# “Truth set” approach to validity and logical consequence: how does it relate to the standard approach? what are the possible drawbacks?

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.

• Both approaches are just different choices of Boolean algebras to interpret propositional logic into. – Derek Elkins Apr 20 at 13:34
• @DerekElkins. Could you please tell me on which sets these two algebras "operate"? Also, what operations are involved? – Eleonore Saint James Apr 20 at 13:45
• If you notice, you don't actually define what an "interpretation" is or what a formula being "true in an interpretation" means. What you actually have is just a set $I$ and a mapping $S$ of formulas into the subsets of $I$, i.e. into the powerset of $I$. The powerset of a set is a Boolean algebra with the usual set-theoretic operations (i.e. the ones you describe). I assume by the standard/"Tarskian" approach you mean interpretation into a two element set, i.e. truth table semantics. Truth table semantics are equivalent to choosing $I$ as any singleton set. – Derek Elkins Apr 20 at 21:57
• @DereckElkins. Thanks. – Eleonore Saint James Apr 21 at 10:44