In

* Note that this is only a logical fallacy when the word "or" is in its inclusive form. If the two possibilities in question are mutually exclusive, this is not a logical fallacy. For example,
#I am either at home or I am in the city. (but not both)
#I am at home.
#Therefore, I am not in the city.

affirming the consequent
Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...

.
An argument could contain both an informal fallacy and a formal fallacy yet lead to a conclusion that happens to be true, for example, again affirming the consequent, now also from an untrue premise:
# If a scientist makes a statement about science, it is correct.
# It is true that quantum mechanics is deterministic.
# Therefore, a scientist has made a statement about it.

On Sophistical Refutations

''De Sophistici Elenchi''. *

Aphorisms concerning The Interpretation of Nature and the Kingdom of Man, XXIIIff

.

The Art of Controversy

''Die Kunst, Recht zu behalten – The Art Of Controversy'' (bilingual)

by

A System of Logic – Raciocinative and Inductive

* C. L. Hamblin

Methuen London, 1970. * Fearnside, W. Ward and William B. Holther

1959. * Vincent F. Hendricks, ''Thought 2 Talk: A Crash Course in Reflection and Expression'', New York: Automatic Press / VIP, 2005, * D. H. Fischer, ''Historians' Fallacies: Toward a Logic of Historical Thought'', Harper Torchbooks, 1970. * Douglas N. Walton, ''Informal logic: A handbook for critical argumentation''. Cambridge University Press, 1989. * F. H. van Eemeren and R. Grootendorst, ''Argumentation, Communication and Fallacies: A Pragma-Dialectical Perspective'', Lawrence Erlbaum and Associates, 1992. * Warburton Nigel, ''Thinking from A to Z'', Routledge 1998. * Sagan, Carl, '' The Demon-Haunted World: Science As a Candle in the Dark''.

philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. So ...

, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (; Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...

for " tdoes not follow") is a pattern of reasoning
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, la ...

rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

.Harry J. Gensler, ''The A to Z of Logic'' (2010) p. 74. Rowman & Littlefield, It is defined as a deductive
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...

argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was int ...

where deduction goes wrong, and is no longer a logical
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
While a logical argument is a non sequitur if, and only if, it is invalid, the term "non sequitur" typically refers to those types of invalid arguments which do not constitute formal fallacies covered by particular terms (e.g., affirming the consequent
Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...

). In other words, in practice, "''non sequitur''" refers to an unnamed formal fallacy.
A special case is a mathematical fallacy, an intentionally invalid mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

proof, often with the error subtle and somehow concealed. Mathematical fallacies are typically crafted and exhibited for educational purposes, usually taking the form of spurious proofs of obvious contradictions.
A formal fallacy is contrasted with an informal fallacy
Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the ''form'' of the argument, as is the case for formal fallacies, but can also be due to their ''content'' and ''context''. Fallac ...

which may have a valid logical form and yet be unsound because one or more premises are false. A formal fallacy; however, may have a true premise, but a false conclusion.
Taxonomy

Prior Analytics isAristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of p ...

's treatise on deductive reasoning and the syllogism. The standard Aristotelian ''logical fallacies'' are:
* Fallacy of four terms (''Quaternio terminorum'');
* Fallacy of the undistributed middle;
* Fallacy of illicit process of the major
Major (commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicators ...

or the minor term;
* Affirmative conclusion from a negative premise.
Other logical fallacies include:
* The self-reliant fallacy
In philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. So ...

