Links below to "Carnap IAB" are links to my notes on Carnap's "Intellectual Autobiography" in his volume of the Library of Living Philosophers [Carnap63].

You may find it helpful to refer to my other account of Carnap's philosophy which this comparison parallels.

Though the verification principle is not adopted by metaphysical positivism, the avoidance of nonsense of various kinds remains a central concern. The methods of metaphysical positivism are intended to permit rigorous analysis and to avoid nonsense and sophistry (witting or not).

Metaphysical positivism has nothing very specific to the foundations of mathematics to add to the broad outlines of Carnap's philosophy in this matter.

Metaphysical positivism sits on a fence here, recognising that there is something interesting to be said in this area, but not going as far as Carnap appears to do.

This early central feature of Carnap's philosophy was later significantly changed by Carnap's shift to semantics.

This is the ameliorisation of foundationalism and the verification principle.

Semantics is of central to metaphysical positivism, but is an area in which methods have advanced considerably since Carnap's work.

Carnap was a linguistic pluralist, advocating not a single language for science but the use of languages appropriate to the domain under consideration. This gives rise to particular difficulties when one has an application involving several branches of science formulated in different languages. This was the problem which he addressed in "Language Planning".

There's a pretty large gulf here. Metaphysical positivism gives no credit to the idea of inductive inference, and is skeptical about the value of confirmation theory.

This approach of Carnap's to the formalisation of science is the one with which metaphysical positivism has greatest sympathy.

Logical positivism divides indicative sentences into three kinds. Analytic sentences, which are supposed to be demonstrable, empirical facts, which must be verifiable, and the rest, which are nonsense. Metaphysics is singled out for obloquy, where metaphysics is construed as a priori claims about the nature of reality (or the synthetic a priori), or as statements which are neither analytic nor verifiable (and hence meaningless).

In Carnap's philosophy through its various stages there are several different attitudes toward metaphysics. In his earlier years he simply regarded metaphysics as useless, in the early years of logical positivism he regarded metaphysics as meaningless, adopting the verificationist view that the meaning of a proposition is its method of verification. Later he adopted a more liberal view on meaning, and expressed the status of metaphysics as lacking cognitive content.

While recognising the same three categories of indicative sentence, the distinction between sense and nonsense is made by different means. Sufficient meaning to support reasoning in a language may be provided by an abstract semantics (for the language under consideration), which will support the demonstration of analytic truths (though formal derivability is likely to be incomplete). Concrete semantics presents greater difficulties, and for science a (nomologico-deductive) modelling methodology is advocated, in which considerations of truth are supplanted by those of utility.

Metaphysics has no special status in this scheme, though in the terminology of metaphysical positivism there are no synthetic a priori truths, and contrary to modern metaphysicians there is no necessity de re. Necessity de re is a product of equivocation about semantics. A statement is necessary de re if it is necessary but not analytic. However, this cannot be the case, since a full semantics for a language must fully capture the truth conditions for the sentences of the language, and if a sentence is true in any possible world then this must be entailed by a full semantics. Of course, it need not be entailed by an incomplete semantics.

This however does not abolish metaphysical problems, it requires that their resolution be incorporated into the semantics.of languages. Carnap's distinction between "internal" and "external" questions is relevant here. Relative to a particular language some metaphysical questions will be analytic, as internal questions. But there will then be a corresponding external question which relates to the legitimacy of the semantics. Carnap tells us that external questions have no objective truth value but should be considered pragmatically. Metaphysical Positivism recognises that many of these are not arbitrary, that some correspond to fundamental objective considerations which we may consider metaphysical, and that there are strong pragmatic reasons for ensuring that the semantics of our languages reflects these considerations.

Carnap's logicism derived initially and primarily from Frege, (from Frege that "all mathematical concepts can be defined on the basis of the concepts of logic and that the theorems of mathematics can be deduced from the principles of logic" and thus, that the truths of mathematics are "analytic in the general sense of truth based on logic alone"). He also studied Principia Mathematica, preferring its notation and building on its theory of relations. He accepted Frege's view that arithmetic is analytic (adding also the mathematical study of geometry), and seemed to regard this as providing an important new way for empiricists to give a satisfactory account of mathematics. (I believe Wittgenstein once made a remark to the effect that the ideas which the logical positivists took from his Tractatus could all have been had from Hume. That the truths of mathematics are truths of reason, expressing relations between ideas, is one of these. Its not clear to me what Carnap thought to be new in his position relative to Hume. Of course there was no detail in Hume, so I guess that's whole lot of new stuff.)

