Not sure what you are saying. Informative content has both intrinsic (randomness deficiency, entropy, kolmogorov sufficient statistic) and relational properties (also randomness deficiency, mutual information, algorithmic mutual information) that can be objectively and quantitatively measured.
Intrinsic/relational may not be the best way of getting at it after all. It would likely regress into an equally involved question on defining "system".
So let me try another approach. We need to distinguish between information and the "semantic content" of information. Here are the definitions I have in mind:
- Data: an arbitrary string
- Information: a numeric property of some data: INFO(d)
- Mutual information: a numeric property of two pieces of data: MI(d1, d2)
- Semantic content: the particular "effect on a system"[0] that results when the system internalizes the data: SC(s, d) ('s' denotes the system.)
My understanding is that you're claiming MI == SC.
There are certain ways we could fiddle with my usage of 'system' above so that it can just be 'data,' too (e.g. representing the system as a string, as with Turing Machines that operate on string reps of Turing Machines). In which case we can at least say that MI and CS both have domains like (data X data), so there is some similarity there.
But, even just the fact that MI is going to evaluate to a single numeric quantity, while SC evaluates to an "effect on a system" (i.e. following a state transition), implies that they're referring to different things.
I think it's also clear that it would be possible for MI(d1, d2) to evaluate to the same number as MI(d3, d4), even though SC(d1, d2) !== SC(d3, d4), since all the informational relations could be equivalent (between (d1, d2) and (d3, d4)), and yet the particular state transitions followed could differ.
[0] We can make this more precise. For instance, in the language of automata theory it would be something like following a particular state transition.
Edit: I should clarify one implicit thing here: this hinges on an assumption that my definition of "semantic content" would be a satisfactory match to common usage of the phrase for most people. Happy to hear an alternate that also captures common usage, and I can expand on my choice of def.
Great definitions. I would make a distinction between the measure of mutual information and the mutual information itself. The former is obviously not the same as the system effect itself. But the latter may be more related to your SC function.
> But the latter may be more related to your SC function
Definitely seems related to me. It's an interesting angle on it I hadn't considered before.
One idea is that it's due to something like compatibility of formats: in order for some information to be 'meaningful' in the context of some other system, its format has to be something recognizable to that other system; and the ability to recognize implies some commonality[0]. That commonality implies a certain amount of mutual information.
I don't feel certain that the mutual information would be sufficient to account for all the effects of semantic content—but I wouldn't be surprised if it was ;)
[0] I wonder if this is something that's been studied: mutual information between patterns and recognizers of patterns... I could see there being some lower bound on mutual information.
