“The total lack of rationality and competence in the White House and the inability of half of the US population to acquire and understand information are far larger threats to Americans than terrorism.”

Paul Craig Roberts


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Constructing a Logical Argument

Introduction
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.

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.

Basic concepts
The building blocks of a logical argument are propositions, also called statements. A proposition is a statement which is either true or false; for example:

“The first programmable computer was built in Cambridge.”
“Dogs cannot see colour.”
“Berlin is the capital of Germany.”

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. 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.

What is an argument?
An argument is, to quote the Monty Python sketch, “a connected series of statements to establish a definite proposition”. There are three stages to an argument: Premises, inference, and conclusion.

Stage one: Premises
One or more propositions will be are necessary for the argument to continue. They must be stated explicitly. They are called the premises of the argument. They are the evidence (or reasons) for accepting the argument and its conclusions.
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.)

Stage two: Inference
The premises of the argument are used to obtain further propositions. This process is known as inference. In inference, we start with one or more propositions which have been accepted. We then derive a new proposition. There are various forms of valid inference.
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”.

Stage three: Conclusion
Finally, we arrive at the conclusion of the argument, another proposition. The conclusion is often stated as the final stage of inference. It 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.

Types of argument
There are two traditional types of argument, deductive and inductive. A deductive argument provides conclusive proof of its conclusions; if the premises are true, the conclusion must also be true. A deductive argument is either valid or invalid.
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:

• Every event has a cause (premise)
• The universe has a beginning (premise)
• All beginnings involve an event (premise)
• This implies that the beginning of the universe involved an event (inference)
• Therefore the universe has a cause (inference and conclusion)

Note that the conclusion of one argument might be a premise in another argument. A proposition can only be called a premise or a conclusion with respect to a particular argument; the terms do not make sense in isolation.

Recognizing an argument
Sometimes an argument will not follow the order described above. For instance, the conclusions might be stated first, and the premises stated afterwards in support of the conclusion. This is perfectly valid, if sometimes a little confusing. 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.

For example:

“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.

Another example:

“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.

Implication in detail
There is one very important thing to remember: The fact that a deductive argument is valid does not imply that its conclusion holds. This is because of the slightly counter-intuitive nature of implication, which we must now consider more carefully.
Obviously a valid argument can consist of true propositions. However, an argument may be entirely valid even if it contains only false propositions.

For example:

• All insects have wings (premise)
• Woodlice are insects (premise)
• Therefore woodlice have wings (conclusion)

Here, the conclusion is not true because the argument’s premises are false. If the argument’s premises were true, however, the conclusion would be true. The argument is thus entirely valid.

More subtly, we can reach a true conclusion from one or more false premises, as in:

• All fish live in the sea (premise)
• Dolphins are fish (premise)
• Therefore dolphins live in the sea (conclusion)

However, the one thing we cannot do is reach a false conclusion through valid inference from true premises.

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
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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.

Fallacies
To delve further into the structure of logical arguments would require lengthy discussion of linguistics and philosophy. It is simpler and probably more useful to summarize the major pitfalls to be avoided when constructing an argument. These pitfalls are known as fallacies.

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.

Argumentum ad baculum / Appeal to force
The Appeal to Force is committed when the arguer resorts to force or the threat of force in order to try and push the acceptance of a conclusion. It is often used by politicians, and can be summarized as “might makes right”. The force threatened need not be a direct threat from the arguer.

For example:

“… Thus there is ample proof of the truth of the Bible. All those who refuse to accept that truth will burn in Hell.”

Argumentum ad hominem
Argumentum ad Hominem is literally “argument directed at the man”.

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.

For example:

“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”.

Argumentum ad ignorantiam
Argumentum ad ignorantiam means “argument from ignorance”. This fallacy occurs whenever it is argued that something must be true simply because it has not been proved false. Or, equivalently, when it is argued that something must be false because it has not been proved true. (Note that this is not the same as assuming that something is false until it has been proved true, a basic scientific principle.)

Examples:

“Of course the Bible is true. Nobody can prove otherwise.”

“Of course telepathy and other psychic phenomena do not exist. Nobody has shown any proof that they are real.”

