Logic is the science of reasoning, proof, thinking, or inference [Concise OED]. Logic will let you analyze an argument or a piece of reasoning, and work out whether it is correct or not. To use the technical terms, logic lets you work out whether the reasoning is valid or invalid.
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 about logical arguments, though, they usually mean the type being described here.
One problem with boolean logic is that people don't have to be consistent in their goals and desires. People use fuzzy logic and non-logical reasoning to handle their conflicting goals; boolean logic isn't good enough. For example:
"John wishes to speak to the person in charge. The person in charge is Steve. Therefore John wishes to speak to Steve."
Logically, that's a totally valid argument. However, John may have a conflicting goal of avoiding Steve, meaning that the answer obtained by logical reasoning may be inapplicable to real life. Garlic tastes good, strawberry ice cream tastes good, but strawberry garlic ice cream is only logically a good idea.
Sometimes, principles of valid reasoning which were thought to be universal have turned out to be false. For example, for a long time the principles of Euclidean geometry were thought to be universal laws.
However, keeping those caveats and limitations in mind, let's go on to consider the basics of boolean logic.
"The first Holden car was built in 1948."
"Ginger cats are always male."
"Canberra is the capital of Australia."
Propositions may be either asserted (said to be true) or denied (said to be false).
Note: This is a technical meaning of the word "deny", not the everyday meaning.
When a proposition has been asserted based on some argument, we usually say that it has been affirmed.
The proposition is generally viewed as the meaning of the statement, and not the particular arrangement of words used. So "An even prime number greater than two exists" and "There exists an even prime number greater than two" both express the same (false) proposition.
Sometimes, however, it is better to consider the wording of the proposition as significant, and use linguistic rules to derive equivalent statements if necessary.
You can think of the premises as the reasons for accepting the argument, or the evidence it's built on. Premises are often indicated by phrases such as "because", "since", "let's assume", and so on.
In inference, you start with one or more propositions which have been accepted. You then use those propositions to arrive at a new proposition. The new proposition can, of course, be used in later stages of inference.
There are various kinds of valid inference -- and also some invalid kinds, but we'll get to those later. Inference is often denoted by phrases such as "implies that" or "therefore".
The conclusion is often stated as the final stage of inference. The conclusion is affirmed on the basis the original premises, and the inference from them. Conclusions are often indicated by phrases such as "therefore", "it follows that", "we conclude" and so on.
Note that the phrase "obviously" is often viewed with suspicion, as it gets used to intimidate people into accepting things which aren't true at all. 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.
1. A deductive argument is one which provides conclusive proof of its conclusions. It is either valid or invalid. A valid deductive argument is defined as one where if the premises are true, then the conclusion must also be true.
2. 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 likely their premises are.
There are forms of argument in ordinary language which are neither deductive nor inductive. However, we'll concentrate 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. Lots of people shower their writing with assertions, without ever producing anything you might reasonably call an argument.
To make the situation worse, some statements look like arguments but are not. For example: "If the Bible is accurate, Jesus must either have been insane, an evil liar, or the Son of God."
The statement above isn't an argument; it's a conditional statement. It doesn't assert the premises which are needed to support what looks like its conclusion. (Even if you add those assertions, it still suffers from a number of other logical flaws.)
Here's another example:
"God created you; therefore obey and worship God."
The phrase "obey and worship God" is neither true nor false. Therefore it isn't a proposition, and the sentence isn't an argument.
Causality is important as well. Suppose we're trying to argue that there's something wrong with the engine of a car. Let's look at two statements of the form "A because B". Here's the first:
"The car won't start because there's something wrong with the engine."
That's not an argument for there being something wrong with the engine; it's an explanation of why the car won't start. We're explaining A, using B as the explanation.
Now consider a second statement:
"There must be something wrong with the engine of the car, because it won't start."
Here we're arguing for A, giving B as evidence. The statement "A because B" is an argument.
The difference between the two cases might not be completely clear. So, remember that "A because B" is equivalent to "B therefore A". The two statements then become:
"There's something wrong with the engine, therefore the car won't start."
