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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 E - Every


 
Every: In logic: term for each, all, any.

The term 'every' is known as a quantifier, i.e. a term that indicates what proportion of a collection is supposed to have some property, and 'every' means that no element of the collection lacks the property.

There are in natural language some subtle differences in the usage or meaning of universally quantifying particles like 'every', 'each', 'all' and 'any' that are not brought out in standard logic.

Also, one difference between 'every' as used in formal logic and as used in natural language is that in formal logic 'every' is used in combination with variables, that do not occur in natural language. Thus, 'every man is rational' gets translated into formal logic on the pattern of 'for every x, if x is a man, then x is rational'.

In formal logic, 'every' is often written as in '(x)(Fx)' i.e. 'for every x Fx' and rules that are adopted for 'every' are often like these:

From (x)(Fx) it follows that (Fa), with a any arbitrary constant.
From (Fx) it follows that (x)(Fx), provided x does not occur elsewhere in the assumptions of the proof.

Thus, what does hold for every variable holds also for any constant, and what does hold for any variable also holds for every variable. The reason for the proviso in the second rule is that if x does occur elsewhere in the proof then there may be additional restrictions imposed on it, but if it does not, and one still can prove (Fx), as for example in 'Fx V ~Fx' or 'Fx --> Fx', then it must hold not just in any but in every case.

 


See also: Quantifier, First-Order Logic


Literature:

Carnap, Cartwright, Hasenjäger, Quine

 Original: Jul 5, 2005                                                Last edited: 12 December 2011.   Top