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.
