In logic and philosophy, a formal fallacy is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy in which deduction goes wrong, and is no longer a logical 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). In other words, in practice, "non sequitur" refers to an unnamed formal fallacy.
A special case is a mathematical fallacy, an intentionally invalid mathematical 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 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.
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!"
Statement 1: Most of the green is touching the red.
Statement 2: Most of the red is touching the blue.
Logical fallacy: Since most of the green is touching red, and most of the red is touching blue, most of the green must be touching blue. This, however, is a false statement.
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.
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 using Venn diagrams. 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 is valid, when in fact it is not:
- All birds 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 and a squid both have beaks, some turtles and cetaceans 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:
Life is life and fun is fun, but it's all so quiet when the goldfish die.
— West with the Night, Beryl Markham
