The above is the basic simplest definition and in fact is relative to some presumed standards of reasoning or notions of when arguments are valid or not.

Also, while everybody makes mistakes of reasoning every day and relies on invalid arguments many times, usually without noticing this, and makes mistakes in good faith, as it were - one believes that the reasoning or argument one used is good, while in fact it isn't, but at least at that time one was not aware of this - there are very many conscious fallacies, that are used to mislead people.

Sections:

1. Introduction: Unconscious and conscious fallacies
    A. Fallacies in general
    B. Fallacies on purpose

2. Some fallacies  
    A. Formal and informal
    B. Threefold taxonomy of fallacies
    C. Examples of fallacies

Ad hominem
Bandwagon
Genetic
Red Herring
False Authority

Equivocation
Straw man

Affirming the Consequent
From ignorance

Begging the question
Cum Hoc
Post Hoc
False Dilemma
Hasty Generalization
Sweeping Generalization
No True Scotsman
False Subjectivism
Tu Quoque

3. Three general problems with arguments

3.1. Only real science is really concerned with the real truth
3.2. Most men know very little about argueing - or anything else
3.3. The media have taken over argumentation, and whorified it

1. Introduction: Unconscious and conscious fallacies

Everybody makes mistakes of reasoning every day - or so it would seem - and most of these mistakes are neither conscious nor important, in that one is not aware of reasoning invalidly nor does it harm, because one does not act on the reasoning.

Such mistakes in reasoning do start to be of importance when they are believed and acted upon, for false beliefs when put into practice often lead to harm.

And next to this, there is - especially in the modern world of TV and advertisements - many fallacies that are propounded consciously, to deceive and mislead people into beliefs, choices or emotions that the people propounding the fallacies want others to have, and try to trick them into.

A. Fallacies in general

Fallacies were first systematically studied by Aristotle, and since then there have been made many lists of fallacies, for which see e.g. Fearnside & Holther, Hamblin or Edwards Ed.

Since one can make mistakes in reasoning in very many ways, and since often, especially with arguments in natural language, it is not quite clear what are the rules of reasoning and assumptions used, there is no complete agreement on fallacies nor on lists of fallacies.

Although this does not make the concept of a fallacy relative in a general way, à la "your truths are my fallacies", this does mean that in general, if one discusses a fallacy, and wants to do this rationally one should indicate the assumptions and rules of reasoning one does suppose, and why the argument is fallacious with reference to those premisses.

In practice - that is, in everyday, political or religious discussions, this is often not really practically feasible, and the best one can do is to come prepared to these, namely with some knowledge of widely known fallacies, so that one at least can recognize them.

For indeed, ever since Aristotle there are a number of fallacies that are well-known, often by agreed upon names for them, such as the ad hominem fallacy, that refers to the practice of attacking one's opponent rather than his arguments; that are clear breakings of standard deductive logic; and that need, at least for an informed audience, little other than a reference and an indication why it applies to the present case.

To this I shall turn below, when I list and discuss some fallacies, but here a further important point about fallacies must be first considered, since so far I have somewhat tacitly presumed that the parties in the argument are in fact concerned with finding the truth about something, and are not intentionally out to deceive each other. This premiss often is factually false.

B. Fallacies on purpose

In fact, not all fallacies are unconscious and indeed in the modern world there are whole industries, lifelong careers and important social institutions that live by them, perpetrate them consciously, and that could not do without them while continuing to be those particular industries, careers or institution. 

Advocates, politicians, religous leaders, rhetoricians of all kinds, journalists and the popular media often use and rely on arguments and techniques of reasoning and of presenting positions and arguments that they know are not valid - in the above sense of: the evidence they present and/or the premisses they articulate, do not logically entail the conclusions they like to establish - but that they use and rely on nevertheless because they (believe they) know that their audiences probably will be deceived, and deception is there aim.