Carnap recognised the difficulty in showing that the axioms of Principia were logical. He preferred to argue that the axiom of infinity is indeed analytic (via a suitable interpretation) but if that failed he would regard mathematical truths as conditional (and hence still analytic) rather than synthetic (and hence empirical, in default of pure intuition).

Carnap also sought to reconcile his logicism with both formalism and intuitionism to as great an extent as was feasible.

Though holding that mathematical truths are analytic, in metaphysical positivism analytic means something like "true in virtue of meaning" (though a more detailed analysis of this is sought), whereas Frege's conception of analyticity seemed more to do with derivability than semantics. (this is perhaps more like Ayer's than Carnap's account of analyticity) This lessens the difficulties arising from ontological axioms, but nevertheless abstract ontology is an important topic for further investigation.

My present position is that deductive inference should be defined as inference conducted exclusively using sound rules (counting axioms as rules with no premises). Logical truth is then coextensive with analyticity and logical necessity, and encompasses mathematics.

This logicism is consistent with the spirit of the moderate formalism held by Hilbert, as one might imagine it would be after acceptance of the brute facts on which Hilbert was mistaken. The central tenet of such a formalism would be that mathematical concepts should be characterised exclusively by formal means, and that mathematical demonstrations should make use only of these formal definitions. This is in the context of some suitable foundational system rich enough to accomplish the required mathematical formalisations. Hilbert's ideas on how the consistency of such systems could be established are more difficult to repair, and I'm not motivated to make the attempt. There is of course an issue about the best way to establish suitable foundations, which is of great interest. On this my preference is to begin with an inductive characterisation of the well-founded sets (a well-founded set is any definite collection of well-founded sets) and to build upwards from that acorn.

Carnap (at various times) advocated and worked on the formal presentation of science in three different ways. They were (I think in the order he considered them):

• phenomenalistic
• physicalistic
• theoretical

The first was for purely philosophical purposes and was associated with the foundationalist principle that all our knowledge of the world is derived from sense data. The second served a different philosophical purpose, and is the language which is relevant to Carnap's view on the unity of science. The third is the least philosophical and the closest to a formalisation for purely scientific purposes using the concepts of the scientist.

A phenomenalistic language speaks about sense data. A physicalistic language speaks about material things, ascribing to them observable properties.

The unity of science principle was a reaction against a view popular in Germany at that time that sciences consisted of two kinds, the natural sciences and the spiritual sciences (covering the social sciences and the humanities). The unity of science movement was connected with physicalism through the thesis that "the total language covering all science can be constructed on a physicalistic basis".

It is for me an interesting question in what sense if any a physicists "theory of everything" provides a basis to which all of science can be reduced. I am sympathetic to the idea that there could be a complete abstract model of the universe broadly along the lines sought by fundamental physics.

However, I advocate a pragmatic attitude to the adoption and application of scientific concepts and theories, and expect that reductionism will remain of limited pragmatic value in many domains. I have no problem with theories modelling aspects of reality using non-physical entities, and am skeptical about the pragmatic value of attempts at physicalistic reduction of mental concepts or events.

I believe that the best level at which to seek a sense of the unity of science is at the logical level. All scientific theories which can be used deductively (as in the "nomologico-deductive" method) can be formulated as abstract models in set theory, or in many other formal or mathematical languages of similar strength. It might be better to talk of "the language of mathematics" as the base for science rather than "set theory", though I think of set theory as admitting models which some mathematicians might not accept as ``mathematical''.

Possibly following Hilbert, Carnap in his early years thought formal syntax could completely replace informal semantic notions. Gödel's incompleteness results show that this is not the case and Carnap later gave considerable attention to semantics.

What Carnap spoke of under the heading "Philosophy and Logical Syntax" was however much broader in its scope than Hilbert's idea that formal axioms suffice for semantics. Carnap sought to adapt Russell's analytic method, applied by Russell to the derivation of mathematics, for use in the empirical sciences, but he also added, following Hilbert, a meta-theoretic dimension which is absent from Russell's work.

