ABSTRACT
Formalisation and Axiomatisation In this chapter we shall discuss formalisation and axiomatisation. We start with a discussion of formalisation.The importance of formalisation lies in the reverse process from providing a full behavioural context for linguistic behaviour. Carnap, Davidson and Chomsky are justified in their approach precisely because they have formalised their theories and examined their consequences in the light of a consideration of language in purely linguistic terms. This is a process of translating from one level of description to a more “precise” level, and entails that we supply the necessary translation rules. It can be carried through (as we shall illustrate) in stages.After we have provided this first stage formalisation, at least as far as the central core of a theory is concerned, we shall proceed to a further stage of formalisation by suggesting, for example, suitable models, such as automata, that might be regarded as appropriate as an underlying structural model for the suggested theory.But before we start the formalisation, let us say a few more general words about the process of formalisation. Pap (1949) says of formalisation: The process of making the logical structure of a language, or of sentences, explicit by replacing descriptive terms with variables. The formalisation of a language also involves the statement of formation rules, transformation rules
Braithwaite (1953) has proposed what he called a “zip-fastener” approach to formalisation, where we can zip down from the theory to the model or up from the model to the theory.The idea of using models in science is by no means new, and Braithwaite, among many others, has analysed in some detail the use of such models and their relation to theories in science. There is a sense in which almost anything can be used as a model for almost anything else, but as in the use of analogies or metaphors, so models, to have predictive value, must bear some measure of similarity to the structure or process being modelled.The idea of modelling (cf. analogy) depends upon similarity of structure and/or function, as in the case of a wind tunnel, a painting, even a person who says “let this piece of paper here represent Africa and this South America . . .”.In science we try to make such models explicit and more precise and tie them to the theory they are supposed to clarify and exemplify by some sets of rules which we usually call translation rules. The operations involved frequently include logic and whereas at the level of the “process” modelled (e.g. a scientific theory or the process it purports to represent) the (descriptive) statements may be empirical, at the level of the model the statements (formulae, sentences, etc.) are formal. This is of course one of the ways in which the formal and factual sciences meet.Following Braithwaite’s line of thought, we might regard scientific theories, for example, as capable of being formalised. This process of formalisation is essentially one of stripping down theories in ordinary language and showing the underlying logic. It is almost, indeed, a matter of making the original theory more precise, where we take some theory and re-write it in greater detail and in more ‘molecular’ fashion; this is the “zipping down” process.We can also argue that a scientific theory is made up of empirical statements which use terms capable of operational definition, whereas the model uses logical statements, which are
not in themselves verifiable. The opposite (interpretative) process of going from model to theory is a “zipping up” process. The implication of the zip-fastener theory is that the alternative processes are interlocked at each stage of the translation from model to theory.An example of a formalisation we might mention is that of Whitehead and Russell of the concept of number in Principia Mathematica. Another famous example from Biology is that of Woodger (1952).By ‘formalising’ then we mean ‘making more precise’ or ‘reducing statements’ to their underlying logic, and clearly ‘formalising’ is itself a vague word, which could well mean ‘to expose the model underlying the theory’. Not only is the process of formalising vague, but it is also a matter of degree.Similarly we can look at the matter the other way around, and interpret models in many different ways to get many different theories. Models and theories are connected by formalising and by interpretation, and theories are linked directly with empirical descriptions. Let us look now at an example of interpretation, e.g. the well-formed formula:
p D (pD q) is from an axiomatic system of the Propositional calculus and can be interpreted so that jfr, q are propositions and D is the traditional logical connective called ‘material implication’. The dot is a bracket as illustrated by our alternative rendering of the formula. This interpretation is a part of the Proposition calculus.But the identical statement p D. p Dq could be interpreted as a statement in a Boolean algebra B. When the second interpretation is adopted, a conventionally different notation is used. So we have:
A^> (4 -> B)
where A' = d f not -Aand AU B = d f the sum of the two classes A and B, then A -> {A -> B)
becomes A'U {A'U B)
The fact is that the same model is being given two different interpretations.Here the process could be described as taking a model or structure, which by itself has no meaning or reference, and supplying the model with the meaning. The word (name) ‘Chicago’ refers to the city Chicago, only by the common agreement that ‘Chicago’ is the name for Chicago. All the problems of naming and predicates as raised by Frege and Russell and mentioned more recently by Geach (we mentioned them ourselves in Chapter 9) are to be borne in mind.
