I begin with a historical sketch, describing first the dogmatic opposition to metaphysics of the most extreme positivists, and then the evolution of Carnap's thought from his phenomenalistic beginings to his liberal conclusion.

The I take one step forward beyond the liberal Carnap, and give an analysis of positivistically acceptable varieties of metaphysics, and touch more generally on aspects of Aristotle's "first philosophy" which might not now be called metaphysics.

Hume's fork supports a rejection of metaphysics, and reappears with similar effect in the philosophy of Rudolf Carnap. A critique of the resulting conventionalist position makes a space for a kind of metaphysics.

Carnap's logical positivism matured into a position a hairs breath from embracing metaphysics.

Why choices about language should not be completely arbitrary.

The re-presentation of metaphyisical positivism, insofar as this can be done, as a choice of idiom rather than as a body of doctrine, preliminary to the use of that language for the statement of various problems.

A discussion of metaphysics from the point of view of metaphysical positivism.

A sketch of a conception of philosophy and of metaphysics closer to those of ancient Greece than to contemporary philosophy.

Some temporary notes about particular metaphysical problems of personal interest.

The approach adopted here is to start from the positivist rejection of metaphysics through Hume's fork and its later presentation in the philosophy of Carnap. This leads to a conventionalist position, in which apparently metaphysical claims are presented as logical consequences of choices about language. A critique of radical conventionalism then make space in the positivist framework for a new kind of metaphysics.

Hume's fork divides matters of objective truth into truths of reason and matters of fact. Hume says that only truths of reason are intuitively or demonstratively certain, and that our conclusions about matters of fact are obtained by non-rational and less reliable means. Hume condemns metaphysics, by which he means any claim to objective truths which do not comply with this classification both in the character of the allegation and the method by which it is to be established.

Hume was inspired by Newton, and placed philosophy as concerned with matters of fact, as an empirical (though not experimental) science of Human nature.

The characterisation of metaphysics in terms of the analytic/synthetic distinction is initiated by Kant in response to Hume, and becomes the dominant simplistic conception of the issue between the Positivists and the Metaphysicians.

Hume not only condemned those matters which might normally have been regarded as metaphysical, but took a Platonic line on what we can know of the world of the senses. His conception of necessity was clearly one of logical necessity, and he first argues that causal connections are not necessary. On the further presumption that all our supposed knowledge of the external world is based on the evidence of our senses, which is at best causally related to the available (sensory) evidence, and the implicit demand that true knowledge can only be had of things which follow necessarily from the evidence he concludes that our knowledge of the external world is illusory, and our tendency to belief is rooted in habitual disposition.

By systematic application of the demand, in effect for demonstrative proof, Hume casts doubt on all knowledge except that of relations between ideas and of sensory impressions. Though he explains our beliefs as rooted in habit, he also effectively re-interprets these beliefs by attempting to construe them in terms which do not go beyond the available evidence.

This sets the pattern for a radical positivist conception of metaphysics as any conclusion about the external world which goes beyond conclusions logically entailed by the available evidence.

Hume's phenomenalism presents his philosophy as a mitigated scepticism. Hume grants a disposition to demonstrative knowledge, allowing by contrast with previous empiricists that this category of a priori knowledge is non-trivial and crucially includes mathematics.

However, the mitigation is confined to the a priori, in respect of empirical knowledge a full-blooded pyrrhonism is applied, denying that we have even probable knowledge of the external world.

This is characteristic of extreme positivism, in which scientists are expected go further than the evidence strictly warrants, and in terms of which any such speculation is regarded as metaphysical.

As positivists become more concerned with meaning, this provides an alternative motivator for the identification and exclusion of metaphysics. At first positivist conceptions of meaning are so closely coupled with the relationship between a proposition and the evidence which would suffice to verify it, that exclusion as meaningless is not clearly distinct from exclusion as going beyond any possible evidence.

Later the coupling is weakened, and the requirement for claims or questions to be meaningful becomes a more moderate demand than that of empirical verifiability.

On the division of propositions Carnap's view is similar to Hume's though the terminology has changed, there is considerably greater detail. The characterisation of a the a priori as concerned with relations between ideas (in Hume) becomes in the early Carnap, via ``semantic ascent'', their characterisation as {\it about syntax} and in the later Carnap as tautological.

In Carnap the notion of analyticity is made precise by a model of language semantics as truth conditions and matches precisely the concept of logical necessity understood as truth in all possible worlds.

Carnap was inspired by Russell, (and influenced by Frege and Wittgenstein), and held that philosophy was analytic. His attitude towards metaphysics was affected by his internal/external distinction. In these terms, the "fork" is a statement about "internal" questions, i.e. questions put in some language with a well defined semantics. Analyticity in such a language is determined by the semantics. Questions about the legitimacy of the semantics are external questions, which Carnap considered meaningless and believed should be settled pragmatically.

For some reason he does not seem to consider the external questions as having meaning determined by the semantics of some meta-language.

Carnap's position is {\it conventionalist} then in this respect. Insofar as true supposedly metaphysical propositions are necessary they must in his conceptual scheme also be analytic. Their truth therefore derives from the conventions which were adopted in defining the semantics of the language, and these are not themselves propositions but definitions.

This aspect of anti-metaphysical positivism is stronger in the early than in the later Carnap, where it is virtually extinguished by the ``principle of tolerance''. Carnap's was concerned with formalisation of the language of science, and he began by attempting a phenomenalistic reduction consistent with a radical nominalism. However, he went on to consider {\it physicalistic} language and {\it theoretical} language. His principle of tolerance effectively admits any metaphysic to be incorporated into the semantics of a language. Once incorporated into a language in this way, the ``internal'' metaphysical questions become analytic, and the ``external'' questions (which relate to the legitimacy of such a semantics) can be dealt with on a purely pragmatic basis.

The description of language suitable for definite statement of certain metaphysical problems, transforming relevant aspects of metaphysical positivism from a position statement to a choice of language.

An outline of a positive conception of metaphysics.

What can be said about the existence of abstract entities.

The analysis of arguments about Space-Time.

Modern conception's of philosophy, particularly those of analytic philosophers in the 20th century are narrow by comparison with those found at the roots of Western philosophy in ancient Greece. Our purpose here is to draw on some of these earlier ideas in a sketch of a liberal modern conception of the nature of philosophy.

Plato's views here are made plain in his similies of the line and of the cave, in which knowledge of Platonic forms is the only true knowledge.

Aristotle's views are found in Book I of his metaphysics. First philosophy lies at one exalted extreme of several linear orderings in which the various kinds of knowledge can be placed.

The core concern of pure positive philosophy is to understand the human predicament, both the cosmos in which we find ourselves, the nature of humanity and place in the universe.

Let us define for present purposes an abstract entity as one which is causally isolated from the material or sensory universe, our knowledge of which arises entirely by inference from the definitions of the entities. What can be said in this context about what abstract entities exist? Well, working with Carnap's distinction between internal and external questions, there are very many internal questions which can be settled, and a lot of this is done in set theory. On the "external" questions, e.g. granted that in the context of set theory we can prove the existence of various sets, but are the axioms of set theory correct descriptions of any part of reality, or is set theory entirely hypothetical?