A framework for the transfer of proofs,lemmas and strategies from classical to non classical logics

There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a backward translation. We propose a framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulasF _TL belonging to the proof in TL into formulasF _SL , such that the formulasF _SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulasF _TL andF _SL can obviously be considered aslemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof keep trace of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given.     

Speakable in quantum mechanics

At the 1927 Como conference Bohr spoke the famous words “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.” However, if the Copenhagen interpretation really adheres to this motto, why then is there this nagging feeling of conflict when comparing it with realist interpretations? Surely what one can say about nature should in a certain sense be interpretation independent. In this paper I take Bohr’s motto seriously and develop a quantum logic that avoids assuming any form of realism as much as possible. To illustrate the non-triviality of this motto, a similar result is first derived for classical mechanics. It turns out that the logic for classical mechanics is a special case of the quantum logic thus derived. Some hints are provided as to how these logics are to be used in practical situations and finally, I discuss how some realist interpretations relate to these logics.     

Quantities in quantum mechanics

The problem of the failure of value definiteness (VD) for the idea of quantity in quantum mechanics is stated, and what VD is and how it fails is explained. An account of quantity, called BP, is outlined and used as a basis for discussing the problem. Several proposals are canvassed in view of, respectively, Forrest's indeterminate particle speculation, the "standard" interpretation of quantum mechanics and Bub's modal interpretation.     

Determinism and locality in quantum mechanics

In current philosophical debate Bell's theorem is often refered to as a proof of the impossibility of determinism in nature. It is argued here that this conclusion is wrong. The main consequence of the theorem is the non-local character of quantum theory itself and it is shown how this quality leads to a contradiction with the theory of relativity. If hidden variable theories are impossible, it is so because no empirically founded interpretation at all can be compatible with both quantum mechanics and relativity.     

Random dynamics and the research programme of classical mechanics

The modern mathematical theory of dynamical systems proposes a new model of mechanical motion. In this model the deterministic unstable systems can behave in a statistical manner. Both kinds of motion are inseparably connected, they depend on the point of view and researcher's approach to the system. This mathematical fact solves in a new way the old problem of statistical laws in the world which is essentially deterministic. The classical opposition: deterministic‐statistical, disappears in random dynamics. The main thesis of the paper is that the new theory of motion is a revolution in the research programme of classical mechanics. It is the revolution brought about by the development of mathematics.     

On the nature of quantum mechanics

Summary It is argued that the EPR paradox cannot be resolved in the context of quantum mechanics. Bell's theorem is shown to be equivalent to a Belinfante theory of zero type. It is concluded therefore that it cannot have as wide a range of applicability in excluding Hidden Variable Theories as commonly alleged. It follows that standard quantum mechanics should not be regarded as a complete theory in Einstein's sense. Indeed, it is argued that a purely probabilistic theory cannot be the basis of a comprehensive understanding of physics. An attempt is made to formulate a deterministic, local Hidden Variable Theory to account for the Bohm-Einstein thought experiment reproducing quantum mechanical predictions.     

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer??s calculus of ??conditional density operators??. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.     

Axiomatic quantum mechanics and radioactive decay

This paper explores the consequences of the orthodox resolution of the measurement problem for the axiomatic base of non-relativistic elementary quantum mechanics. It is argued that the standard resolution of the measurement problem generates a paradox whose dissolution may be achieved through an enrichment of the axiomatic foundations of quantum mechanics. These results are also linked to some recent creative proposals by Nancy Cartwright concerning the nature of the so-called reduction of the wave packet.