References : I think the " truth set approach" to validity and logical consequence can be linked to the name of R. Carnap ( who defines L-truth and L-implication in this way in his Introduction to Symbolic Logic, except that Carnap calls "range" of a formula what others call " truth-set"). The standard approach to these concepts is more related to Tarski ( according to MacKeon, "Logical consequence", Internet Encyclopedia of Philosophy).
By " the truth set of a formula X" I mean the set of all interpretations in which X is true. With I as universal set ( set of all possible interpretations of the language) and S(X) denoting the truth set of the formula X, one could write :
S(X) = { i belonging to I| X is true in i }.
With some definitions, such as
S(X&Y) = S(X) Inter S(Y)
S (XvY) = S(X) Union S(Y)
S(~X)= Complement of S(X)= I\S(X)
X is a tautology iff S(X) = I
X is an antilogy iff S(X) = Empty set,
X |= Y iff S(X) is included in S(Y) etc. ...
one can apply set theory to prove basic metalogicalfacts such as
" (Av~A) is a tautology" that is :
S(Av~A) = S(A) Union S(~A) = S(A) Union Complement of S(A) = I
or
X|= Y iff |= (X--> Y)
Indeed
(1) X |= Y
(2) iff S(X) is included in S(Y)
(3) iff there is no i such that ( X is true in i and Y is false in i)
(4) iff there is no i such that ( the formula X & ~ Y is true in i )
(5) iff S( X & ~Y) = Empty set
(6) iff S ( ~ (X & ~Y) ) = Complement of Empty set = I
(7) iff S ( X --> Y) = I
(8) iff the formula (X --> Y) is a tautology.
My question :
(1) is the truth set approach to validity and logical consequence different from the standard approach ?
(2) what are the possible drawbacks of this approach?
(3) does the use of set theory and of quantification ( over I) to prove metalogical facts imply a risk of circularity?
(4) is this approach totally rigorous? for example, in the previous reasoning, am I really allowed to go from line (3) to line (4)? ( Do I tacitly use something like an illegal "meta &-Introduction" rule? )
(5) are the definitions given at the beginning derivable from the general definition of the truth set of a formula X.