Note that this fallacy does not apply in a court of law, where one is generally assumed innocent until proven guilty.

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.

For example:

“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

Argumentum ad misericordiam
This is the Appeal to Pity, also known as Special Pleading. The fallacy is committed when the arguer 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. To commit this fallacy is to 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:

“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?”

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.

Argumentum ad verecundiam
The Appeal to Authority uses the admiration of the famous to try and win support for an assertion. For example:

“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.

The fallacy of accident
The Fallacy of Accident is committed when a general rule is applied to a particular case whose “accidental” circumstances mean that the rule is inapplicable. It is the error made when one goes from the general to the specific. For example:

“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.

Converse accident / Hasty generalization
This fallacy is the reverse of the Fallacy of Accident. It occurs when one forms a general rule by examining only a few specific cases which are not representative of all possible cases. For example:

“Jim Bakker was an insincere Christian. Therefore all Christians are insincere.”

Sweeping generalization / Dicto simpliciter
A sweeping generalization occurs when a general rule is applied to a particular situation in which the features of that particular situation render the rule inapplicable. A sweeping generalization is the opposite of a hasty generalization.

Non causa pro causa / Post hoc ergo propter hoc
These are known as False Cause fallacies.

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.”

Cum hoc ergo propter hoc
This fallacy is similar to Post Hoc Ergo Propter Hoc. It asserts that because two events occur together, they must be causally related, and leaves no room for 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.

Circulus in demonstrando
This fallacy occurs when one assumes as a premise the conclusion which one wishes to reach. Often, the proposition will be rephrased so that the fallacy appears to be a valid argument. For example:

“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.”

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 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.

Ignoratio elenchi
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 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.

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 will always be freely redistributable.”

Amphiboly
Amphiboly occurs when the premises used in an argument are ambiguous because of careless or ungrammatical phrasing.

Accent
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”

and

“We should not speak ill of our friends”

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 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.”

“A is based on B” fallacies / “…is a type of…” fallacies / Fallacy of the Undistributed Middle
These fallacies occur when one attempts to argue that things are in some way similar without actually specifying 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.

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. The latter 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 is not that the implication is invalid; rather it is that the falseness of A does not allow us to deduce anything about B.

Converting a conditional
This fallacy is an argument of the form “If A then B, therefore if B then A”.

Argumentum ad antiquitatem
This is the fallacy of asserting that something is right or good simply because it is old, or because “that’s the way it’s always been.”

Argumentum ad novitatem
This is the opposite of the Argumentum ad Antiquitatem; it is 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.

Argumentum ad lazarum
The fallacy of assuming that because someone is poor he or she is sounder or more virtuous than one who is 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 the more often it is heard. An “argumentum ad nauseam” is one that employs constant repetition in asserting something.

Bifurcation
Also referred to as the “black and white” fallacy, bifurcation occurs when one presents a situation as having only two alternatives, where in fact other alternatives exist or can exist.

Plurium interrogationum / Many questions
This fallacy occurs when a questioner demands a simple answer to a complex question.

Non sequitur
A non-sequitur is an argument where the conclusion is drawn from premises which are not logically connected with it.

Red herring
This fallacy is committed when irrelevant material is introduced to the issue being discussed, so that everyone’s attention is diverted away from the points being 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 making an assertion or proposition. 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 being made. The source of the fallacy is the assumption that something is true unless proven otherwise.

Straw man
The straw man fallacy is to misrepresent someone else’s position so that it can be attacked more easily, then to knock down that misrepresented position, then to conclude that the original position has been demolished. It is a fallacy because it fails to deal with the actual arguments that have been made.

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!”

Tu quoque
This is the famous “you too” fallacy. It occurs when an action is argued to be acceptable because the other party 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 do not bother to state. The principle of Audiatur et Altera Pars is that all of the premises of an argument should be stated explicitly. It is not strictly a fallacy to fail to state all of one’s assumptions; however, it is often viewed with suspicion.

Ad hoc
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 is an explanation.

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.”

Argumentum ad logicam
This is the “fallacy fallacy” of arguing that a proposition is false merely on the grounds that it has been presented as the conclusion of a fallacious argument. Remember always that fallacious arguments can arrive at true conclusions.