[ Having established the distinction between logical and empirical truths we now turn to a closer consideration of logical truth.
There are two main considerations. One is the scope and relevance of logical truth. This should include a discussion of its place in analytic philosophy, in mathematics, empirical science, and engineering. A principle aim in this discussion is both to re-emphasize, as Hume did, the severe limits to what can be established purely by deduction, and to show that it nevertheless is of the greatest practical significance. The limitations and potential are to be made real by illustrations of a different character to those of Hume but which are connected with the various concerns of my own positive philosophical thinking, theoretical and practical. Specifically in relation to philosophy, some discussion of the nature of philosophical analysis making the distinction between a claim being analytic and a claim being ``about language'', i.e. between subject matter and epistemic status.
I would like to introduce here the notion of an analytic oracle, which can then be refined in the next chapter to the FAn oracle.
We have seen that the words ``analytic'' and ``synthetic'' acquired in the philosophy of Kant a sense distinct both from their use outside academic philosophy and from previous philosophical and mathematical usage.
In non-philosophical use the related terms analysis and synthesis have diverse application, generally concerned with taking apart or putting together the parts of some complex whole. In the special domain of logical proof, the usage in classical Greece was similar, connoting two methods of proof. An analytic proof proceeded by analysis of the proposition to be proven, ultimately reducing it to principles which can be known without proof. A synthetic proof begins instead with axiomatic principles reasoning forward until the desired theorem is eventually reached. In contemporary automation of reason, these different methods, which we will consider further in a later chapter, are sometimes known as backwards and forwards proof respectively.
Kant introduced a new usage in which analytic and synthetic are applied to propositions (in the Aristotelian sense), classifying them along the lines of Hume's fork. Kant gave two criteria for this classification, one proof theoretic (concerning the manner in which such a proposition might be established), and the second semantic concerned with meanings. However characterized, analyticity in this sense is closely connected with deductive reason, and it is our purpose in this chapter to relate this precise technical concept with the very general notion of analysis, both in its academic and its more worldly applications.
Though we are concerned here with analysis in general, there is one particular kind of analysis which will have special place. This kind of analysis is closely coupled with the notion of analyticity through the concept of deductive soundness.
The notion of soundness is applicable both to formal logical systems which are supplied both with a semantics (an account of the meanings of the sentences of the language) and a formal notion of derivation or proof, or to particular inferences in any language for which the semantics is well-defined. In the former case the system is sound if the inference rules respect the semantics in such a manner that from true propositions only true conclusions can be derived. In the latter case, without reference to any particular deductive system we may say that an inference is sound if the premises entail the conclusion. A set of sentences (the premises) entails some other sentence (the conclusion) if under all circumstances whenever the premises are true the conclusion will also be true.
The connection with analyticity is then that all claims about entailments are true if and only if analytic. An alternative statement of the connection is that all propositions derivable from the empty set of premises, the theorems, of a sound deductive system are analytic.
Because of this connection, Rudolf Carnap and the logical positivist held that philosophy, which they thought of as an a priori science consisted of analytic truths. This involves a position on the demarcation of philosophy which we will not adopt here. Instead we simply note the scope of this particular kind of analysis, and of the kind of philosophy which employs it. Philosophy falling within the scope of such methods can now be made as rigorous and reliable as mathematics by the use of modern formal languages and computer software which assists in the construction of formal proofs and automatically checks that the proofs are correct.
Abstract logical analysis consists in the application of deductive reasoning to analysis of arguments, of concepts, of theories or doctrines. The application to arguments may be considered to be primary, and in this case the idea is that in any domain in which reason is thought to be applicable, modern logical methods can be used to improve the rigour of the reasoning. The idea here is much as it was in Plato, it is that the standards of rigour which are generally found in mathematics and generally lacking elsewhere, can be made more widely available by appropriate methods. In particular, by focussing on the concepts involved, by ensuring that we have clear definitions of these concepts, we can reason within the relevant domain rigorously.
Plato, Aristotle, and the many logicians who followed them until very recent times, failed to realize this ideal of logically rigorous reasoning beyond mathematics. The greater difficulty in pinning down non-mathematical concepts may have been a factor here, and it is not until the century that logic was progressed to the point at which the logic itself could be used in making precise definitions from which one can reason with formal rigour.
Other points of controversy important to this project remain, concerned with the scope, applicability and importance of analytic truth. These are important to us here for two distinct kinds of reason.
The theoretical core of Positive Philosophy, Metaphysical Positivism, is primarily an analytic philosophy, and it is essential in presenting a conception of analytic philosophy to address some of the reasons why a purely analytic philosophy might be thought to be of narrow scope and limited value.
The project we are considering is of broader scope, just as were the projects of Leibniz and Carnap, encompassing not merely an approach to philosophical analysis, but also important and substantial parts of mathematics, empirical science, engineering and other activities in which deductive reasoning might play a significant role.
