**
Some **: In
logic: term for asserting that there are
things satisfying a predicate.
The term '**some**' is known as a
quantifier, i.e. a term that indicates
what proportion of a collection is supposed to have some property, and
'some' means that not every element of
the collection lacks the
property.
In standard logic, the term '**some**' is used as an equivalent of
'**there is**' and '**there are**', in that each is taken in the
sense that a predicate does hold of at least one element in the domain,
and this is taken as the sufficient and necessary reason to say that
some things have the predicate, and that there is a thing or there are
things that have the predicate.
In natural language there may be subtle differences in the usage or
meaning of 'some' and 'there is', notably the following: Most speakers
of English have no trouble with 'Some Greek gods dallied sexually with
human virgins', since this is part of the tales told about Zeus and
other Greek gods, but many speakers of English do not believe this
implies that 'There is a Greek god who dallied sexually with human
virgins', since they do not believe there really are or ever were any
Greek gods.
These differences can be accounted for by other means, such as
introducing a predicate for existence, which allows the formulation of
statements to the effect that some things exist and some things do not
exist.
In standard logic, 'some' and 'every' are interdefinable, in that
'some' amounts to 'not every not' and 'every' to 'not some not', as in
'some women are blond iff not every woman is not blond' and 'every man
is rational iff not some man is not rational' and 'every prime has one
factor iff it is not true there is a prime with more or less than one
factor'.
This interdefinability of 'some' and 'every' is not the case in
intuitionist logic, where 'some' has the reading 'there is' and 'there
is' requires some sort of proof that one can **construct** a thing
that is said to be amongs those things that there is. This makes sense
in many cases, but leads to problems with statements like 'There is a
number no one ever thought of' or 'There are specific numbers so large
that no one will ever mention them'. Also, it excludes constructions and
proofs in standard mathematics that
conclude that there is a such-and-such merely because one can prove that
if this is not so one can derive a
contradiction.
Also, one difference between 'some' as used in formal logic and as used in
natural language is that in formal logic 'some' is used in combination with
variables, that do not occur in natural language. Thus, 'some man is rational'
gets translated into formal logic on the pattern of 'for some x, x is a man
and x is rational'.
In formal logic, 'some' is often written as '(jx)(Fx)' and rules that are
adopted for 'some' are often like these:
From (jx)(Fx) it follows that (Fa), provided a
does not occur elsewhere in the proof.
From (Fa) it follows that (jx)(Fx), provided x
does not occur elsewhere in (F).
Thus, what holds for some variable also holds for some constant, and
what holds for some constant also holds for some variable. The reason
for the proviso in the first rule is that if the constant a does occur
in elsewhere in the proof additional restrictions may be imposed on it,
but if it does not one can use it. (See below). And the reason for the
proviso in the second rule is that if the variable x occurs elsewhere in
(F) writing (jx)(Fx) may be false, but if
it does not one can use the variable.
Both provisos are meant to exclude possibilities that may make the
conclusion invalid. In the inference of (Fa) from (jx)(Fx)
the constant a is often called a quasi-constant, because it tends to
function not as a real constant but as a place-taker for a constant, as
in 'Suppose there is a largest number in the set of things that are F.
Let's call that number a.' |