, the term ''logical fallacy'' properly refers to a formal fallacy—a flaw in the structure of a deductive
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...

argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

, which renders the argument invalid.
It is often used more generally in informal discourse to mean an argument that is problematic for any reason, and encompasses informal fallacies
Informal fallacies are a type of incorrect argument in natural language. The source of the error is not just due to the ''form'' of the argument, as is the case for formal fallacies, but can also be due to their ''content'' and ''context''. Fallac ...

as well as formal fallacies—valid but unsound claims or poor non-deductive argumentation.
The presence of a formal fallacy in a deductive argument does not imply anything about the argument's premises or its conclusion (see fallacy fallacy). Both may actually be true, or even more probable as a result of the argument (e.g. appeal to authority), but the deductive argument is still invalid because the conclusion does not follow from the premises in the manner described. By extension, an argument can contain a formal fallacy even if the argument is not a deductive one; for instance an inductive argument that incorrectly applies principles of probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...

or causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...

can be said to commit a formal fallacy.
Affirming the consequent

Any argument that takes the following form is a non sequitur #If A is true, then B is true. #B is true. #Therefore, A is true. Even if the premise and conclusion are both true, the conclusion is not a necessary consequence of the premise. This sort of non sequitur is also calledaffirming the consequent
Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...

.
An example of affirming the consequent would be:
#If Jackson is a human (A), then Jackson is a mammal. (B)
#Jackson is a mammal. (B)
#Therefore, Jackson is a human. (A)
While the conclusion may be true, it does not follow from the premise:
# Humans are mammals.
# Jackson is a mammal.
# Therefore, Jackson is a human.
The truth of the conclusion is independent of the truth of its premise – it is a 'non sequitur', since Jackson might be a mammal without being human. He might be an elephant.
Affirming the consequent is essentially the same as the fallacy of the undistributed middle, but using propositions rather than set membership.
Denying the antecedent

Another common non sequitur is this: #If A is true, then B is true. #A is false. #Therefore, B is false. While B can indeed be false, this cannot be linked to the premise since the statement is a non sequitur. This is calleddenying the antecedent
Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form:
:If ''P'', then ''Q''.
:Therefore, if not ...

.
An example of denying the antecedent would be:
#If I am Japanese, then I am Asian.
#I am not Japanese.
#Therefore, I am not Asian.
While the conclusion may be true, it does not follow from the premise. The statement's declarant could be another ethnicity of Asia, e.g., Chinese, in which case the premise would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.
Affirming a disjunct

Affirming a disjunct is a fallacy when in the following form: #A or B is true. #B is true. #Therefore, A is not true.* The conclusion does not follow from the premise as it could be the case that A and B are both true. This fallacy stems from the stated definition of ''or'' in propositional logic to be inclusive. An example of affirming a disjunct would be: #I am at home or I am in the city. #I am at home. #Therefore, I am not in the city. While the conclusion may be true, it does not follow from the premise. For all the reader knows, the declarant of the statement very well could be in both the city and their home, in which case the premises would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.Denying a conjunct

Denying a conjunct is a fallacy when in the following form: #It is not the case that A and B are both true. #B is not true. #Therefore, A is true. The conclusion does not follow from the premise as it could be the case that A and B are both false. An example of denying a conjunct would be: #I cannot be both at home and in the city. #I am not at home. #Therefore, I am in the city. While the conclusion may be true, it does not follow from the premise. For all the reader knows, the declarant of the statement very well could neither be at home nor in the city, in which case the premise would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true.Illicit commutativity

Illicit commutativity is a fallacy when in the following form: #If A is the case, then B is the case. #Therefore, if B is the case, then A is the case. The conclusion does not follow from the premise as unlike other logical connectives, the implies operator is one-way only. "P and Q" is the same as "Q and P", but "P implies Q" is not the same as "Q implies P". An example of this fallacy is as follows: #If it is raining, then I have my umbrella. #If I have my umbrella, then it is raining. While this may ''appear'' to be a reasonable argument, it is not valid because the first statement does not logically guarantee the second statement. The first statement says nothing like "I do not have my umbrella otherwise", which means that having my umbrella on a sunny day would render the first statement true and the second statement false.Fallacy of the undistributed middle

The fallacy of the undistributed middle is afallacy
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was int ...

that is committed when the middle term in a categorical syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...

is not distributed Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a varia ...

. It is a syllogistic fallacy
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...

