IDEAL

Iterative Dialectic Engine for Automated Learning


General principles and particular instances

What constitutes 'learning'? Is it the accumulation of known generalisations about the physical world - termed 'general principles' in the definition of induction - or is it the accumulation of notable experiences - termed 'particular instances'? In his attempt to reconcile the theories of Popper (1) and Kuhn (2), Lakatos stresses the importance of both theory and experiment:

Mature science consists of research programmes in which not only novel facts but, in an important sense, also novel auxiliary theories, are anticipated; (3)

Both, then. But how do general principles and particular instances relate to one another? Specifically, do general principles imply particular instances, or the other way round? Lakatos, like Popper before him, employs modus tollens in the defence of falsification, such that te, i.e. theory implies experiment (where 'implies' is taken to mean logical entailment, using the propositional calculus). However, it can be argued that Lakatos and Popper are wrong and that et, as follows:

  1. The axioms of propositional logic correspond precisely to the axioms of basic set theory (4);

  2. As part of this equivalence, AB (A implies B) corresponds to AB (A is a subset of B);

  3. Sets may be represented in Venn diagrams, where, loosely-speaking, the size of a set is proportional to its generality. Thus, for instance, on the Venn diagram the set of black things might occupy more space than the set of crows or the set of Model T Fords. Accordingly, one is more likely to suppose that the set of crows and the set of Model T Fords are subsets of the set of black things, rather than the other way round;

  4. Likewise, one might suppose that 'particular implies general', i.e. et.

Falsificationists may well contest the legitimacy of step 3, but the fact remains that their modus tollens argument is not as dependable as they might have supposed. For the rest of us, this dispute indicates that mathematical logic on its own does not help our understanding of the relationship between general principles and particular instances. In any case, it turns out that we never compare general principles with particular instances, whether in Venn diagrams or anywhere else. If we were to set down the informal procedures by which we undertake practical problem-solving, it is likely that we would identify the following:

  1. Faced with a new problem (i.e. a situation requiring our response), we first look for similarities with earlier experiences (i.e. particular instances stored in memory);

  2. The retrieved particular instances are down-selected to a manageable number having the greatest correspondence with the new situation;

  3. For each of the particular instances selected from memory, the general principles governing the associated responses are recalled;

  4. These general principles are applied to the new situation, with minor alterations to take account of the differences between the new situation and the remembered particular instances;

  5. If the new situation and/or our response to it are particularly novel then we commit both to memory, as a particular instance and a general principle (or set of general principles) respectively. These memories are correlated so that this experience may contribute towards future problem-solving (step 3).

In other words, particular instances are compared with particular instances, and general principles are compared with general principles, but 'never the twain shall meet', i.e. particular instances may be correlated with general principles but they will never be compared with them.

It is concluded that any design of an automated learning engine must be able to distinguish general principles and particular instances, and it must be able to compare and correlate input data and stored data.

References

  1. Karl Popper, The Logic of Scientific Discovery (Hutchinson 1959).

  2. Thomas Kuhn, The Structure of Scientific Revolutions (Chicago 1970).

  3. Imre Lakatos, Falsification and the Methodology of Scientific Research Programmes, in Imre Lakatos and Alan Musgrave (eds.), Criticism and the Growth of Knowledge (Cambridge 1970).

  4. Nick Earle, Logic (Macmillan 1973).

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Copyright © Roger Kingdon 2004