|Tractatus Logico-Philosophicus by Ludwig Wittgenstein + comments by Maarten Maartensz|
Here W. seems to confuse statements and their referents (as is often the case when the term "proposition" is used).
In general terms - not W.'s - an interpretation of a statement comes about through some function that assigns set-theoretical entities to the parts of the statement.
In the case at hand, such as - to take a fairly
specific example - "(x)(A(x)) |- (Ex)(A(x))", in words: if every thing
is A, it follows that some thing is A", we might use a function i() with
the properties that
In words, this may be summarized as: i() assigns to the expression "(x)(A(x))" a set i(A) and to the expression "(Ex)(A(x))" a non-empty subset of i(A). Now if we also stipulate that "p |- q" iff "i(q) inc i(p)", i.e. q follows from p iff the set of things that serves as the interpretation of q is included in the set of things that serves as the interpretation of p, and if one stipulates that i( (x)(A(x)) ) is to non-empty sets exclusively, one has - in principle, and with a considerable amount of detail left unstated - a tolerably clear explanation of "follows from" in the logical sense, at least in the case of the present example, of explaining why "some things are A" would "follow from" "every thing is A". (There are other ways, but I wanted to avoid discussing the semantics of variables.)
To recapitulate: "some things are A" does "follow from" "every thing is A" if we use "every thing is A" to refer to A's indiscriminately, and we use "some things are A" to refer to A's discriminately (and reconstruct this in terms of set-theory, where one uses a function to map parts of speech to sets of real (and possible) things).
But this type of explanation - that owes most to
Tarski - is not what W. had in mind. It may serve to show what W. confused: The
relation between the statements "p" and "q", such that e.g.
"q" contains only variables also contained in "p", may or
may not be somehow be correlated to the relation between the (sets of) things
the statements "p" and "q" represent.