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A quantifier shift is a
logical fallacy In logic and philosophy, a formal fallacy is a pattern of reasoning rendered invalid by a flaw in its logical structure. Propositional logic, for example, is concerned with the meanings of sentences and the relationships between them. It focuses ...
in which the quantifiers of a statement are erroneously transposed during the rewriting process. The change in the logical nature of the statement may not be obvious when it is stated in a
natural language A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...
like English.


Definition

The fallacious deduction is that: ''For every A, there is a B, such that C. Therefore, there is a B, such that for every A, C.'' :\forall x \,\exists y \,Rxy \vdash \exists y \,\forall x \,Rxy However, an inverse switching: :\exist y \,\forall x \,Rxy \vdash \forall x \,\exist y\, Rxy is logically valid.


Examples

1. Every person has a woman that is their mother. Therefore, there is a woman that is the mother of every person. :\forall x \,\exists y \,(Px \to (Wy \land M(yx))) \vdash \exists y \,\forall x \,(Px \to (Wy \land M(yx))) It is fallacious to conclude that there is ''one woman'' who is the mother of ''all people''. However, if the major premise ("every person has a woman that is their mother") is assumed to be true, then it is valid to conclude that there is ''some'' woman who is ''any given person's'' mother. 2. Everybody has something to believe in. Therefore, there is something that everybody believes in. :\forall x \,\exists y \,Bxy \vdash \exists y \,\forall x \,Bxy It is fallacious to conclude that there is ''some particular concept'' to which everyone subscribes. It is valid to conclude that each person believes ''a given concept''. But it is entirely possible that each person believes in a unique concept. 3. Every
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
n has a successor m = n + 1, the smallest of all natural numbers that are greater than n. Therefore, there is a natural number that is a successor to all natural numbers. :\forall n \,\exists m \,Snm \vdash \exists m \,\forall n \,Snm It is fallacious to conclude that there is a single natural number that is the successor of every natural number.


References

* Robert Audi (General Editor), The Cambridge Dictionary of Philosophy (Second Edition), 1999, pp. 272–3. *A. R. Lacey, Dictionary of Philosophy (Third Revised Edition) (
Barnes & Noble Barnes & Noble Booksellers is an American bookseller with the largest number of retail outlets in the United States. The company operates approximately 600 retail stores across the United States. Barnes & Noble operates mainly through its B ...
, 1996). *Introduction to
Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, Harry J. Gensler, p. 220 * Antony G. Flew, Dictionary of Philosophy: Revised Second Edition *Harry J. Gensler, Historical Dictionary of Logic {{Formal Fallacy Quantificational fallacies