"The car won't start, therefore there's something wrong with the engine."
We're supposed to be arguing that there's something wrong with the engine, but now it's obvious that the first statement doesn't do that at all. Only the second statement is arguing that there's something wrong with the engine.
The fact that a deductive argument is valid doesn't necessarily mean that its conclusion holds.
That may seem confusing, but it's because of the slightly counter-intuitive nature of how implication works.
Obviously you can build a valid argument out of true propositions. But you can also build a completely valid argument using only false propositions. For example:
More subtly, you can reach a true conclusion from false premises -- even ludicrously false ones:
Look at the following truth table:
Ultimately, the conclusion of a valid logical argument is only as compelling as the basic premises it is derived from. Logic in itself does not solve the problem of verifying the basic assertions which support arguments. The only way to verify basic assertions is by scientific enquiry.
So for the purposes of this discussion, a fallacy is a logical argument which looks correct, but which can be seen to be incorrect when examined more carefully. If fallacies are recognized they can be pointed out as being fallacious and will therefore be less likely to mislead people.
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.
Many of the examples below are commonly found, though some have been rephrased for the sake of clarity.
"Violent crime is on the increase because you hear a lot more about it on the news these days."
It's quite valid to use personal experience to illustrate a point; but such anecdotes don't really 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.
"If you don't turn to Jesus Christ, you'll 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?"
If you argue against some assertion by attacking the person who made the assertion, then you have committed the abusive form of argumentum ad hominem. A personal attack isn't a valid argument, because the truth of an assertion doesn't depend on the virtues of the person asserting it. For example:
"No intelligent person could believe in Creation."
Sometimes in a court of law doubt is cast on the testimony of a witness. For example, the prosecution might show that the witness 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 doesn't demonstrate that the witness's testimony is false.
If you argue that someone should accept the truth of an assertion because of that person's particular circumstances, then you have committed the circumstantial form of argumentum ad hominem. 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. The fallacy can also be used as a means of rejecting a particular 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 you allege that someone is rationalizing a conclusion for selfish reasons, is also known as "poisoning the well".
"Of course telepathy and other psychic phenomena do not exist. Nobody has shown any proof that they are real."
Note that this fallacy doesn't apply in a court of law, where you're generally assumed innocent until proven guilty.
Argumentum ad misericordiam
This is the Appeal to Pity, also known as Special Pleading. The fallacy is committed when someone appeals to pity for the sake of getting a conclusion accepted. For example:
"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."
Argumentum ad populum
This is known as Appealing to the Gallery, or Appealing to the People. You commit this fallacy if you attempt to win acceptance of an assertion by appealing to a large group of people. This form of fallacy is often characterized by emotive language. For example:
"If we allow religion in schools all our children will get brain- washed."
Argumentum ad numerum
This fallacy is closely related to the argumentum ad populum. It consists of asserting that the more people who support or believe a proposition, the more likely it is that that proposition is correct. For example:
"The vast majority of people in this country believe that capital punishment has a noticable 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."
Argumentum ad verecundiam
The Appeal to Authority uses admiration of a famous person to try and win support for an assertion. For example:
"Bultmann didn't believe in a physical resurrection of Christ"
This line of argument isn't always completely bogus; 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"
"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.
Argumentum ad antiquitatem
This is the fallacy of asserting that something is right or good simply because it's old, or because "that's the way it's always been." The opposite of Argumentum ad Novitatem. "This interpretation has been accepted for hundreds of years. It must be correct."
Argumentum ad novitatem
This is the opposite of the Argumentum ad Antiquitatem; it's the fallacy of asserting that something is more correct simply because it is new, or newer than something else.
Argumentum ad crumenam
The fallacy of believing that money is a criterion of correctness; that those with more money are more likely to be right. The opposite of Argumentum ad Lazarum.
Argumentum ad lazarum
The fallacy of assuming that someone poor is sounder or more virtuous than someone who's wealthier. This fallacy is the opposite of the Argumentum ad Crumenam.
