Possible Worlds quote:

“In the second place, the concept of conjunction can be conveyed without using any sentence connective whatever. One way – indeed one of the commonest of all ways – of expressing the conjunction of two propositions is simply to use first the sentence expressing one and then the sentence expressing the other. If we want to assert both that there are five oranges in the basket and that there are six apples in the bowl, then we need only utter, one after the other, the two separate sentences “There are five oranges in the basket” and “There are six apples in the bowl.” We will then be taken, correctly, to have asserted both that there are five oranges and that there are six apples in the bowl. The fact that someone who asserts the first one proposition and the another has thereby asserted both of them, licenses the Rule of Conjunction […]”¹

Notice that there is an ambiguity in the quote between asserting that P∧Q and asserting P and asserting Q. One might think there is no difference but there is. From the two assertions P and Q, one can infer the proposition P∧Q of course. But that does not imply that someone, S, who asserts both P and Q individually implies that S asserts the compound prop. P∧Q. Why not?

The inference rule from two props. to a conjunction of them is not surprisingly called conjunction. However what I want to attack is the stronger thesis that “someone, S, who asserts P and asserts Q” implies that “S asserts all props. that are logically implied by P and Q”. It might appear to be the case with respect to a single case of a single inference rule, but it is not the case with other cases. Let’s see two examples.

Think of the inference rule of addition (aka. disjunction introduction). Suppose now that S asserts P. Does that imply that S asserts all the infinite number of props. that are logically implied by P? I think not. That S asserts that “The Moon is round” does not imply that S asserts that “The Moon is round or his dog is president in the USA”. Notice that one can add any number of extra disjuncts to this proposition and it is still logically implied by that “The Moon is round”.

Return to the inference rule of conjunction. Suppose that S asserts that “The Earth is round”. Does that imply that S asserts all the props. that are logically implied by “The Earth is round”? No. For instance that S asserts that “The Earth is round” does not imply that S asserts “The Earth is round and the Earth is round and the Earth is round”.

So why might it appear as though “Someone who asserts P” implies that “S asserts all props. that are logically implied by P”? Perhaps because one confuses two usages of “and”. One is the one used to bind things together grammatically (the grammatical “and”) and the “and” which is used inside proposition expressing sentences to express the concept of conjunction. Consider

1. S asserts that P and Q.

(1) is perhaps ambiguous between

2. S asserts the compound proposition P∧Q. and

3. S asserts that P and S asserts that Q.

The authors seem to fail to notice this distinction.

Alternative theory of beliefs

One might alternatively accept the thesis, that is, “S asserts P” implies that “S asserts all propositions logically implied by P”. If that is the case, then when we assert something we also assert an infinite number of other proportions. I use this result as a basis for an MT argument but one could simply accept it as a fact and use another theory of assertions, perhaps something like the the dispositional theory about beliefs. The analogy with the case with beliefs is quite strong.² If S asserts P implies that S believes P, then this is extremely similar. However one can assert something without believing it, so the conditional from before is false. One could crate a true version of it by adding “sincerely”, so if S sincerely asserts that P, then S believes that P (at the time of the assertion).