. More specifically it is also a form of non sequitur.
The fallacy of the undistributed middle takes the following form:
#All Zs are Bs.
#Y is a B.
#Therefore, Y is a Z.
It may or may not be the case that "all Zs are Bs", but in either case it is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that "all Bs are Zs," which is ignored in the argument.
An example can be given as follows, where B=mammals, Y=Mary and Z=humans:
#All humans are mammals.
#Mary is a mammal.
#Therefore, Mary is a human.
Note that if the terms (Z and B) were swapped around in the first co-premise then it would no longer be a fallacy and would be correct.
In contrast to informal fallacy

Formal logic is not used to determine whether or not an argument is true. Formal arguments can either be valid or invalid. A valid argument may also be sound or unsound: * A ''valid'' argument has a correct formal structure. A valid argument is one where ''if'' the premises are true, the conclusion ''must'' be true. * A ''sound'' argument is a formally correct argument that ''also'' contains true premises. Ideally, the best kind of formal argument is a sound, valid argument. Formal fallacies do not take into account the soundness of an argument, but rather its validity. Premises in formal logic are commonly represented by letters (most commonly p and q). A fallacy occurs when the structure of the argument is incorrect, despite the truth of the premises. As ''modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...

'', the following argument contains no formal fallacies:
# If P then Q
# P
# Therefore, Q
A logical fallacy associated with this format of argument is referred to as affirming the consequent
Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dar ...

, which would look like this:
# If P then Q
# Q
# Therefore, P
This is a fallacy because it does not take into account other possibilities. To illustrate this more clearly, substitute the letters with premises:
# If it rains, the street will be wet.
# The street is wet.
# Therefore, it rained.
Although it is possible that this conclusion is true, it does not necessarily mean it ''must'' be true. The street could be wet for a variety of other reasons that this argument does not take into account. If we look at the valid form of the argument, we can see that the conclusion must be true:
# If it rains, the street will be wet.
# It rained.
# Therefore, the street is wet.
This argument is valid and, if it did rain, it would also be sound.
If statements 1 and 2 are true, it absolutely follows that statement 3 is true. However, it may still be the case that statement 1 or 2 is not true. For example:
# If Albert Einstein makes a statement about science, it is correct.
# Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

states that all quantum mechanics is deterministic.
# Therefore, it's true that quantum mechanics is deterministic.
In this case, statement 1 is false. The particular informal fallacy being committed in this assertion is argument from authority
An argument from authority (''argumentum ab auctoritate''), also called an appeal to authority, or argumentum ad verecundiam, is a form of argument in which the opinion of an authority on a topic is used as evidence to support an argument. Some con ...

. By contrast, an argument with a formal fallacy could still contain all true premises:
#If an animal is a dog, then it has four legs.
# My cat has four legs.
# Therefore, my cat is a dog.
Although 1 and 2 are true statements, 3 does not follow because the argument commits the formal fallacy of Common examples

"Some of your key evidence is missing, incomplete, or even faked! That proves I'm right!" "The vet can't find any reasonable explanation for why my dog died. See! See! That proves that you poisoned him! There’s no other logical explanation!" In the strictest sense, a logical fallacy is the incorrect application of a valid logical principle or an application of a nonexistent principle: # Most Rimnars are Jornars. # Most Jornars are Dimnars. # Therefore, most Rimnars are Dimnars. This is fallacious. And so is this: # People in Kentucky support a border fence. # People in New York do not support a border fence. # Therefore, people in New York do not support people in Kentucky. Indeed, there is no logical principle that states: # For some x, P(x). # For some x, Q(x). # Therefore, for some x, P(x) and Q(x). An easy way to show the above inference as invalid is by usingVenn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships ...

s. In logical parlance, the inference is invalid, since under at least one interpretation of the predicates it is not validity preserving.
People often have difficulty applying the rules of logic. For example, a person may say the following syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
...

is valid, when in fact it is not:
#All bird
Birds are a group of warm-blooded vertebrates constituting the class Aves (), characterised by feathers, toothless beaked jaws, the laying of hard-shelled eggs, a high metabolic rate, a four-chambered heart, and a strong yet lightweig ...

