Logic is the science of reasoning, proof, thinking, or inference [Concise OED]. Logic allows us to analyze a piece of reasoning, and determine whether it is correct or not. To use the technical terms, we determine whether the reasoning is valid or invalid.
One does not need to study logic in order to reason correctly. However, a little basic knowledge of logic is often helpful when constructing or analyzing an argument.
Note that I am not claiming that logic is universally applicable. That issue is very much open to debate. This document only explains how to use logic; you must decide whether logic is the right tool for the job.
Note also that this document deals only with simple boolean logic. Other sorts of mathematical logic, such as fuzzy logic, obey different rules. When people talk of logical arguments, though, they generally mean the type being described here.
"The first programmable computer was built in Cambridge."Propositions may be either asserted (said to be true) or denied (said to be false). Note that this is a technical meaning of "deny", not the everyday meaning.
"Dogs cannot see colour."
"Berlin is the capital of Germany."
The proposition is the meaning of the statement, not the particular arrangement of words used. So "A God exists" and "There exists a God" both express the same proposition.
Premises (or assertions) are often indicated by phrases such as "because", "since", "obviously" and so on.
(The phrase "obviously" is often viewed with suspicion, as it can be used to intimidate others into accepting dubious premises. If something doesn't seem obvious to you, don't be afraid to question it. You can always say "Oh, yes, you're right, it is obvious" when you've heard the explanation.)
The propositions arrived at by inference may then be used in further inference. Inference is often denoted by phrases such as "implies that" or "therefore".
A valid argument is defined as one where if the premises are true, then the conclusion is true.
An inductive argument is one where the premises provide some evidence for the truth of the conclusion. Inductive arguments are not valid or invalid, but we can talk about whether they are better or worse than other arguments. We can also discuss how probable their premises are.
There are forms of argument in ordinary language which are neither deductive nor inductive. However, this document concentrates on deductive arguments, as they are often viewed as the most rigorous and convincing.
Here is an example of a deductive argument:
Arguments are harder to recognize than premises or conclusions. Many people shower their writing with assertions without ever producing anything which one might reasonably describe as an argument. Some statements look like arguments, but are not.
"If the Bible is accurate, Jesus must either have been insane, an evil liar, or the Son of God."The above is not an argument, it is a conditional statement. It does not assert the premises which are necessary to support what appears to be its conclusion. (Even if we add the assertions, it still suffers from a number of other logical flaws -- see the section on this argument in "Alt.Atheism Frequently Asked Questions".)
"God created you; therefore do your duty to God."The phrase "do your duty to God" is neither true nor false. Therefore it is not a proposition, and the sentence is not an argument.
Causality is important. Suppose we are trying to argue that there is something wrong with the engine of a car. Consider two statements of the form "A because B". The first statement:
"My car will not start because there is something wrong with the engine."The statement is not an argument for there being something wrong with the engine; it is an explanation of why the car will not start. We are explaining A, using B as the explanation. We cannot argue from A to B using a statement of the form "A because B".
However, we can argue from B to A using such a statement. Consider:
"There must be something wrong with the engine of my car, because it will not start."Here we are arguing for A, offering B as evidence. The statement "A because B" is then an argument.
To make the difference clear, note that "A because B" is equivalent to "B therefore A". The two statements then become:
"There is something wrong with the engine, therefore my car will not start."And:
"My car will not start, therefore there is something wrong with the engine."If we remember that we are supposed to be arguing that there is something wrong with the engine, it is clear that only the second statement is a valid argument.
Obviously a valid argument can consist of true propositions. However, an argument may be entirely valid even if it contains only false propositions.
More subtly, we can reach a true conclusion from one or more false premises, as in:
We can therefore draw up 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 ---------------------------- F F T F T T-- If the premises are false and the inference valid, the conclusion can be true or false.
T F F-- If the premises are true and the conclusion false, the inference must be invalid.
T T T-- If the premises are true and the inference valid, the conclusion must be true.
A sound argument is a valid argument whose premises are true. A sound argument therefore arrives at a true conclusion. Be careful not to confuse sound arguments with valid arguments.
Of course, we can criticize more than the mere soundness of an argument. In everyday life, arguments are almost always presented with some specific purpose in mind. As well as criticizing the argument itself, one can criticize the apparent intent of the argument. Such criticism is outside the scope of this document, however!
In everyday English the term "fallacy" is used to refer to mistaken beliefs as well as to the faulty reasoning that leads to those beliefs. This is fair enough, but in logic the term is generally used to refer to a form of technically incorrect argument, especially if the argument appears valid or convincing.
