Chris Mortensen

Anything is possible

This paper criticises necessitarianism, the thesis that there is at least one necessary truth; and defends possibilism, the thesis that all propositions are contingent, or that anything is possible. The second section maintains that no good conventionalist account of necessity is available, while the third section criticises model theoretic necessitarianism. The fourth section sketches some recent technical work on nonclassical logic, with the aim of weakening necessitarian intuitions and strengthening possibilist intuitions. The fifth section considers several a prioristic attempts at demonstrating that there is at least one necessary proposition and finds them inadequate. The final section emphasises the epistemic aspect of possibilism.

Inconsistent Models for Relevant Arithmetics

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peanos axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R#, was absolutely consistent. It was pointed out that such a result escapes incautious formulations of Godels second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not ordinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle.The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein constitutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R#. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P#. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here).

The Delta Function

In Chapter 6, differentiating the function g(t) = k 1 t for all t ≤ 0 and g(t) = k 2 t for all t ≥ 0, where k 1 and k 2 are classically distinct real numbers, led to the inconsistent continuous function f(t) = g′(t) = k 1 for all t ≤ 0 and f(t) = k 2 for all t ≥ 0. If a dynamic system is described by g(t), then the derivative takes an instantaneous jump at t = 0. That it is instantaneous rather than taking an infinitesimal amount of time, is represented by the inconsistent continuity. The aim in this chapter is to differentiate one step further, finding f′(t). The new derivative can naturally be thought of as (k 2 − k 1).Δ(t), where Δ(t) has the two properties (i) Δ(t) = 0 for all t ≠ 0, and (ii) \(\int_{ - \infty }^{ + \infty } {\Delta (t)dt = 1} \). Property (ii) means that a constant times the integral recovers the precise amount of the jump from k 1 to k 2. The delta function occupies an interesting niche in the history of mathematics. Long regarded as problematic but useful in elementary quantum theory and quantum field theory, it was eventually ‘solved’ in Schwartz’ Theory of Distributions; but at the cost of a considerable increase in complexity, as well as an increase in the size of the function space for quantum mechanics.

Inconsistent Nonstandard Arithmetic

This paper continues the investigation of inconsistent arithmetical structures. In §2 the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In §3 several nonisomorphic inconsistent models with identity which extend the (=, Z m , are briefly considered. In §5 two models modulo an infinite nonstandard number are considered. In the first, it is shown how to model inconsistently the arithmetic of the rationals with all names included, a strengthening of earlier results. In the second, all inconsistency is confined to the nonstandard integers, and the effects on Fermats Last Theorem are considered. It is concluded that the prospects for a good inconsistent theory of fields may be limited.

Chunk and permeate III: : the Dirac delta function

Dirac’s treatment of his well known Delta function was apparently inconsistent. We show how to reconstruct his reasoning using the inconsistency-tolerant technique of Chunk and Permeate. In passing we take note of limitations and developments of that technique.

Topological Separation Principles And Logical Theories

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity.

Models for inconsistent and incomplete differential calculus.

Nonclassical model theory is sketched and an incomplete model is defined. It is shown that the elements of equational differential calculus hold in this model, and a comparison with synthetic differential geometry is made. An inconsistent theory is defined with many, though not all, of the same properties

On Logical Strength and Weakness

First, we consider an argument due to Popper for maximal strength in choice of logic. We dispute this argument, taking a lead from some remarks by Susan Haack; but we defend a set of contrary considerations for minimal strength in logic. Finally, we consider the objection that Popper presupposes the distinctness of logic from science. We conclude from this that all claims to logical truth may be in equal epistemological trouble.

Reply to Burgess and to Read.

Reponse aux critiques de cercle vicieux formulees par Burgess et Read a propos du syllogisme disjonctif

Plato's Pharmacy and Derrida's Drugstore.

Abstract In a long essay Platos Pharmacy , Jacques Derrida attacked Western metaphysics. This paper undertakes to defend Western philosophy from Derridas arguments. It is shown that Derridas arguments are very unsatisfactory.