Argumentum ad nauseam
This is the incorrect belief that an assertion is more likely to be true, or is more likely to be accepted as true, the more often it is heard. So an Argumentum ad Nauseam is one that employs constant repetition in asserting something; saying the same thing over and over again until you're sick of hearing it.
This is a common technique used by preachers (usually with a very shaky argument!).
The fallacy of accident / Sweeping generalization / Dicto simpliciter
A sweeping generalization occurs when a general rule is applied to a particular situation, but the features of that particular situation mean the rule is inapplicable. It's the error made when you go from the general to the specific. For example:
"Most aborigines have been in trouble with the law. You are an aborigine so you must have been in trouble with the law as well."
This fallacy is often committed by people who try to decide moral and legal questions by mechanically applying general rules.
Converse accident / Hasty generalization
This fallacy is the reverse of the Fallacy of Accident. It occurs when you form a general rule by examining only a few specific cases which aren't representative of all possible cases. For example:
"Jim Bakker was an insincere, immoral Christian. Therefore all Christians are insincere."
Non causa pro causa
The fallacy of Non Causa Pro Causa occurs when something is identified 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."
This is known as a false cause fallacy.
Post hoc ergo propter hoc
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 that event. This is another type of false cause fallacy.
Cum hoc ergo propter hoc
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.
Petitio principii / Begging the question
This fallacy occurs when the premises are at least as questionable as the conclusion reached. For example:
"Aliens are abducting innocent victims every week. The government must know what is going on. Therefore the government is in league with the aliens."
Circulus in demonstrando
This fallacy occurs if you assume as a premise the conclusion which you wish to reach. Often, the proposition is rephrased so that the fallacy appears to be a valid argument. For example:
"Real scientists are evolutionists. Evolutionists propose scientific theories of evolution. Therefore real scientists are evolutionists."
Note that the argument is entirely circular. 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.
Complex question / Fallacy of interrogation / Fallacy of presupposition
This is the interrogative form of Begging the Question. One example is the classic loaded question: "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 Prime Minister 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.
Ignoratio elenchi / Irrelevant conclusion
The fallacy of Irrelevant Conclusion consists of claiming that an argument supports a particular conclusion when it is actually logically nothing to do with that conclusion.
For example, a Bhuddist may begin by saying that he will argue that the teachings of the Bhuddahs are undoubtably true. If he then argues at length that Bhuddism is of great help to many people, no matter how well he argues he will not have shown that Bhuddist 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.
Equivocation / Fallacy of four terms
Equivocation occurs when a key word is used with two or more different meanings in the same argument. For example:
"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 it 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.
Amphiboly occurs when the premises used in an argument are ambiguous because of careless or ungrammatical phrasing.
Accent is another form of fallacy through shifting meaning. In this case, the meaning is changed by altering which parts of a statement are emphasized.
For example, consider:
"We should not speak *ill* of our friends"
"We should not speak ill of our *friends*"
Be particularly wary of this fallacy in written communication, where it's easy to mis-read the emphasis of what's written.
Fallacies of composition
One Fallacy of Composition is to conclude that a property shared by the parts of something must apply to the whole. For example:
"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."
Fallacy of division
The fallacy of division is the opposite of the Fallacy of Composition. Like its opposite, it exists in two varieties. The first is to assume that a property of some thing must apply to its parts. For example:
"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."
The slippery slope argument
This argument states that should one event occur, so will other harmful events. There is no proof made that the harmful events are caused by the first event. For example:
"If we allow people to sing anything other than hymns in church then we'll start having rock bands, then heavy-metal bands and then the whole place will turn into disco."
"A is based on B" fallacies / "...is a type of..." fallacies / Fallacy of the Undistributed Middle
These fallacies occur if you attempt to argue that things are in some way similar, but you don't actually specify in what way they are similar. Examples:
"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?"
Affirmation of the consequent
This fallacy is an argument of the form "A implies B, B is true, therefore A is true". To understand why it is a fallacy, examine the truth table for implication given earlier. Here's an example:
"If I fall into the swimming pool, I get wet. I am wet, so I must have fallen into the swimming pool."
This is the converse of Denial of the Antecedent.