As I indicated, the trade in fallacies on purpose is a big industry, and indeed it is likely that most prose ordinary average folks in the West read in their papers and hear on TV is intentionally fallacious in that those who produced those arguments - as in advertising, as in political speeches, as in religious arguments - were out to try to convince their audiences by whatever means, and fallacies, especially when carefully crafted or served with lots of trimmings (from halfnaked girls to loud music and popular persons smiling recommendations of the speaker's points), often work in many audiences, in the sense that the majority of the audience is taken in, as was the aim of the fallacious reasoning presented in the first place.

This is most blatant in advertisements, but is hardly less common, if somewhat less blatant, in politics, the popular media, and religion.

2. Some fallacies  

Now some fallacies will be considered, based on the assumption that arguments involve - logically speaking, at least - three kinds of items:

Premisses or assumptions, that serve as reasons or evidence for conclusions; rules of inference, that allow one to infer a certain kind of conclusion from certain kinds of premisses; and conclusions, that state the belief that the argument - using those premisses, relying on such and such inferential principles - is said to lead to, and claimed to establish

The assumption just articulated is logical, and outside logic books, where many premises and rules of inference tend to be tacit, and where in fact the belief that such and such is a good car, which is the belief that the company want its potential buyers to adopt, may mostly depend on the curves, blondness and beauty of the model draped over the hood (rather than some techtalk that is also offered).

This model - of premisses, rules of inference and conclusion - keeps applying when discussing fallacies, but it is important to see that in everyday life and argumentation it must used with some liberalism, e.g. allowing and seeing that the premisses in ads are often little better than the creation of positive emotions (about a model, about accompanying music, about a wellknown person mentioned or shown); that the rules of inference may be little better than mere association, suggestion, or blatant saying it is so, with or without repetition, beautiful models displaying their curves, pleasant music by wellknown bands, and the wellknown face and sympathetic smile of a media-celebrity; and that the conclusion aimed at may be little other than making the consumers of the argument feel better about something ("Our Great Candidate!"), or bad about something ("The Evil Opponents"), or indeed sometimes confusing them (as in political arguments).

Fallacies have been sorted in groups ever since Aristotle. Here are two broad groups.

A. Formal and informal

One way to classify fallacies is to sort them into two groups, namely the formal fallacies, that are fallacious because they involve some mistake in deductive logic, in the sense that the premisses offered do not deductively entail the conclusion they said (and possibly but not necessarily: believed) to imply, and the informal fallacies, that are fallacious for other reasons, such as having false premisses (while leading in a perfectly valid way to the conclusion, for example).

The distinction is useful, but not precise, and is probably best made in psychological terms. In these terms, arguments - premises, inferences, conclusions - are attempts to move people to (more) belief (acceptance, support, credence) in the conclusion of the argument than they had without it.

And in these terms, a formal fallacy can be pinpointed by showing that the premisses offered do not really, in formal deductive logic, validly imply the conclusion inferred from the premisses, whereas an informal fallacy rather leads to a conclusion that cannot be rationally supported in a wider sense.

The "wider sense" may involve a lot (what do the qualities of a car have to do with the qualities of the halfnaked beauties surrounding it?) but one fairly good sense that often applies is probabilistic: The conclusion offered by an informally fallacious argument is simply not as probable (likely, plausible, frequent, well supported, credible, based on good evidence, rational, certain) as the argument tries to make out it is.

Most fallacies seem to be informal in the above sense, which may be restated thus: An informal fallacy is an argument (or mere semblance or mockup thereof, as in advertisements) that makes a conclusion appear better founded than in fact it is, on known evidence.

B. Threefold taxonomy of fallacies

A widely accepted taxonomy of fallacies divides them into three groups

• Fallacies of relevance: Attempts to establish a conclusion on grounds that are not relevant to its truth (or probability).
• Fallacies of ambiguity: Attempts to establish a conclusion on grounds that involve misleading language.
• Fallacies of presumption: Attempts to establish a conclusion on grounds that are false.

