Martin Luther King."
"Are you saying that cryptography legislation is as important as the struggle for Black liberation? How dare you!"
For example, a Christian may begin by saying that he will argue that the teachings of Christianity are undoubtedly true. If he then argues at length that Christianity is of great help to many people, no matter how well he argues he will not have shown that Christian teachings are true.
Sadly, these kinds of irrelevant arguments are often successful, because they make people to view the supposed conclusion in a more favorable light.
"The natural world is characterized by competition; animals struggle against each other for ownership of limited natural resources. Capitalism, the competitive struggle for ownership of capital, is simply an inevitable part of human nature. It's how the natural world works."Another form of appeal to nature is to argue that because human beings are products of the natural world, we must mimic behavior seen in the natural world, and that to do otherwise is 'unnatural':
"Of course homosexuality is unnatural. When's the last time you saw two animals of the same sex mating?"Robert Anton Wilson deals with this form of fallacy at length in his book "Natural Law". A recent example of "Appeal to Nature" taken to extremes is The Unabomber Manifesto.
This is an example of an ad hoc change being used to shore up an assertion, combined with an attempt to shift the meaning of the words used original assertion; you might call it a combination of fallacies.
"I took an aspirin and prayed to God, and my headache disappeared. So God cured me of the headache."This is known as a false cause fallacy. Two specific forms of non causa pro causa fallacy are the cum hoc ergo propter hoc and post hoc ergo propter hoc fallacies.
"Since Egyptians did so much excavation to construct the pyramids, they were well versed in paleontology."(Non sequiturs are an important ingredient in a lot of humor. They're still fallacies, though.)
"Aliens are abducting innocent victims every week. The government must know what is going on. Therefore the government is in league with the aliens."
"Are higher taxes an impediment to business or not? Yes or no?"
"The Soviet Union collapsed after instituting state atheism. Therefore we must avoid atheism for the same reasons."This is another type of false cause fallacy.
"You may claim that the death penalty is an ineffective deterrent against crime -- but what about the victims of crime? How do you think surviving family members feel when they see the man who murdered their son kept in prison at their expense? Is it right that they should pay for their son's murderer to be fed and housed?"
"I noticed you described him as 'evil'. Where does this 'evil' exist within the brain? You can't show it to me, so I claim it doesn't exist, and no man is 'evil'."
For further discussion of this idea, see the "Introduction to Atheism" document.
"OK, so if you don't think the grey aliens have gained control of the US government, can you prove it?"
"If we legalize marijuana, then more people would start to take crack and heroin, and we'd have to legalize those too. Before long we'd have a nation full of drug-addicts on welfare. Therefore we cannot legalize marijuana."
"To be an atheist, you have to believe with absolute certainty that there is no God. In order to convince yourself with absolute certainty, you must examine all the Universe and all the places where God could possibly be. Since you obviously haven't, your position is indefensible."The above straw man argument appears at about once a week on the net. If you can't see what's wrong with it, read the "Introduction to Atheism" document.
"You're just being randomly abusive."This is a personal attack, and is therefore a special case of Argumentum ad Hominem."So? You've been abusive too."
"Isn't history based upon faith? If so, then isn't the Bible also a form of history?""Islam is based on faith, Christianity is based on faith, so isn't Islam a form of Christianity?"
"Cats are a form of animal based on carbon chemistry, dogs are a form of animal based on carbon chemistry, so isn't a dog a form of a cat?"
NOTE from Skeptic X: History isn't based upon faith -- we recognize the error potentials -- but the logic fails even if we accept that premise.