### on William Lane craig on whether A logically implies B.

Moreover, Howard-Snyder seems to assume
that truth-making is closed under logical
implication […] But that assumption is false. For example, “[a cat] has
retractable claws” implies that “Grass is green,” since both are true, but they
obviously have different truth-makers. (§3)

I have nothing to say about whether
truth-making is closed under logical implication. I do have something to say
about Craig’s argument that

*A*A cat has retractable claws.

(logically) implies

*B*Grass is green.

“Since”, Craig says, “both are true”.
Craig is almost certainly confused here.

By my lights, a
sentence/statement/proposition 𝒫

*logically implies 𝒬**, abbreviated 𝒫 ⊨ 𝒬, just in case 𝒬 is a logical consequence of 𝒫, which is to say that there is no truth-value assignment or “valuation” in which 𝒫 is true and 𝒬 is false. But, if you were to put**A*and*B*on a truth table, there will be one line in which*A*is true and*B*is false. Hence A does not logically imply B, or*A*⊭

*B*

It may be, however, that all that Craig
intended is to say is that, given that

*A*and*B*are true,*A*“materially” implies*B*, or*A*⊃*B*. Indeed, since there is no truth value assignment in which*A*and*B*are true and*A*⊃*B*if false, the former logically implies the latter, or*A*&

*B*⊨

*A*⊃

*B*

Note, however, that this doesn’t affect
my previous point. It’s one thing to say that,

given

*A*&*B*, the material conditional*A**⊃**B*is true,
and it’s another to say that,

given

*A*&*B*,*A*logically implies*B*.
The former says something true about
the conjunction,

*A*&*B*, in relation to the material conditional,*A*⊃*B*, whereas the latter says something false about the conjunction,*A*&*B*, and its relation to another relation: the relation between*A*and*B*per se.
Now, by my understanding,

Relation ℛ is

*closed under logical implication*just in case, if*x*stands in relation ℛ to 𝒫, and if 𝒫 logically implies 𝒬, that is 𝒫 ⊨ 𝒬, then*x*stands in relation ℛ to 𝒬.
For example: some have thought, but
many deny, that the

*knows that*relation between an agent and a proposition is closed under logical implication. This view has the consequence that, if Craig knows that*A*, then Craig knows that: either*A*or ¬*B*, for*A*⊨*A*v ¬*B*. Notice that if the*knows that*relation were closed under logical implication, and we understood logical implication as “material” implication, then those who think that the*knows that*relation is closed under logical implication would be saddled with the view that, if Craig knows*A*, then Craig knows*B*, for*A*⊃*B*. As dubious as the view that the*knows that*relation is closed under logical implication might be, it's not*that*dubious.
Similarly, if we suppose that the

*makes true*relation is closed under logical implication, then this would require only that, e.g., if*x*makes it true that*A*, then*x*makes it true that: either*A*or ¬*B*, for again*A*⊨*A*v ¬*B*. It would not require that, if*x*is makes it true that*A*, then*x*makes it true that*B*, as*A*⊭*B*, and this despite the fact that*A*and*B*are both true. In summary, then, it appears that Craig’s argument against the view that the*makes true*relation is closed under logical implication rests upon the following false conditional: If*A*and*B*, then*A*logically implies*B*.
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