Note that - again - these are not precise distinctions, and that some fallacies belong to several groups, and indeed some kinds of fallacious argumentation, such as used in advertising and the media, especially TV, tend to involve all three and as a matter of course.

C. Examples of fallacies

In this section I will present a choice of some fallacies using the above threefold taxonomy, by giving a brief informal description and examples, followed by an explanation in smaller letters

1. Some fallacies of relevance

Ad hominem: This replaces the argument for a conclusion by an argument for or against persons who have supported or contradicted the conclusion. ("Look who's (also) saying it!", "He would say so, wouldn't he?" etc.)

Note that this may be quite justified and true in some cases, but often it is not. And in any case an ad hominem argument does not address the truth or falsity of the conclusion, but the character of those who are said to hold it. (See Red Herring.)

Bandwagon: This replaces the argument for a conclusion by an argument that appeals (falsely or not) to wide (popular, scientific etc.) support for the conclusion. ("The silent majority approves of it; ten milion Frenchmen can't be wrong".)

This often succeeds and indeed may be argued for - see  Equivocation  - in the name of Democracy, but does not address the truth or falsity of the conclusion but only its (im)popularity in some group. (It may make sense - "many qualified scientific experts agree" - if one can rationally move from such support to the truth of the conclusion.)

Genetic: This replaces the argument for a conclusion by an argument that appeals to the (de)merits of its source. ("The pope said it; the nazis also wanted it; the professor can't be wrong, and he says it")

Like the previous two.

Red Herring: This replaces the argument for a conclusion by an argument that appeals to some other topic. ("So you must in fact be saying X, which clearly is false.")

This is the most often used fallacy of relevance, and indeed is changing the topic. It is especially confusing, in that in ordinary argumentation, what are the premisses is often not clear. (See also Straw man.)

False Authority: This replaces the argument for a conclusion by an argument that appeals to the authority of someone who also supports it, but whose authority does not derive from expertness about that kind of conclusions.

Appeals to authority may be justified if indeed the authorities are authorities as regards the conclusion argued about. However, if they are authorities in another subject than the conclusion belongs to - "Oprah Winfrey says quantum mechanics is baloney, so there!" - the argument is bogus.

2. Some fallacies of ambiguity

Equivocation: This argues for a conclusion by using the same words in two distinct senses.

This may be blatant but often it is subtle, in the sense that a word is used in the premisses in one sense, and in the conclusion in a slightly different sense. One way of achieving this is by accenting: A argues that X is possible, because he can imagine it; B replies that sure, he can imagine it is possible. (So mental conceivability gets altered to mere fantasy, by innuendo).

Straw man: This argues against a conclusion by using a restatement of the premisses (often an exaggeration of them): "What you're really saying is so and so, which is ridiculous.")

This is very common in ordinary discussions, in which the positions of opponents are often stated in exaggerated forms.

3. Some fallacies of presumption

Affirming the Consequent: This argues for the truth of a conclusion by insisting that it implies something that is true.  

This is a deductive fallacy for it simply is not true that from (P implies Q) is true and (Q) is true it follows that (P) is true, but it is not necessarily invalid if restated: from (P implies Q) is true and (Q) is true it follows that (P) is more probable than it was before knowing that (Q) is true. 

From ignorance: This argues for the truth of a conclusion by insisting that it is not known to be false. ("It is true that God exists, for everybody knows that it is false there is a proof that he does not.")

Note this fallacy often is from real ignorance, and many people argue that X is (probably so) because they are ignorant of evidence to the contrary (or incapable of understanding it). It also is popular in medical circles, and there runs like so: "What I don't know - or what is not recorded in reputable books of medicine - is not so (and therefore you must be imagining things)".

Begging the question: This argues for the truth of a conclusion by assuming it, possibly in other words.