Metaphysical positivism starts rather than ends with a concern for semantics. The hinterland of logical truth beyond the limits of accepted axiom systems is of particular interest, and, as a study in abstract ontology is one of the elements of "metaphysics" of interest to metaphysical positivism. Metaphysical positivism continues the concern with methods for formal logical analysis applicable wherever deductive conclusions are sought.

Metaphysical positivism is conceived in a world of digital electronics. Digital information processing has a radical effect upon the feasibility of methods for formal analysis. It takes away an increasing part of the drudgery of detailed formal derivation, and in that way makes the application of formal techniques (rather than their metatheory) more tractable. At the same time these technologies continually transform the ways in which those we now call "knowledge workers" work, and promise that hitherto exclusively human activities (knowing and reasoning) may in the future be done by wholly inhuman entities using radically novel methods.

To be relevant to the future, analytic method must now anticipate advances in software capabilities the nature of which is itself a matter of speculation.

At the time of the Logischer Aufbau Carnap supposed all empirical knowledge decidably derived with certainty from the indubitable immediately given. In addition the principle of verifiability asserted that all meaningful sentences were in principle susceptible of definite verification or refutation.

This position Carnap perceived to be in conflict with other important principles which he accepted. These included an emphasis on the hypothetical character of laws of nature. Other aspects also, for example the indubitability of the immediately given, came to be doubted, under the influence of Neurath and Popper. The "Left Wing" of the Circle (Carnap, Hahn, Neurath) doubting the verification principle sought a better criterion of significance. This took place over an extended period during the 30s.

This resulted in Carnap abandoning the verification principle and accepting as significant sentences which are "confirmable", i.e. if observation sentences can contribute either positively or negatively to its confirmation. He also abandoned the requirement that the concepts of science be explicitly definable in terms of observation concepts (in a physicalistic language), "more indirect methods of reduction could be used". Similar liberalisations were also desirable for phenomenalistic accounts.

Metaphysical positivism maintains a foundationalist stance both in terms of a priori and a posteriori knowledge, but this is quite a long way removed from even the more liberal foundationalisms of Carnap. Possibly closer to a causal theory of perception.

The a posteriori foundationalism recognises that our knowledge of the external world is mediated by certain sense data. This data is what it is, indubitably! Insofar as it is taken to represent something other than itself the correlation will be contingent and hence dubitable. The nature of the sense data depends upon where the boundaries are drawn between self and other. If the self is a digital computer then the sense data could simply be the content of an area of memory into which data from sensors is transferred directly by the sensory peripherals. In the case of human beings, there is no reason to believe either that the sense data corresponds to some conscious experience, or that the process of derivation whereby beliefs about the external world are "inferred" from the data is wholly, or even partly conscious. Furthermore, it is certain that this process is not deductive, for by deduction from sense data it will not be possible to obtain information about things whose relation with the sense data is contingent rather than logical. Nevertheless, this is very probably what happens. The epistemological significance is not to give is reason to trust conclusions obtained from sense data about the external world, but rather simply to affirm that allegations about the external work which are based on no relevant sense data should be completely un-trustworthy.

Metaphysical positivism advocates the formalisation of science, and of any domain in which deductive reasoning is achievable. and has something to say about how this might be done. However, it retains no vestige in this programme of formalisation of the original empiricist goals, or of Carnap's phenomenalistic and physicalistic languages.

The starting point for such a formalisation is the formalisation of classical mathematics using computer based tools. Physics and other sciences can then be formalised as mathematical models of various aspects of physics (or perhaps the whole "TOE"). This serves no epistemological purpose.

In terms of "criteria of significance" metaphysical positivism adopts neither the verification principle nor the confirmation principle. It is advocated that abstract and concrete semantics be separated. This corresponds to the practice suggested above of formalisation of science as mathematical models. The models are abstract structures and their semantics is pinned down by giving an abstract semantics to the mathematical language in which the models are defined, and by the practice of defining models exclusively be conservative extension.

This gives to scientific models a meaning sufficient to allow deductive reasoning about the models.

The non-abstract part of the criterion of significance is pragmatic. Firstly, the concrete semantics is given by informal accounts of the correspondence between structures and concepts in the abstract model and the relevant aspects of the physical world. Then the model is evaluated for its fidelity to the real world, and may also be evaluated in terms of its utility for various purposes. In this way we obtain some information about the domain in which the model is applicable. Whether the model is "true" in any sense, is a matter for metaphysics not science.

