The Metaphysical Basis of Logic: The Law of Non-Contradiction as Basic Knowledge.

"The firmest of all principles": thus Aristotle introduced the principle later known as the Law of Non-Contradiction (LNC), according to which contradictions cannot obtain, or be true, in any possible circumstance. Aquinas, Leibniz, Hume, Kant, Popper, Lewis, and many others took the LNC as the supreme cornerstone of knowledge and rational thought.

However, the Law has come under attack by contemporary philosophers called "dialetheists". A 'dialetheia' is a sentence, A (or the expressed proposition), such that A and its negation, not-A, are true. Dialetheists take some logical paradoxes, paradoxes of absolute generality, and quantum phenomena, as delivering dialetheias, or peculiar situations in which the same thing can both be and not be, against the LNC.

Dialetheists use 'paraconsistent' logical systems, "paraconsistent" meaning "beyond the consistent": non-classical logics rejecting the law called 'ex contradictione quodlibet' (ECQ), stating that a contradiction entails any arbitrary claim. ECQ has devastating consequences for any self-contradictory theory. By rejecting ECQ, however, a dialetheist can accept *some* contradictions without accepting everything.

In 2004 Oxford U.P. published "The Law of Non-Contradiction", a 450 page book with contributions by David Lewis, Graham Priest, R.M. Sainsbury, Stewart Shapiro, and other major philosophers. What the logician Jan Lukasiewicz called "our undisputed faith" in the Law has turned into a rich debate, taking place in reviews like 'Mind', 'The Philosophical Quarterly', the 'Australasian Journal of Philosophy'.

This debate, however, may turn into a conflict of intuitions. One main problem is in our notion of 'reductio ad absurdum'. A rule of minimal logic, 'reductio' is a tool of rational criticism: a theory entailing both A and not-A *must* be revised, for it hosts a falsity. But a dialetheist can keep her entire theory, and the criticism that it involves an inconsistency: she may accept the entailed contradiction.
This is an aspect of what John Woods called Philosophy's Most Difficult Problem: what, for one philosopher, is a sound argument with a counterintuitive conclusion, for another is a refutation of some premise. Dialetheists take some instances of 'reductio' as *proof* of their inconsistent conclusions.

This project will establish (1) the conditions for a non-question-begging debate on the LNC, and (2) a metaphysical basis for non-contradictoriness. A key role is played by an operator for logical negation, drafted in my previous works, there labelled "NOT", and based on the notion of *exclusion between features of the world*. The operator has a technical definition entailing the following intuitive features:

a) It does not refer to the controversial concept 'truth': dialetheists doubt its being exclusive with respect to 'falsity' (for them, some truth-bearers exemplify both). It is characterized via the concept of exclusion as a basic primitive;
b) NOT captures a key expressive function in communication: to convey determinate information we need an exclusion-expressing device to *rule out* that some circumstance obtains. Natural language negation plays such function (among others); NOT makes it formally precise.
c) Once NOT has been given a semantics, we can establish a version of the LNC formulated via it. Dialetheists can accept it, as based on a concept they share: our sense of exclusive possibilities is, in the Kantian foundational jargon, a condition of possibility of our experience of the world.

An early career researcher, I have already published a book and essays on the LNC in 'The Philosophical Quarterly', 'The Australasian Journal of Philosophy', 'Philosophical Studies', the 'Stanford Encyclopedia of Philosophy'. This research will allow a decisive advancement of our understanding of the role of contradictions in rational inquiry and, most importantly, to help the career of a promising post-doc in this burgeoning field.

This project investigates a key foundational notion of human knowledge. As often happens with foundational research, the impact is difficult to foresee, but at the same time potentially momentous.

In the short run, the immediate products of the research will include not only technical publications, such as peer-reviewed essays individually and/or jointly written by me and the post-doc, but also non-technical outputs. These will publicize the outcomes of the project among non-specialists, and shall appear in widely accessible web sites and on line resources, like the 'Stanford Encyclopedia of Philosophy' (http://plato.stanford.edu, where I am already responsible for two entries on "Dialetheism" and "Impossible Worlds"), 'Wikipedia', and the 'Internet Encyclopedia of Philosophy' (www.iep.utm.edu, where I already act as a referee).

Further on line content, in the shape of lectures on contradictions and the role they play in our rational thought, shall be prepared, in cooperation with the chosen post-doctoral fellow. The inspiring model, on a smaller scale, is the one of the Kahn Academy (http://www.khanacademy.org): a free web site where practitioners have access to learning tools, exercises, and recorded lectures in an interactive environment - more information on this is in the "Pathways to Impact" attachment, and information on the technical management of the web site is in the Technical Appendix. The project web site, hosted by the servers of Aberdeen University (www.abdn.ac.uk), will also include the summaries of our project's research seminar and the draft papers produced by me and the post-doc.

I have already set up a similar framework in a different project for a private research enterprise. A book on non-standard computation was commissioned by iLabs (www.ilabs.it), the research sub-section of Diagramma srl (www.diagramma.it), a private company selling commercial software in Milan. Co-authored by me and two Computer Science specialists, and published in the Computing Series of the King's College Publications of London, the book was accompanied by Java and NetLogo applications and by on line interactive content prepared by me, simulating the computing environment and automaton characterized in the book - see: http://www.mmdr.it/defaultEN.asp

In the long run, the cognitive-metaphysical model produced by the research is expected to have applications in theoretical computer science and AI. Specifically, it should allow the modeling of a cognitive agent, whose inputs from the external environment are shaped by a consistency requirement (the "exclusionary possibilities" constraint described above); which nevertheless can receive inconsistent input, and efficiently elaborate the incoming contradictory information *despite* the inviolability of the consistency requirement itself. The study of the computational complexity of algorithms characterizing such processes may constitute a further interesting spin-off of the research.