From Science to Language

Doc, let us here go down the "technical" route, and take, as a starting point, the list of definitions contained in your latest comment (Sunday 2:14 pm). Among these notions, let us first isolate the following sub-list : Mathematics, Observation, Modeling, Model Non-Uniqueness, Science and Model Selection. I have no difficulty with them if they are considered as an account of science stricto sensu, that is to say of a specific human activity practiced by trained people, working in laboratories and conducting experiments or creating theories.

Now, what happens when we add the three other notions you mention to this basic list ? Let us first only add Logic. Here, it seems that some difficulty appear, even if we stay within the bounds of science, especially in relation to Model Non-Uniqueness. In theory, you can indeed have infinitely many models that agree exacly with a given set of observations or measurements. In practice, however, there is often a small number of models in competition, each approximately consistent with the set of measurments gathered so far. What if you have, say, two accounts of how complex ceramics manage to become supraconducting at "high" tempearture. For a given experiment, one will predict a value of 12.458, for the material's resisitvity (let us call it R), and the other 12.743. Measurement yield values hovering between 12.465 and 12.580. It seems that the first model is better than the second thus, if many other experiments confirm it, the first theory will be chosen to become the established one. We will then have people routinely using the following logical entailments : let us call E the set of statements fully describing the experiment above, and P the statement "R = 12.458". According to the chosen theory, we have E => P. Then let P' be the statement "R between 12.465 and 12.580" (the experiment results). We have P' => ~P. If we conduct the experiment, we suposedly make E and P' true. Then, by logical entailment, we have E & (E => P), thus P and also P' & (P'= > ~P), thus ~P. Hence, we have P and ~P at the same time !

Well, I am not the first one to point out that formal logic has applicability problems. What is striking is how difficult it is for us to see these problems. We have been trained, on the one hand, into believing that a proposition must be true or false, so we say "yes obviously, we cannont hold simultaneously that P and ~P". On the other hand, thankfully, we continue to behave in a pragmatic fashion, as the example above shows. But our training has become so ingrained that we do not notice that our own practice violates the logical theory.

Now, let us consider the full list : the items on science + Logic and the last two, Completeness and Natural Language. Before proceeding further, I feel some additional input on your part would help me. First, does 'natural language' cover all of it, with all its actual and possible uses, or just a subset ? Also, what becomes of 'meaning' ? It seems to me that an account of natural language cannot do without some explanation of that notion. Lastly, how do you intend to use 'Completeness' ? Will there be some sort of test or procedure to decide, for a given statement, whether it is complete or not ?