on William Lane craig on whether A logically implies B.

William Lane Craig (“Trinity Monotheism once more...”) says that,

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.