“The words ‘to imply’ in the English language mean the same as ‘to contain’ or ‘to involve’…a confusion of implication with the consequence relation…” - Rudolf Carnap (1937)
If A has no meaning (or intension), then how can it imply or entail B? If A is just an inscription (or is self-referential), then what connection could it have to B?
On one reading, A implies B because A “contains” or “involves” something that's also in B. This is the standard Kantian view of of conceptual analyticity. B, on the other hand, can be a consequence of A without A “containing” or “involving” something that's common to A.
We can ask what exactly does it mean to say that A is “contained” within B. Quine accused Kant of speaking at a metaphorical level when he used the word “containment”. However, what non-metaphorical way of describing what's at issue here have we got? If A is simply an inscription (or "syntactic form"), then of course it can’t contain B – it can’t really contain anything except itself.
A will require content if it's to imply B. In that case, it all depends on what the symbol ‘A’ stands for. Is it a concept, sentence, statement or a proposition? All these possibilities have content.
If the symbol ‘A’ stands for the concept [politician], then what content will it have? We can say that contained within the concept [politician] are the macro-concepts [human being], [person]; as well as the micro-concepts [professional], [Member of Parliament], etc.
In a certain sense it's quite arbitrary to categorise certain concepts as micro-concepts and others as macro-concepts because it all depends on context. However, we can say that in one context we can categorise the concept [politician] as a micro-concept.
There's a simple way to decide what's what. We can ask ourselves this question:
Is it necessary for a politician to be a person or a human being?
The answer is 'yes'.
Is it necessary for a human being or person to be a politician?
The answer is 'no'.
Thus, in this simple sense, the macro-concepts encompass the micro-concepts. Of course there are also yet higher levels of concept.
Take the concept [animal]. This would include the concepts [human being] and [person]. And there are yet higher order concepts than that: e.g., [living thing]. This could go on until we reach the concepts [object], [thing], [entity], [spatiotemporal slice] and so on.
Thus if A is taken to be a concept, then it may well have a huge amount of content. It could imply all sorts of things. Indeed it's a strange thing to take A as simply standing for a single concept. (It's hard to make sense of a concept all on its own.) Thus we need to fill in the dots ourselves.
If the symbol A is a sentence, then things become a little clearer and not as broad-ranging. The sentence may of course include concepts; though concepts within a sentential framework will probably be more finely delineated and circumscribed. Something will be said about the concepts contained and they'll also be contextualised.
To say that the concept [politician] implies the concepts [human being] and [person] sounds strange. In a sense, the bare concept [politician] isn't actually saying of implying anything. Thus the idea of containment must be taken more literally in the case of concepts standing for A than if A stands for a statement or sentence.
How would B, therefore, be a consequence of A if the relation between A and B were not one of conceptual containment? In nature, B can be caused by A without sharing anything with A. Perhaps B can also be deduced from A without sharing anything. If so, how would that deduction actually come about?
Carnap, Rudolf. (1937) The Logical Syntax of Language