Carnap broadens his meta-lingual theory to encompass semantics, with particular concern for the semantic definition of logical truth and the distinction between logical and factual truth.

A separation is argued between abstract semantics and concrete semantics. An abstract semantics can be realised formally in set theory or some comparable foundation system, and suffices for the establish of the relation of entailment and the property of analyticity or logical truth. Concrete semantics concerns the connection between the abstract entities in the abstract semantics and any concrete entities in the real world with which they may be intended to correspond. It is neither so easy, nor so important, from the point of view of formal analysis, to get the concrete semantics done with formal precision. The concrete semantics has no impact on the conclusions which can be drawn from the formal analysis, but does determine whether and how the analysis is relevant to any real world problem.

This position connects with the idea that hard science consists in the construction of mathematical models of aspects of the world. The construction of such models and their formal analysis, including the derivation of particular conclusions given sufficient particular assumptions (e.g. behaviours resulting from various initial conditions), can all be conducted by conservative extension in the context of a mathematical foundation system (such as set theory) involving only abstract entities (the existence of which need only be consistently presupposed).

The principle of tolerance leads to linguistic pluralism and to the need for planning how a number of languages can be fitted together yielding a system fulfilling given desiderata. Carnap, like Leibniz before him, was interested both in the construction of formal languages and in that of devising a new international informal language. These two problems suggested to him "utterly different" methods of solution.

Metaphysical Positivism is also pluralistic (as well as foundationalist). The simplest solution to the language planning problem is to adopt a semantic foundation system and develop suitable tools which support reasoning in the language of that system or in other languages via semantic embedding into the foundational language.

The other solution comes with a complex of ideas connected with XML called X-Logic. This is along the same lines but oriented to fit with the standards for multilingual markup on the Internet.

Carnap's work on probability and inductive logic were connected with the liberalisation of empiricism as a part of which he abandoned the verification principle. The idea was to replace the black-and-white notion of verifiability with the more subtle tones of confirmability. Carnap sought a notion of probability suitable for this purpose, the frequency notion of probability not being suitable. This he called logical or inductive probability, and is used in giving an exact numerical value for the degree of confirmation which bodies of evidence confer upon scientific hypotheses. "Inductive logic", by which Carnap means any system of inference in which conclusions do not hold with deductive necessity, is essentially the rules whereby these logical or inductive probabilities are assigned to conclusions.

Doubts the utility of "inductive logic" or confirmation theory. Instead it considers scientific theories as providing models of aspects of the world which need not be considered "true" or "false". Even models which are known to be imperfect (and hence presumably if true or false must be false) can be very useful, and probably most accepted scientific theories are known to be only approximations.

It therefore seems to me more rational to compile useful data about the scope of applicability, and the reliability and precision in its various applications of a scientific theory or model, rather than attempting to quantify confidence in its truth. What confidence can we have in the truth of Newton's laws of motion or Boyle's law? We know presumably that they and many other useful scientific laws are in fact false. If we are to give a useful comment about these theories it must surely be to provide some information which will help to determine when they are adequate approximations for some practical application.

I do not wish to rule out the possibility that confirmation theory or inductive probabilities might be useful in evaluating scientific models. More pragmatic measures seem to me to be more worthwhile, but not a topic which I propose to consider myself.

Tenuously connected with this is the question of assurance, which concerns the confidence we may be entitled to in the truth of some definite proposition.

Insofar as we speak of the use of formal methods for scientific rather than philosophical purposes then the starting point should be the formalisation of the theoretical language used by science. Formalisation would expose scientific theory to a microscope and would demand greater care for consistency (e.g. it would be a bit picky about how to handle theories like relativity which break down under certain circumstances).

In Carnap's treatment of the theoretical language there remains concern to connect the theory with observations. The connection between the two is softened from early desire for definition in terms of observable's to a notion of reducibility. This still required however that applications of theoretical concepts are entailed by observation statements, which in fact will rarely be the case.

In metaphysical positivism the formalisation of science is cut free from the epistemological desire to make clear the connection between scientific theory and observable phenomena. The impact and value of formalisation considered to lie elsewhere, and the connection such as there is between general theories and particular events is that the theories entail entailments between events (e.g. "Newton's theory of gravitation entails that if the sun and earth have certain masses and are such a distance apart then there will be a gravitational attraction between them of a certain magnitude).