There is a circle to be squared here. There is one perspective from which the entire enterprise is without content, and represents a preoccupation with academic trivia, and another diametrically opposed perspective in which it is of the greatest practical significance and may be thought to warrant substantial and energetic prosecution.
I have presented a history of the evolution of the concept of analyticity and related concepts, and have adopted in Metaphysical Positivism a conception which is similar to that described by Hume ``relations between ideas''. A good recent account of this concept may be found in the writings of Rudolf Carnap, but more importantly analyticity is a characteristic of the theorems of a large class of modern tools for constructing and reasoning in formal logical theories. Further sharpening remains of interest from a philosophical point of view, and will be discussed later, but for practical purposes, even its relevance to applications demanding the very highest standards of rigour, the concept and our technologies for checking when it applies are sufficient.
It remains a matter of controversy how significant analyticity, analytic truths, and methods in which they figure prominently are or might be.
Analytic truths may be thought of as falling into two principal groups.
The first group consists of true claims in languages whose subject matter is entirely ``abstract''. For this we understand abstract entities as mere ideas, about which we reason by offering a ``definition'' of the relevant domain of entities and then reasoning logically from the definition to conclusions about that domain which will be true by definition. This group encompasses the whole of mathematics. It also encompasses reasoning about abstract models of any domain whatever, whether or not one considers the model to be ``mathematical'', so long as the model is well-defined and the method is deductive.
This is a Platonic conception of the scope of deductive knowledge, and can be broadened very broadly in the direction which Aristotle took, to yield a body of analytic truths which might be thought to be about the material world rather than purely concerned with abstract entities. There are several ways in which this can be done in the context of modern logic without embracing the complexities of Aristotelian metaphysics.
Carnap's approach is to adopt formal languages whose subject matter is the material world, to define the semantics of the languages by giving the truth conditions, and to define analyticity in terms or such a truth conditional semantics. The analytic truths are then those sentences in such languages whose truth conditions are invariably satisfied.
The term analytic philosophy was first applied in the century to a kind of philosophy which began at the turn of the century as Bertrand Russell and G.E.Moore . These two men, though sharing the idea that philosophy should be in some sense analytic, had quite different conceptions of the kind of analysis involved, and their differences remained significant as methods of analysis evolved throughout the century.
Russell conception of analysis was shaped by the new methods in logic in the development of which he played an important part. He believed that failures of rigour in philosophical reasoning resulted in some cases from imperfections in ordinary language, and could be avoided if a special ideal language were adopted along the lines of the Theory of Types which he devised with A.N. Whitehead for the formalization of mathematics in Principia Mathematica. He did not advocate or practice the adoption of such formality in philosophical rather than mathematical reasoning, but did advocate the adoption of similar logical methods. An example the kind of method he envisaged is the use of logical constructions. By such means Russell advocated a ``scientific'' philosophy in which logical methods transformed philosophy into a rigorous deductive discipline progressively advancing in a manner similar to that of mathematics and science, rather a perpetual sequence of conflicting theories and a lack of solid progress. We may recall Plato's prescription here, in which the subject matter of philosophy and the only place where true knowledge might be attained is in the world of ideal forms. Even though Russell was an empiricist, his conception of philosophy is as a deductive discipline.
On the other hand Moore's conception of analysis concerned natural languages and consisted in the clarification of concepts in those languages, yielding illumination consistent with common sense. There is a connection here with Socratic method, but neither Socrates nor Plato conferred such authority upon ordinary language and common sense. Socrates sought the true nature of concepts such as justice and virtue, and though he believed that ordinary men were in some sense possessed of these true concepts, they nevertheless might not correctly grasp them until they are induced by a Socratic dialogue to properly recall this knowledge.
Is it the case that analytic philosophy in the century was narrower in its scope than philosophical tradition from which it grew, and if so is this a necessary consequence of the conception of philosophy as analytic, or of some particular ideas of what kind of analysis was at stake.
Two mid century conceptions of the nature of analytic philosophy illustrate the issue, those of Rudolf Carnap, exhibited in his Philosophy of Logical Syntax and the subsequent developments of his ideas, and those of the ``linguistic philosophy'', exemplified by J.L.Austin, which prevailed briefly in post war Oxford.
A central thesis of Carnap's philosophy of logical syntax was that philosophy consists of logical analysis and yields results insofar as they are established truths, which are themselves analytic. This apparent narrowing of scope is confirmed by explicit exclusion of fields such as ethics, not only from the domain of analytic philosophy, but from scientific discourse altogether (bearing in mind that this kind of philosophy is itself conceived of as scientific in character, though not empirical). The severity of the apparent narrowing here is mitigated by the predominance in the actual philosophy of Rudolf Carnap of work which is methodological, including work which ensues in some proposal for the use of language. Though philosophy has relinquished any claim to offer factual enlightenment about the world, it may nevertheless make material contributions to the advancement of such knowledge by contributions to the methods of science.
Linguistic philosophy in its most extreme form conceives of philosophy as the analysis of natural language.
Roger Bishop Jones 2012-09-23