Webster's Dictionary, for example, shows us the following loop of definitions:
defines information through knowledge
defines knowledge through fact
defines fact through information
Is it possible to define the term “information” without a loop, that is, with the help of some fundamental term?
(1) In a dictionary we find impressionistic characterizations of various semantic meanings of words and phrases - meanings that may be tied to very different contexts. If you just pick and choose arbitrary characterizations it's pretty easy to find the kind of loops that you pointed out. Those are really not relevant, since that's not the way a dictionary should be/is intended to be used. A dictionary is not intended as a comprehensive theory of all words in a language.
(2) It's not possible to define every word in terms of other words. Or rather, it is possible, in principle, but it's trivial to see that this would lead to circular chains - like the one you found.
(3.1) It is possible to define information without using a circular chain of definitions. The simplest way, is just stop after the first or second definition. Is that cheating? No. What I'm getting at is: What matters is whether the definition makes sense to you, whether it is a useful definition for whatever you want to do with that term. -- This is not to deny that if you still need clarifications after two steps, and someone would propose the third step, you're perfectly right in saying: "Sorry, you just made a circular definition, I'm still not in the clear about what you meant with your original term."
Ultimately, what matters most, perhaps, is how powerful a definition is (in how many contexts, for how many people, it is useful and brings things together).
(3.2) Claude Shannon gave a powerful definition of information, or rather of what it means to be informative or carry information, without circularity. His definition of x is informative (to somebody, in some context) boils down to x is surprising (to that person, in that context). Something is more informative in sofar as it is more surprising (less probable).
Information (physics, computer science, cryptography) is a characteristic of the latter of a pair of measurements or a pair of sets of related measurements, whose numerical value is:
I = -log(p(E|C))
Read: The information obtained by measurement of event E under measured conditions C is equal to opposite the logarithm (base 2) of the probability of E given C.
By a convention artfully designed to confuse undergraduates, "probability of event E given conditions C" is abbreviated "probability of event E" and expressed:
I = -log(p(E))
For instance, if we roll two fair dice and sum them, the information obtained by measurement of the number 2 (which will come up on 1 out of 36 possible rolls of a fair pair of dice) is
I = -log(1/36) = 5.170
While if we roll one die first and observe that it is a 1, the information obtained by measurement of the number 2 on the sum of the two dice (which will come up if we roll another 1) is
I = -log(1/6) = 2.585
The mathematical formulation is arbitrary, but it is chosen this way to conform to common use and professional convenience in the following ways:
a message Z times as long will have Z times bigger I for a given specificity of outcome. Note how when we rolled two dice, requiring the two dice to both come up a particular value, I was double the value as when we fixed one of the dice and rolled just one of them.
more surprising outcomes (such as rolling a 2 on two dice, versus rolling a 7) are represented by bigger Information.
the formulation allows for mathematical treatment similar to thermodynamic entropy in mathematical physics.
using a base-2 logarithm makes the maximum information obtained by a measurement whose result is expressible as a base-2 string (such as a computer file), equal to the length of that string. Since most of our information transfer and storage is done in base-2 strings at the moment, it's a convenient choice of logarithm base.
Definition of information depends heavily on the field of study. Information theory says it is the reduction of uncertainty. In computer science it is processed , organized and structured data. In quantum mechanics it is the state of system. In biology it is DNA , RNA . In communication theory it is the content delivered through communication channel.
However we are missing one field of study which is consciousness. In consciousness, the information develops with the question “what?”. What is that exists ? Earth exists , food exists , smell exists , fire exists, God exists etc etc … In consciousness we are the consumers of that information. Given the base knowledge we ask further questions like Why that exist ? What should we do about it ? Why should we do that ? How should we do that ? When should we do that ? Etc… Information builds up starting with the question “what exists?”
Information at least answers one fundamental question: What exists ?
And both the question and answer depends on understanding in the consciousness. Information can not be seperated from consciousness. Only conscious beings understand what information is.
Unequivocally, yes, information can be defined rigorously. Four starting places to study a rigorous notion of information are Shannon, Chaitain, Floridi, and Shagrir and Piccinini.
Bumble's comment is both canonical and strong. Information theory as a description of a problem space in deciphering messages goes back quite a way to Shannon, a contemporary of Turing, and eventually was adopted into disciplines like control theory. The gist of the position is that a mathematical analysis can be conducted on codes and used to understand meaning. Gregory Chaitain is another towering figure in the study of information. He has pushed a number of important ideas on the relationship of randomness and measures of information. In addition, Luciano Floridi has a sweeping theory of information called The Philosophy of Information (GB) which develops a metaphysically rigorous account of information. Lastly, if you are interested in a rigorous notion of information, you might be interested in the notion of physical computation (SEP), which views information through the lens of physical systems that compute.
