What does it mean for a set of sentences $\mathcal{T}$ to "secure" a set of sentences $\Delta$?

Question

I know the standard interpretation is:

$\mathcal{T}$ secures $\Delta$ iff every interpretation that makes all members of $\mathcal{T}$ true makes at least one member of $\Delta$ true.

However, I'm having trouble understanding the following results:

$\mathcal{T} \models \Delta$ iff $\mathcal{T}$ secures $\{\Delta\}$

$\mathcal{T}$ is unsatisfiable iff $\mathcal{T}$ secures $\emptyset$

$\Delta$ is valid iff $\emptyset$ secures $\{\Delta\}$

In particular, the second and third results are hard to interpret. For the second, if $\mathcal{T}$ secures $\emptyset$, that means that every interpretation that makes all of $\mathcal{T}$ true makes at least one member of $\emptyset$ true. Does this imply that $\mathcal{T}$ is unsatisfiable because there is no interpretation in which $\emptyset$ can be true, so there is no interpretation which makes all of $\mathcal{T}$ true? Does "satisfiable" for a set of sentences imply that all of it's members must be true? I thought this was what "valid" was used for.

Answer 1

Yes, the important issue in the second and third items is that $\emptyset$ has no members. Thus is it impossible to secure $\emptyset$, because the definition of "secure" requires satisfying at least one member. On the other hand, every structure satisfies $\emptyset$, because the definition of "satisfies" is that the structure satisfies every formula in the set, and it is trivially the case that every formula in $\emptyset$ is satisifed.

Answer 2

This terminology is used in George Boolos & John Burgess & Richard Jeffrey, Computability and Logic (5th ed - 2007), Ch.14 : Proofs and Completeness, regarding sequent calculus, page 167 :

We say that one set of sentences $\Gamma$ secures another set of sentences $\Delta$ if every interpretation that makes all sentences in $\Gamma$ true makes some sentence in $\Delta$ true.(Note that when the sets are finite, $\Gamma = \{ C_1, . . . ,C_m \}$ and $\Delta = \{ D_1, . . . , D_n \}$, this amounts to saying that every interpretation that makes $C_1 \land. . . \land C_m$ true makes $D_1 \lor · · · \lor D_n$ true: the elements of $\Gamma$ are being taken jointly as premisses, but the elements of $\Delta$ are being taken alternatively as conclusions, so to speak.)

The condition is clearly a version of the semantical valuation of sequents : $\Gamma \rightarrow \Delta$ [compare with Stephen Cole Kleene, Mathematical Logic (1967), page 290 : an interpretation falsify (or refute) a sequent $\Gamma \rightarrow \Delta$ when all of $\Gamma$ are true and all of $\Delta$ are false; thus, the negation of this definition will be : if all of $\Gamma$ are true, then at least one of $\Delta$ is true].

BBJ states in Table 14-1 the equivalences between Metalogical notions [page 168] :

$D$ is a consequence of $\Gamma$ if and only if $\Gamma$ secures $\{ D \}$

$\Gamma$ is unsatisfiable if and only if $\Gamma$ secures $\emptyset$

$D$ is valid if and only if $\emptyset$ secures $\{ D \}.$

The basic metalogical notions are the standard ones :

$D$ is a consequence of $\Gamma$ if, for every interpretations $I$ such that all $\gamma \in \Gamma$ are true in $I$, also $D$ is true in $I$

$\Gamma$ is satisfiable if there is an interpretation $I$ such that all $\gamma \in \Gamma$ are true in $I$

$D$ is valid if it is true in all interpretations.