So for the purposes of this discussion, we define a fallacy as a logical argument which appears to be correct, but which can be seen to be incorrect when examined more closely. By studying fallacies we aim to avoid being misled by them.
Below is a list of some common fallacies, and also some rhetorical devices often used in debate. The list is not intended to be exhaustive.
"... Thus there is ample proof of the truth of the Bible. All those who refuse to accept that truth will burn in Hell."
The Abusive variety of Argumentum ad Hominem occurs when, instead of trying to disprove the truth of an assertion, the arguer attacks the person or people making the assertion. This is invalid because the truth of an assertion does not depend upon the goodness of those asserting it.
"Atheism is an evil philosophy. It is practised by Communists and murderers."Sometimes in a court of law doubt is cast upon the testimony of a witness by showing, for example, that he is a known perjurer. This is a valid way of reducing the credibility of the testimony given by the witness, and not Argumentum ad Hominem; however, it does not demonstrate that the witness's testimony is false. To conclude otherwise is to fall victim of the Argumentum ad Ignorantiam.
The circumstantial form of Argumentum ad Hominem is committed when a person argues that his opponent ought to accept the truth of an assertion because of the opponent's particular circumstances. For example:
"It is perfectly acceptable to kill animals for food. How can you argue otherwise when you're quite happy to wear leather shoes?"This is an abusive charge of inconsistency, used as an excuse for dismissing the opponent's argument.
This fallacy can also be used as a means of rejecting a conclusion. For example:
"Of course you would argue that positive discrimination is a bad thing. You're white."This particular form of Argumentum ad Hominem, when one alleges that one's adversary is rationalizing a conclusion formed from selfish interests, is also known as "poisoning the well".
"Of course the Bible is true. Nobody can prove otherwise."Note that this fallacy does not apply in a court of law, where one is generally assumed innocent until proven guilty.
"Of course telepathy and other psychic phenomena do not exist. Nobody has shown any proof that they are real."
Also, 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 did not occur.
"A flood as described in the Bible would require an enormous volume of water to be present on the earth. The earth does not have a tenth as much water, even if we count that which is frozen into ice at the poles. Therefore no such flood occurred."In science, we can validly assume from lack of evidence that something has not occurred. We cannot conclude with certainty that it has not occurred, however. See also Shifting the Burden of Proof
"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."
"Pornography must be banned. It is violence against women."
"The Bible must be true. Millions of people know that it is. Are you trying to tell them that they are all mistaken fools?"
"Isaac Newton was a genius and he believed in God."This line of argument is not always completely bogus; for example, reference to an admitted authority in a particular field may be relevant to a discussion of 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.
"Christians generally dislike atheists. You are a Christian, so you must dislike atheists."This fallacy is often committed by moralists and legalists who try to decide every moral and legal question by mechanically applying general rules.
"Jim Bakker was an insincere Christian. Therefore all Christians are insincere."
The fallacy of Non Causa Pro Causa occurs when one identifies something as the cause of an event but it has not actually been shown to be the cause. For example:
"I took an aspirin and prayed to God, and my headache disappeared. So God cured me of the headache."The fallacy of Post Hoc Ergo Propter Hoc occurs when something is assumed to be the cause of an event merely because it happened before the event. For example:
"The Soviet Union collapsed after taking up atheism. Therefore we must avoid atheism for the same reasons."
"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."
"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 EC 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.
For example, a Christian may begin by saying that he will argue that the teachings of Christianity are undoubtably 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, such fallacious arguments are often successful because they arouse emotions which cause others to view the supposed conclusion in a more favourable light.
"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."
"We should not speak ill of our friends"and
"We should not speak ill of our friends"
"The bicycle is made entirely of low mass components, and is therefore very lightweight."The other Fallacy of Composition is to conclude that a property of a number of individual items is shared by a collection of those items. For example:
"A car uses less petrol and causes less pollution than a bus. Therefore cars are less environmentally damaging than buses."
"You are studying at a rich college. Therefore you must be rich."The other is to assume that a property of a collection of items is shared by each item. For example:
"Ants can destroy a tree. Therefore this ant can destroy a tree."
"If we legalize marijuana, then we would have to legalize crack and heroin and we'll have a nation full of drug-addicts on welfare. Therefore we cannot legalize marijuana."
"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 aren't dogs a form of cat?"
This fallacy is best explained using a real example from a 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!"
"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."
The Ad Hoc fallacy is to give an after-the-fact explanation which does not 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."