The Debate over the Consequence Argument 89counterexamples to this. There are cases in which it is true that Np, and true that
Nq, but it would be mistaken to infer N(p&q), because, as it happens, a person
who does not have a choice about p or about q does have a choice about (p&q),
and so N(p&q) is false.^20
We’ll now explain each of these two key points: first, that Rule β strictly
implies agglomerativity, and, second, that there are counterexamples to it. So
first, consider the following complex proposition:
[p → (q → (p&q))]Treat “p” and “q” as any random propositions. Notice the following about the
complex proposition: Whatever combination of values true and false is assigned
to p and q, the complex proposition comes out as true. There is no interpretation
of the truth values for p and q that will yield any result other than truth for the
entire proposition.^21 Now, when a complex proposition in the language of first-
order logic is true on any interpretation of the truth values of all of its constitu-
ents, it is a necessary truth. Hence:
□[p → (q → (p&q))]Apply Rule α, and we get:
N[p → (q → (p&q))]Setting this inference aside for the moment, consider any two propositions such
that no one has a choice about them, Np and Nq. Grant them for argument’s
sake. Given the preceding presentation, here, in compressed form, is Thomas
McKay and David Johnson’s proof that Rule β, along with a few simple logical
ingredients, strictly implies agglomerativity (1996: 115):
- Np premise granted
- Nq premise granted
- □[p → (q → (p&q))] necessity of a logical truth
- N[p → (q → (p&q))] apply Rule α to step 3
- N(q → (p&q)) apply Rule β to steps 1 and 4
- Therefore, N(p&q) apply Rule β to steps 2 and 5
This proof is sound and decisive. From Np and Nq, with an application of some
simple logical truths, Rule α (which seems innocent here), and Rule β, we get
N(p&q).
Here, now, is McKay and Johnson’s counterexample to agglomerativity for
“no one has a choice about.” Suppose that Jones does not toss a coin but could
have. He kept it in his pocket, untossed. Set aside all assumptions about deter-
minism or the free will debate. Now consider these three propositions, and note
that, as it happens, all three are true, given that Jones kept the coin in his pocket: