Formal Logical Analysis

The method here described is intended to assist in the acquisition, promulgation and exploitation of knowledge.
It concerns the kinds of knowledge which can be viewed as the possession of a useful model of some subject matter, and which is exploited by reasoning or computation, about or with the model, to draw conclusions which prove reliable in relation to the intended subject matter. |
The method contributes to this process first by providing ways of constructing models which are precise and unambiguous, and secondly by providing ways of ensuring that the conclusions drawn from the models are correct.
An important feature of the method is that by enabling the precise formulation of models it facilitates the use of computers both in the construction and the exploitation of the models. |

The method of formal logical analysis is intended for application wherever arguments are under consideration which are though to be, in part or in whole, deductive.
A deductive argument is an
A deductive argument may also be Once these rules have been shown to be sound with respect to the semantics of the notations, checking the correctness of arguments is reduced to a mechanical process. |
The premises from which the conclusions have been deduced represent our knowledge of the subject matter under consideration, and should be examined carefully to secure a proper understanding of the scope of applicability of the conclusions.
Usually informal arguments fail to make clear what their premises are. This makes it difficult to check the soundness of the argument, and makes it difficult to determine in any particular circumstance whether the argument is applicable and its conclusion true. Formal analysis involves exposure of the premises of the argument so that their consistency can be established, so that the soundness of the argument can be checked and so that the scope of the conclusions is made clear. |

The process of establishing the premises for a deductive argument can be beneficially thought of as defining or constructing an abstract model of the intended subject matter.
This process, which may be thought of either as creative abstract modelling, or as conceptual analysis, and which may extend to the development of new notations, is the most creative and beneficial aspect of logical analysis. It is the failure to give proper attention to this activity which is the most damaging defect of wholly informal methods. |
Any deductive argument will proceed from some finite set of premises about its subject matter.
These will normally be only a partial characterisation of the subject matter and will then be true of a wider range of systems.
These system will be the models of an abstract theory which treats the essence of the concepts whose logic is exploited in the argument.
In all areas of knowledge, advances are dependent on the development of new terminology, concepts, or notations. These permit expression of those finer distinctions upon which the advance of knowledge so often depends. |

The use of formality in logical analysis, which may be thought of primarily as permitting the detailed mechanical checking of deductive arguments, is in fact much more significant in enabling the meaning of new concepts and notations to be located more precisely than they can be by informal means.
In the development of modern mathematics, Frege first devised notations suitable for formal development of mathematics. The greater precision of these notation made it possible for Russell to locate a defect in the reasoning which they supported (now known as Russell's paradox). This in turn prompted further developments in our understanding of the formal theory. In particular, they resulted in the axiomatisation of well-founded set theories backed by the semantic intuitions constituting the iterative conception of set. |
Without the formal activities it is doubtful that we could have arrived at that clearer appreciation of the universe of well-founded sets which has been the basis for the development of mathematics in the 20^{th} century.
However, once the notion of set has been pinned down in this way (notwithstanding that there remains some looseness), the precision can be imported back into the less formal notations employed by working mathematicians.
The central plank of the method of formal logical analysis is the use of mathematical models to pin down the meaning of the concepts under consideration. This brings semantics into the foreground and encourages recognition that when we wish to use language precisely we must chose the precise meanings of concepts rather suppose that precise meanings can be distilled from precedent or can be done without. |

© created 1997/10/29 modified 1997/12/1