Denial of the antecedent
This fallacy is an argument of the form "A implies B, A is false, therefore B is false". The truth table for implication makes it clear why this is a fallacy.
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 I fall into the swimming pool, I get wet. I did not fall into the swimming pool, therefore I am not wet."
This is the converse of the fallacy of Affirmation of the Consequent.
Converting a conditional
This fallacy is an argument of the form "If A then B, therefore if B then A".
"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."
"If it's raining outside and I don't have an umbrella I get wet. So if I get wet, then it's raining outside and I don't have an umbrella."
This fallacy is similar to the Affirmation of the Consequent, but phrased as a conditional statement.
Also referred to as the "black and white" fallacy, bifurcation occurs if you present a situation as having only two alternatives, where in fact other alternatives exist or can exist.
Plurium interrogationum / Many questions
This fallacy occurs when someone demands a simple (or simplistic) answer to a complex question.
A non sequitur is an argument where the conclusion is drawn from premises which aren't logically connected with it. For example:
"Since Egyptians did so much excavation to construct the pyramids, they were well versed in paleontology."
This fallacy is committed when someone introduces irrelevant material to the issue being discussed, so that everyone else's attention is diverted away from the points made, towards a different conclusion.
Reification / Hypostatization
Reification occurs when an abstract concept is treated as a concrete thing.
Shifting the burden of proof
The burden of proof is always on the person asserting something. Shifting the burden of proof, a special case of Argumentum ad Ignorantiam, is the fallacy of putting the burden of proof on the person who denies or questions the assertion. The source of the fallacy is the assumption that something is true unless proven otherwise.
"OK, so if you don't think aliens have gained control of the US government, can you prove it?"
It should be noted though, that if no "proof" can be offered for a particular assertion, that in no way implies the assertion is not true. There is a danger in any form of "positivism" which holds that ONLY propositions that can be proved beyond doubt should be accepted as true.
The straw man fallacy is when you misrepresent someone else's position so that it can be attacked more easily, then knock down that misrepresented position, then conclude that the original position has been demolished. It's a fallacy because it fails to deal with the actual arguments that have been made.
"Textual criticism is wrong because it is inconsistent. Aleph and B are accepted as the best manuscripts available but they disagree so often in the gospels."
The above is straw man argument because the person that uses this kind of argument does not really understand Textual Criticism. Textual decisions are very complex and are not simply a matter of manuscript preference. Aleph and B disgree in the gospels because B is Alexandrian whereas Aleph is Western in the gospels and Alexandrian in the rest of the NT.
The extended analogy
The fallacy of the Extended Analogy often occurs when some suggested general rule is being argued over. The fallacy is to assume that mentioning two different situations, in an argument about a general rule, constitutes a claim that those situations are analogous to each other.
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!"
This is the famous "you too" fallacy. It occurs if you argue that an action is acceptable because your opponent has performed it. For instance:
"You're just being randomly abusive."
"So? You've been abusive too."
This is a personal attack, and is therefore a special case of Argumentum ad Hominem.
Audiatur et altera pars
Often, people will argue from assumptions which they don't bother to state. The principle of Audiatur et Altera Pars is that all of the premises of an argument should be stated explicitly. It's not strictly a fallacy to fail to state all of your assumptions; however, it's often viewed with suspicion.
For example, many scholars reject the doctrine of inspiration and inerrancy. This will of course dramatically affect their reasoning and conclusions.
There is a difference between argument and explanation. If we're interested in establishing A, and B is offered as evidence, the statement "A because B" is an argument. If we're trying to establish the truth of B, then "A because B" is not an argument, it's an explanation. 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."
Argumentum ad logicam
This is the "fallacy fallacy" of arguing that a proposition is false because it has been presented as the conclusion of a fallacious argument. Remember always that fallacious arguments can arrive at true conclusions.
"Take the fraction 16/64. Now, cancelling a 6 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?"
The "No True Scotsman..." fallacy
Suppose I assert that no Scotsman puts sugar on his porridge. You counter this by pointing out that your friend Angus likes sugar with his porridge. I then say "Ah, yes, but no true Scotsman puts sugar on his porridge. 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.