Synthetic Philosophy and Deductive Engineering
The SPaDE project takes a foundational approach to the representation of declarative knowledge. To explain what that means in the context of the SPaDE project requires some new terminology and this page is intended to introduce some of the new terminology, in particular that of a foundational ontology.
This terminology and the discussion around it belongs to the philosophical foundations underpinning the SPaDE project, which in its broadest sense we call Synthetic Philosophy. When we think foundationally, we are looking for a simple kernel upon which a larger structure can be built. In this case, the larger structure of interest is declarative knowledge, its representation and exploitation, especially that which enables the design and construction of benign self-proliferating intelligent systems.
As we approach the kind of foundational kernel for the philosophical underpinning of such an enterprise, we find a number of interrelated aspects of philosophy, no one of which can be independently addressed without reference to the others. These aspects include metaphysics, language, logic and epistemology.
Declarative knowledge is expressed in or represented by certain kind of language, which we also call declarative. Declarative sentences, in order to express knowledge, must be meaningful. In Synthetic Philosophy we are concerned, not so much with discovering meaning, as with defining or prescribing meaning. In order to do that, one must have some conception of what exists, which is the province of metaphysics, of that part of metaphysics which is called ontology.
Synthetic Philosophy largely falls within the positivist tendency in analytic philosophy, and therefore shares its antipathy to much that passes as metaphysics. That position is moderated by some sympathy for the position of Rudolf Carnap, who primarily abjured metaphysics as meaningless, and was content with things which might have been thought metaphysical but for their having definite sense or role in a linguistic framework. This did give Carnap’s apparently metaphysical positions (notably on ontology)a conventionalist rather than an absolute flavour (which he tried to elucidate by distinguishing between internal and external questions). The bottom line there is that in effect Carnap was, in relation to abstract ontology, a conventionalist, a position which is shared by Synthetic Philosophy (which tends to a broader conventionalism in ontology).
It should be noted that the conventionalism of Synthetic Philosophy is not a denial of there being a reality independent of our conceptions of it. However, it does extend to any attempt to describe that reality, the limitations of language and of our ability to describe reality are substantial, reality is likely too complex for any finitary language to capture the whole. The effective use of language to progress the proliferation of intelligence across the cosmos depends upon pragmatic choices.
An ontology is an account of what exists.
For the purposes of assigning meaning to declarative languages, a factorisation into abstract semantics and concrete syntax is advocated, giving meaning in the first instance in terms of abstract entities, ultimately rendered concrete by a correlating abstract with concrete entities.
Abstract entities have some similarities with fictions. They have just those properties which we ascribe to them, and any which logically flow from those ascriptions, subject only to the pragmatic constraint that our ascriptions be consistent.
The idea of logical foundations for knowledge, and the role of ontology in those foundations, is of very recent origin within the two millennia since the explicit study of logic was initiated by Aristotle. Despite enthusiasm on the part of Leibniz for the use of Aristotle’s logic as a foundation for knowledge, it was not until the late 19th century that the limitations of Aristotelian logic were surpassed by Frege’s Begriffsschrift (concept script), and the idea of a logical foundation for knowledge was seriously entertained. At this stage the significance and difficulty of ontology was not fully appreciated, and Frege’s logical foundations for mathematics were found to be inconsistent for lack of a clear underlying ontology.
Nevertheless, Frege’s work was seminal, and set the ball rolling. In this I mention two aspects of his ambition which were of particular importance.
Frege was the first to attempt not merely the provision of a small set of principles (axioms) from which inference should begin, but also to define precisely the rules of inference themselves governing the derivation of new truths from those axioms.
Frege enunciated the motto: mathematics = logic + definitions
that is, that all mathematical truths could be derived deductively from definitions of mathematical concepts
The latter conception was controversial but influential, and is the substance of the position in the philosophy of logic which came to be called logicism. The principle attack on that conception of mathematics and its foundations was connected with the difficulties associated with ontology, and hence with whether those logical systems which were ontologically adequate could properly be considered to be purely logical.
Here we sidestep that issue as terminological, by adopting the term logical foundation systems to refer to those systems which provide logical foundations for mathematics, regardless of whether they are purely logical or not. The motto then becomes: mathematics = logical foundation system + definitions
In 1908, terminology apart, we had on the table, two candidates for “logic” which realised these elements of Frege’s ambition: Zermelo’s Axiomatic Set Theory and Russell’s Theory of Types. Neither of these, as published in 1908 satisfied Frege’s desire for precisely defined rules of inference, but both were subsequently refined to meet that requirement. These two systems are prototypes for the logical foundation systems which are most important in this context, and we may identify more specifically the later derivatives of these two systems in the first order theory known as Zermelo-Fraenkel Set Theory with Choice (ZFC), and the higher order theory known as Church’s Simple Theory of Types (STT) augmented primarily by polymorphism in the logic of Cambridge HOL.
The failure of Frege’s logical foundations for mathematics led to a period of reflection on the nature of logic and ontology, culminating in the development of two different approaches to logical foundations for mathematics, both published in 1908.
The one most closely related to Frege’s approach was Russell’s Theory of Types, which sought to avoid the Fregean antinomy (“Russell’s paradox”) by stratifying entities into types.