By Terry Oldberg
Introduction
In building climate models, climatologists generalize. Can the means by which they generalize be improved?
Yes they can. The means can be improved by replacement of intuitive rules of thumb called “heuristics” by the principles of reasoning.
A model is a procedure for making inferences. In each instance in which an inference is made, there are many candidates for being made; often these candidates are infinite in number. Which inference among the candidates is the one correct inference? The model builder must decide, but how?
In theory the one correct inference is identified by the principles of reasoning. Logic is the science of these principles.
In reality there is a stumbling block. While Aristotle left us the principles of reasoning for the deductive branch of logic he failed to leave us the principles of reasoning for the inductive branch of logic.
The inductive branch contains the principles that govern generalization. In building a model, the model builder must generalize.
The problem of extending logic from its deductive branch and through its inductive branch is called “the problem of induction.” Many scientists and philosophers believe this problem to be unsolved. In 2005 the professor who taught Logic I at M.I.T. told his students that the problem was unsolved.
Model builders who hold this belief cope with the apparent absence of the principles of reasoning; they cope through the use of intuitive rules of thumb called “heuristics” in discriminating the one correct inference from the many incorrect ones. Maximum parsimony (Ockham’s razor) is example of a heuristic. Maximum beauty is another example. However, in each instance in which a particular heuristic identifies a particular inference as the one correct inference a different heuristic identifies a different inference as the one correct inference. In this way, the method of heuristics violates the law of non-contradiction. Non-contradiction is a principle of reasoning. Thus, the method of heuristics is inconsistent with the principles of reasoning.
I’m going to argue that those scientists and philosophers who believe the problem of induction to be unsolved are wrong in this belief. I’ll argue that they are among the victims of a long-running lapse in communications, for the engineer-physicist Ronald Christensen solved the problem 47 years ago.
After solving the problem, Christensen published his findings but few people recognized their significance. When he submitted a paper to a philosophical journal, it was rejected. When he submitted a book for review to a journal of mathematical statistics, the reviewer stated that he couldn’t think of a single person who would profit from reading it.
A few years ago, while in the library of a renowned research university, I stumbled upon the set of books by which Christensen had tried to communicate some of his findings. In the period of 25 years over which the books had lain on a shelf each of them had been checked out only a few times.
Later, I delivered an offer of help in the replacement of the method of heuristics by the principles of reasoning to most of this university’s scientists; 100% of the recipients spurned my offer. When I offered to address the university’s philosophy department, this department spurned my offer. A professor at the same university spurned my offer of help by stating that he could conceive of no use for my ideas in his field of research. Another professor explained the behavior of his colleagues by suggesting that, like mules, his colleagues were “set in their ways.”
I’ve gotten similar responses from the faculties of many additional universities. The evidence seems clear that, like other scientists and philosophers, academic scientists and philosophers have not yet received Christensen’s message.
In meteorology, replacement of the method of heuristics by the principles of reasoning has produced great advances in predictability. In climatology, advances of a similar magnitude may be available.
With this essay, I begin a two part series on the principles of reasoning. In part I, I set up the problem of induction for solution. In part II, I solve this problem. Solving it identifies the principles of reasoning.
Abstraction
Part I features the idea that is implied by the semantics of the word “model.” This is that a model is not reality itself but rather is an abstraction from reality.
Abstraction is a logical process. It operates on two or more of the states of nature that I’m going to call “constituent states” to produce the state of nature which I’m going to call an “abstracted state.” Abstraction produces an abstracted state by placing the constituent states in inclusive disjunction. An illustrative example follows.
In the example, the constituent states are male and female. Placement of these states in inclusive disjunction produces the abstracted state male OR female where “OR” designates the inclusive disjunction operator.
A “state” is a description of an entity in the real world. To attach the adjective “abstracted” to a state that is formed by abstraction is apt because this state is abstracted (removed) from the differences among the constituent states; for example, the abstracted state male OR female is removed from the gender differences between its constituent state male and its constituent state female. In this way, abstraction removes detail from the description.
In the construction of and testing of a model, abstraction has the merit of increasing the level of statistical significance. It increases this level because the descriptions of more observed statistical events are apt to match an abstracted state than to match either of its constituent states. For example, the descriptions of about twice as many people match the abstracted state male OR female than match the constituent state male or the constituent state female.
To continue I need to reveal what I mean by a “model.” Associated with every such model is the set of this model’s dependent variables and the set of this model’s independent variables. Each such variable is an observable feature of nature or can be computed from one or more observable features.
In making the following description, I use the words tuple, Cartesian product space and partition as they are used in mathematics. Each dependent variable takes on the values in a specified set of values. The set that contains one of the values of each dependent variable is an example of a tuple. The complete set of these tuples is an example of a Cartesian product space.
This space can be divided into parts. The complete set of parts is an example of a partition. Each element of this partition contains a number of tuples. By placement of these tuples in inclusive disjunction, one forms an abstracted state. I’ll call this state an “outcome.”
A complete set of outcomes can be formed by abstraction from the tuples belonging to each element of the partition. This set is a kind of state-space. An example is {rain in Milwaukee in the next 24 hours, no rain in Milwaukee in the next 24 hours}.
By a process that is similar to the one described previously, a complete set of abstracted states may be formed from a model’s independent variables. I’ll call this set the “conditions.”
The set of conditions is a kind of state space. An example of one is {cloudy in Milwaukee, not cloudy in Milwaukee}.
The complete set of descriptions of the statistical events for the model is generated by taking the Cartesian product of the set of conditions and the set of outcomes outcomes contains the complete set. For example, taking the Cartesian product of the set of conditions {cloudy in Milwaukee, not cloudy in Milwaukee} with the set of outcomes {rain in Milwaukee in the next 24 hours, no rain in Milwaukee in the next 24 hours} generates 4 event descriptions; one of them is {cloudy in Milwaukee, rain in the next 24 hours}.
An “observed” event is a consequence from an observation that was made in nature. The complete set of a study’s observed events is an example of a statistical population.
From a population of this kind, a sample may be drawn and the observed events in this sample which match a particular event description may be counted; for example, observed events matching the description {cloudy in Milwaukee, rain in the next 24 hours} may be counted. This count is called the “frequency” of the description. The ratio of the frequency of a particular event description to the sum of the frequencies of all of the various event descriptions is called the “relative frequency” of the event description. The relative frequency in the limit of an infinite number of observations is called the “limiting relative frequency.”
A “prediction” is an extrapolation from the known condition of an event to the uncertain outcome of this event in which the outcome lies in the future of the condition. A prediction states a falsifiable claim about the numerical value of the limiting relative frequency of each of outcome given each condition.
Posing the problem of induction
I’ve completed my description of the mathematical entity which I call a “model.” The completion places me in a position to pose the problem of induction.
The problem is a consequence of the fact that partitions of the Cartesian product space of the model’s independent variables are of infinite number. Each of these partitions is associated with a different set of conditions. Among the various sets of conditions, a single set is associated with a model that makes no incorrect inferences. Conditions belonging to this uniquely determined set are called “patterns.” The problem of induction is to discover the rules under which the patterns are discovered. These rules are the principles of reasoning.