s have beaks.
#That creature has a beak.
#Therefore, that creature is a bird.
"That creature" may well be a bird, but the conclusion does not follow from the premises. Certain other animals also have beaks, for example: an octopus
An octopus ( : octopuses or octopodes, see below for variants) is a soft-bodied, eight- limbed mollusc of the order Octopoda (, ). The order consists of some 300 species and is grouped within the class Cephalopoda with squids, cuttlef ...

and a squid
True squid are molluscs with an elongated soft body, large eyes, eight arms, and two tentacles in the superorder Decapodiformes, though many other molluscs within the broader Neocoleoidea are also called squid despite not strictly fitting ...

both have beaks, some turtles and cetaceans
Cetacea (; , ) is an infraorder of aquatic mammals that includes whales, dolphins, and porpoises. Key characteristics are their fully aquatic lifestyle, streamlined body shape, often large size and exclusively carnivorous diet. They propel them ...

have beaks. Errors of this type occur because people reverse a premise. In this case, "All birds have beaks" is converted to "All beaked animals are birds." The reversed premise is plausible because few people are aware of any instances of ''beaked creatures'' besides birds—but this premise is not the one that was given. In this way, the deductive fallacy is formed by points that may individually appear logical, but when placed together are shown to be incorrect.
Non sequitur in everyday speech

In everyday speech, a non sequitur is a statement in which the final part is totally unrelated to the first part, for example:See also

* * * * * * * * * * * * *References

;Notes ;Bibliography *Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of p ...

On Sophistical Refutations

''De Sophistici Elenchi''. *

William of Ockham
William of Ockham, OFM (; also Occam, from la, Gulielmus Occamus; 1287 – 10 April 1347) was an English Franciscan friar, scholastic philosopher, apologist, and Catholic theologian, who is believed to have been born in Ockham, a small vi ...

, ''Summa of Logic'' (ca. 1323) Part III.4.
* John Buridan, ''Summulae de dialectica'' Book VII.
* Francis Bacon, the doctrine of the idols in ''Novum Organum Scientiarum''Aphorisms concerning The Interpretation of Nature and the Kingdom of Man, XXIIIff

.

The Art of Controversy

''Die Kunst, Recht zu behalten – The Art Of Controversy'' (bilingual)

by

Arthur Schopenhauer
Arthur Schopenhauer ( , ; 22 February 1788 – 21 September 1860) was a German philosopher. He is best known for his 1818 work '' The World as Will and Representation'' (expanded in 1844), which characterizes the phenomenal world as the pro ...

* John Stuart MillA System of Logic – Raciocinative and Inductive

* C. L. Hamblin

Methuen London, 1970. * Fearnside, W. Ward and William B. Holther

1959. * Vincent F. Hendricks, ''Thought 2 Talk: A Crash Course in Reflection and Expression'', New York: Automatic Press / VIP, 2005, * D. H. Fischer, ''Historians' Fallacies: Toward a Logic of Historical Thought'', Harper Torchbooks, 1970. * Douglas N. Walton, ''Informal logic: A handbook for critical argumentation''. Cambridge University Press, 1989. * F. H. van Eemeren and R. Grootendorst, ''Argumentation, Communication and Fallacies: A Pragma-Dialectical Perspective'', Lawrence Erlbaum and Associates, 1992. * Warburton Nigel, ''Thinking from A to Z'', Routledge 1998. * Sagan, Carl, '' The Demon-Haunted World: Science As a Candle in the Dark''.

Ballantine Books
Ballantine Books is a major book publisher located in the United States, founded in 1952 by Ian Ballantine with his wife, Betty Ballantine. It was acquired by Random House in 1973, which in turn was acquired by Bertelsmann in 1998 and remains ...

, March 1997 , 480 pp. 1996 hardback edition: Random House
Random House is an American book publisher and the largest general-interest paperback publisher in the world. The company has several independently managed subsidiaries around the world. It is part of Penguin Random House, which is owned by Germ ...

,
External links

{{Authority control Barriers to critical thinking Deductive reasoning Philosophical logic