Note this is somewhat subtle, in that the argument may be perfectly valid, for (P implies P) is valid as is (Q is equivalent to P, so Q implies P). The problem with it is that what really gets proved is not (P), as is pretended or believed, but (P implies P), which is a tautology that says nought.

Cum Hoc: This argues that A and B are causally related because they (often) occur together. ("Cum Hoc" = "Together with it")

This may be quite true, but need not be. Indeed, if stated in a considerably weaker form it is not a fallacy but a verity: That A and B often occur together is some evidence that they may be causally connected.

Post Hoc: This argues that A and B are causally related because they (often) occur Aafter another. ("Post Hoc" = "After it")

As in the previous fallacy, this may be quite true, but need not be: That B followed A repeatedly or often is some evidence that they may be causally connected.

False Dilemma: This argues for the truth of a conclusion by insisting that there are fewer possibilities than in fact there are. ("You are either for or against us").

This can take subtle forms, and one pertinent consideration is that most listings of possibilities should include a last category "other possibilities".

Hasty Generalization: This argues for the truth of a conclusion that all or most Xs are Ys from one or a few Xs that are Ys. ("Women want careers, for I am a woman who wants a career.")

This is very common in politics, and indeed often is perpetrated by speaking or writing without quantifying terms: Instead of saying things like "all", "most", or "over 65%" of Xs are Ys, it is simply said "Xs are Y" (as in "Women want careers"). See Equivocation.

Sweeping Generalization: This argues for the truth of a conclusion that this x that is an X also must be an Y since (most) Xs are Ys. ("John Stuart Mill did not know Greek when he was an English five year old, because English five year olds don't know Greek.")

As it were, the opposite of the previous fallacy. Here they are combined: "Osama is a Muslim and a terrorist, so Muslims are terrorists; since Muslims are terrorists, this 4-year old Muslim is a terrorist".

No True Scotsman: This argues for the falsity of a refutation by redefining the premisses. ("You say that all Scotsman drink. He is a Scotsman who doesn't drink, so you're mistaken." "No, I am not: he is evidently no true Scotsman, for all true Scotsman drink, as I have been saying all along.")

Often resorted to when people are refuted, and also called ad hoc. Note firstly that while it may be true that the premisses, when qualified, entail the conclusion, the perpetrator of the ad hoc line of reasoning still stands refuted as regards his original argument, which is what he started from. And note secondly that every premiss or conclusion can be qualified - but then the argument changed to a different one, and the qualification implies the unqualified one has been given up. 

False Subjectivism: This argues for the falsity of a factual conclusion by insisting that it is not factual but a mere matter of subjective opinion. ("That's just your opinion.")

This may be called The Postmodern Fallacy, or Postmodernism in short, for it is the favorite gambit of postmodernists: All statements - including all factual statements - are personal opinions only. The same who indulge in that trick (they use it to disqualify the ideas of those they disagree with as mere personal opinion, but usually do not at all qualify their own beliefs as such: These usually are  presented as Gospel) tend to indulge in a condescending sort of politeness: "It's just your opinion, of course - but hey: everybody is entitled to their opinion". (That is: you, me and the flat-earthers and postmodernists.)

Tu Quoque: This argues for a conclusion on the ground that others do it too.

A favorite with small children, and also with bank directors, arms traders, and torturers. And note also that "If I would not do it, someone else would do it" (an arms traders' favorite) seems to be equivalent with "If someone else would not do it, I would do it" - i.e. they would do it anyway.

3. Three general problems with arguments

Human beings style themselves the rational animal but most animals that are human do not care much for rationality, nor do they know much about it. This has always been so, in human history so far, but in this age this has become a serious problem, for reasons that are now briefly outlined.

3.1. Only real science is really concerned with the real truth

One general problem with arguments is that most people argue not because they are interested in finding out the truth (about something), but because they are interested in having their opinion prevail.

In fact, about the only place where people argue because they are interested in finding out the truth is in science, and even there it is far more common in the hard sciences (mathematics, physics, chemistry) than in the soft sciences (sociology, politicology, psychology).

