Nature of information

A piece of information I is a representation R of some fact F for some recipient X who did not know it was a fact. Consequently, I has the following properties:

If X already knows F then receiving R does not constitute information for him (although it may still constitute information for somebody else).
As I is a representation of a fact, it purports to be true. If X learns that what he took to be a piece of information was not true, then it ceases to be information for him. (He may, of course, reframe it, taking it as a piece of information on the reliability of its sender; but that is a different matter.)
The person S who conveys R to X typically presupposes that X does not know F. By conveying R to X, S purports to inform X.¹ Again, if X already knows F, then that intention of S's fails.

Moreover, the notion of information is typically restricted to contingent truths, i.e. to such Fs that could be otherwise. For instance, if at primary school I learn that three times three is nine, this may be considered information for me by the above definition. However, since it could not be otherwise, such a truth is typically not treated as information.

Amount of information

The amount of information conveyed by some message may be measured. This presupposes its analysis in terms of propositional calculus:

The message consists of a set of propositions. These are connected with each other by propositional connectives. The most important connections for present purposes are conjunction and disjunction.

Assume that p_i conveys an amount of information equal to 1.

Then the amount of information conveyed by p₁ ∧ p₂ ∧ ... p_n (n conjoined propositions) is equal to n;
whereas the amount of information conveyed by p₁ ∨ p₂ ∨ ... p_n (n disjoined propositions) is equal to 1/n.

Exemplifying: Suppose some of my pupils have painted the classroom. The proposition that Sam painted the classroom constitutes one bit of information. Compared to this,

the proposition that both Sam and Bill painted the classroom constitutes two bits of information;
while the proposition that either Sam or Bill painted the classroom constitutes just half of the information that Bill painted the classroom.

Joining more propositions by disjunction leads to reduction of information. A complex proposition of the logical form of the last example, joining the names of all of my pupils by disjunction, amounts to no information at all.

Temporality

As seen in §1, information is essentially news that is contingently true. From this it follows that a piece of information is bound up with time. In particular, it may expire. There are at least two ways that this may happen:

The fact represented by I may cease to be a fact. More precisely: Since a fact is a representation of some real-world state of affairs, it has certain spatio-temporal coordinates. Once the point or period of time T embodied by R is over, then R, stripped of its temporal coordinates T, is not a fact.
- Assume that the number of speakers of Lacandón Maya is 200 in the year 1989 (F), but 60 in the year 2009. Suppose further that I learn F in 1989. Then F is information for me. Suppose, instead, that I learn in 2009 that the number of speakers of Lacandón is 200. That would have been true 20 years ago. If I take it to be information that is true in 2009, then I am mistaken.
In order to avoid this, I want information to be provided by some time-stamp (or an expiration date, as the case may be). Information that is not tied down to some point or period of time tends to be worthless.
- For instance, I find a flyer of my local public transport company showing the net of bus connections. It contains no date of printing, and I have no other information to infer when it might have been produced. It would be unwise of me to wait for a bus on the basis of this “information”.
A piece of information may expire in the sense that it ceases to be relevant for me. It may still be true, but I can no longer make use of it.
- Suppose my wife is waiting for me at the main entrance of the train station to take me home. She sends a message to that effect to my cellphone. However, not checking my messages, I leave the station by the back entrance and take a bus home. Not finding my wife there, I check my cellphone messages. And in fact, she is still waiting there. However, now it is no longer relevant, and she will come home alone.

The two kinds of expiration must be distinguished.

The first case shows that if we receive some piece of information without a time-stamp, then by default we assume it to be true now. If that is not so, then the R we received was no piece of information.
Since the second case does not concern the truth of R, its status as a piece of information is not affected by its expiration. However, the typical role that a piece of information plays for X is for X to make use of it in his actions. If that is precluded, then I, although true, is of no value for X.

Although the two kinds of expiration are, thus, logically distinct, the consequence for X is in both cases that R – whether it be called information or not – is of no value for him.

¹ S may, of course, intend to mislead X.