The paper presents a proof of the consistency of Peano Arithmetic (PA) that does not lie in deducing its consistency as a theorem in an axiomatic system. PA’s consistency cannot be proved in PA, and to deduce its consistency in some stronger system PA+ is self-defeating, since the stronger system may itself be inconsistent. Instead, a semantic proof is constructed which demonstrates consistency not relative to the consistency of some other system but in an absolute sense.
A common criticism of free will or origination theories is that if what we do is not the result of an unbroken sequence of causes and effects, then it must to some degree be the product of chance. But in what sense can a chance act be intentional or deliberate, in what sense can it be based on reasons, and in what sense can a person be held responsible for it? If free and responsible action is incompatible with determinism, must (...) it not equally well be incompatible with indeterminism? Professor McCall says no. He argues that a new idea, that of a controlled indeterministic process, resolves a variety of classical dilemmas and opens the way to a new understanding of the relationship between actions, reasons, causes, and responsibility. Does he succeed? All of this, like a related line of argument by Professor McCann to which you can turn, is a long way from what seems to me the continuing arguableness of determinism and the unavoidableness of the proposition that both incompatibilism and compatibilism about freedom are false. But we all need to remember, with Cromwell, in our own bowels if not by those of Christ, that we may be mistaken. I guess that given the proportion of false to true views in the world, we need to remember it is arguable that we are more likely to be mistaken. (shrink)
The evolution of a single trapped ion exhibiting intermittent fluorescence and dark periods may be described either as a continuous process, using differential rate equations, or discretely, as a Markov process. The latter models the atom as making instantaneous transitions from one energy eigenstate to another, and is open to the objection that superpositions of energy states will form which are not covered by the Markov process. The superposition objection is replied to, and two new mathematical elements, Markov vectors and (...) Markov matrices, are proposed as additions to quantum theory. The paper concludes by attributing the cause of dark periods in the ion's history to instantaneous transitions in the ion itself, rather than to photon detection or other components of measurement. (shrink)
Over the last 25 years, experimental findings published by Benjamin Libet have indicated that conscious acts of will are preceded by a characteristic kind of brain event of which the agent is not conscious. It, Libet says, rather than the will, is what causes actions. His discoveries, if correct, would seem to imply that the notion of a free, conscious will is an illusion, and that actions are initiated by neural processes not under conscious control. In what follows it is (...) argued that Libet’s conclusion is incorrect, and that other evidence points to the essential causal role of consciousness in voluntary action. (shrink)
In his paper ‘‘Bad luck once again’’ Neil Levy attacks our proof of the consistency of libertarianism by reiterating a time-worn compatibilist complaint.1 This is, that what is not determined must be due to chance. If A has a choice of X or Y, neither X nor Y being causally determined, then if A chooses X it can only be by chance, never for a reason. The only ‘‘reason’’ that could explain the choice of X over Y would have to (...) be a causally sufficient reason, which would rule out A’s having a genuine choice in the first place. Either X is causally necessitated or X is realized by sheer luck. But that these are the only alternatives is untrue. The exercise of deliberative reason opens the way between the Scylla of causal necessitation and the Charybdis of chance, as we shall try to make clear. The central core of Levy’s argument is that any attempt to give a reasons-based explanation of a contrastive fact must fail. A contrastive fact is a fact of the kind ‘‘Jane decides to vacation in Hawaii rather than Colorado,’’ or ‘‘Jane assigns a greater weight to surfing that to white-water rafting.’’ In the last three paragraphs of his paper Levy argues that Jane’s assigning more weight to surfing than to rafting cannot be a reasons-based assignment, because, as he puts it, ‘‘the reasons that would explain the weighting are the weighted reasons themselves.’’ Similarly, prior to making her Hawaii⁄ Colorado decision, Jane has.. (shrink)
The aim of the paper is to prove the consistency of libertarianism. We examine the example of Jane, who deliberates at length over whether to vacation in Colorado (C) or Hawaii (H), weighing the costs and benefits, consulting travel brochures, etc. Underlying phenomenological deliberation is an indeterministic neural process in which nonactual motor neural states n(C) and n(H) corresponding to alternatives C and H remain physically possible up until the moment of decision. The neurophysiological probabilities pr(n(C)) and pr(n(H)) evolve continuously (...) according to the different weights Jane's judgement attaches to C and H at different times during the deliberation. The overall process is indeterministic, since Jane's exact judgemental weighting would vary slightly were the process to be repeated from the same initial conditions. The weighting is however rational, and entirely under Jane's control. This controlled, rational, indeterministic process shows that libertarianism is a consistent philosophical thesis. (shrink)
The thesis of 3D/4D equivalence states that every three-dimensional description of the world is translatable without remainder into a four-dimensional description, and vice versa. In representing an object in 3D or in 4D terms we are giving alternative descriptions of one and the same thing, and debates over whether the ontology of the physical world is "really" 3D or 4D are pointless. The twins paradox is shown to rest, in relativistic 4D geometry, on a reversed law of triangle inequality. But (...) considering the twins as 3D beings who age through time, the paradox implies that time passes at different rates in different reference frames, and therefore that the concept of a single global or Absolute time is unsustainable. (shrink)
The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra–ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. (...) The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities. (shrink)
Combining quantum mechanics with special relativity requires (i) that a spacetime representation of quantum states be found; (ii) that such states, represented as extended along equal-time hyperplanes, be invariant when transformed from one frame to another; and (iii) that collapses of states be instantaneous in every frame. These requirements are met using branching spacetime, in which probabilities of outcomes are represented by the numerical proportions of branches on which the outcomes occur. Quantum states of systems are then identified with the (...) probability values, built into spacetime along spacelike hypersurfaces, of all possible outcomes of all possible tests to which the systems can be subjected. (shrink)
Storrs McCall presents an original philosophical theory of the nature of the universe based on a striking new model of its space-time structure. He shows that this theory can illuminate a wide variety of hitherto unresolved philosophical problems. These include: the direction and flow of time; the nature of scientific laws; the interpretation of quantum mechanics; the definition of probability; counterfactual semantics; and the notions of identity, essential properties, deliberation, decision, and free will. A particular instance of the explanatory powers (...) of the proposed space-time model is its account of quantum non-locality in the EPR and GHZ experiments. -/- Professor McCall argues that the fact that the model explains and throws light on such a broad range of problems constitutes strong evidence that the universe is as the model portrays it. (shrink)
A theory of temporal passage is put forward which is "objective" in the sense that time flow characterizes the universe independently of the existence of conscious beings. The theory differs from Grunbaum's "mind-dependence" theory, and is designed to avoid Grunbaum's criticisms of an earlier theory of Reichenbach's. The representation of temporal becoming is accomplished by the introduction of indeterministic universe-models; each model representing the universe at a time. The models depict the past as a single four-dimensional manifold, and the future (...) as a branched structure of such manifolds. Time flow is relativistic in that it manifests itself in a frame-dependent (but not observer-dependent) way. The indeterministic character of the universe-models is mirrored in a "temporal" theory of truth which rejects the principle of bivalence, and suitable semantics are provided for this theory. Finally, an account of physical law is given which defines it in terms of physical possibility, rather than vice versa. (shrink)