This means that outside real science, argument is often not what it would seem and is claimed to be; often is not about the truth (even if it is claimed it is); and often proceeds on the "anything goes" principle: What defeats or convinces one's opponents ipso facto is a good argument.

Now, all of this so far is human-all-too-human, but here the other two general problems enter, of which the first is minor if also important:

3.2. Most men know very little about argueing - or anything else

The second problem is that while all men (and women and children) that can speak can argue and often do, very few of them have any or much special knowledge about argueing or logic, or about psychology (that enters because of questions related to what convinces people - which often is not logical validity), or indeed about what they are argueing about.

There is little one can do about this, except recognize and admit the problem and try to do something about it in one's own case - and indeed to realize that virtually all public argumentation that is not in real science is not concerned with finding the real truth about things, but with trying to make people believe or desire something, usually because this serves the interests of those who try to convince one, or of those they get paid by.

3.3. The media have taken over argumentation, and whorified it

The third problem - and this is a major and modern problem - is that in the modern world fallacious argumentation, especially on TV, advertisement and in political propaganda, has become a profesionnal trade and has become a major industry, a maker and breaker of opinions, and a very important influence in what people think, feel and want: Nearly all of it has been influenced by intentional fallacious and professional crafted lies, deceptions and frauds.

Most of what the people see on TV is some kind of deception, even - and sometimes especially - if it is The News, and nearly everything that happens in the media - which is where in the present time the vast majority of people get their opinions, values, and tastes from - is intentional deception, conscious  misrepresentation, or is partial or biased in some way, and is there to serve someone's interests, even if that is not obvious.

The same goes for advertisements, which these days very well may make up the majority of what ordinary men read in their lives.

The reason why I use strong terms - "whorified" - is because I feel strongly about it:

What the vast majority of humanity consumes and gets served in terms of ideas, values and information these days comes by way of the media, which are in the hand of a very small number of people, who tend to be totally out of control by anyone (Maxwell, Murdoch, Berlusconi: that manner of men) and which systematically have replaced (prostituted) real argument about the real world by surreal argument about an infotainment world.

In this infotainment world, in which most people really believe they really live, simply because what they think and feel and know comes from it, and not from private education, schools or universities anymore, and that not because these don't work anymore (though this also may be so, and often is so, since postmodernism destroyed the Western education-systems to a large extent, between 1970 and 2000, and replaced them by academies for the study of and education in fashionable lying), but because the vast majority spend most of their lives, insofar as they have a choice, happily in front of their TV-dreambox, lapping up whatever it conveys, as long as it entertains.

This is much unlike what the world was like before, when human opinions normally were also not made in a more rational way, but when opinions also could not be broadcast over the billions for pay, and could not be manufactured by professional opinion-makers, spin-doctors, and media-personalities, and could not be endlessly sauced by and pimped up with all manner of emotionally pleasing irrelevancies, intentional misdirections, false poses by people specializing in false poses that look very real and cuddly and nice, and beautiful ever broadly smiling assistants of kind, clever and very attractive Media-Personalities (some botoxed, but that too you can't really see), and were not yet behind the scenes directed by someone (who serves someone's interests by) high;y trained and highly paid  professional lying, professional misrepresenting, intentional biasing, and conscious saucing and colouring of any kind of information or topic.

The underlying problem is not that people are, by and large, more stupid than they used to be (though on average they very well may be) nor that they lie more than they used to do (religion and politics, at least, have always been mostly composed of falsehoods and deceptions, as has sales-talk), but that these days (1) there is very much more of this by the media; (2) it is especially crafted by professionals; (3) it is broadcast over and reaches far more men, women and children than any form of propaganda ever did, and does so far more effectively, pleasingly and convincingly than any form of earlier propaganda could do and (4) it is in the hands of very few very powerful men, who are virtually or factually completely beyond control.