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.
