The aim of this paper is to argue about the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm that represent the human capacity of recognizing mathematical truths. More specifically, based on a direct argument from Gödel's First Incompleteness Theorem, we discuss the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm to the human capacity of recognition of first-order arithmetical truths and so of mathematical truths in general.

We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic. We first study FML-definable frames and give a version of the Goldblatt–Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt–Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindström type theorems for first-order modal logic. We show that (...) FML has the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property. (shrink)

A search is under way for a theory that can accommodate our intuitions in population axiology. The object of this search has proved elusive. This is not surprising since, as we shall see, any welfarist axiology that satisfies three reasonable conditions implies at least one of three counter-intuitive conclusions. I shall start by pointing out the failures in three recent attempts to construct an acceptable population axiology. I shall then present an impossibility theorem and conclude with a short discussion (...) of how it might be extended to pluralist axiologies, that is, axiologies that take more values than welfare into account. (shrink)

According to a wrong interpretation of the Bell theorem, it has been repeatedly claimed in recent times that we are forced by experiments to drop any possible form of realism in the foundations of quantum mechanics. In this paper I defend the simple thesis according to which the above claim cannot be consistently supported: the Bell theorem does not concern realism, and realism per se cannot be refuted in itself by any quantum experiment. As a consequence, realism in (...) quantum mechanics is not something that can be simply explained away once and for all on the basis of experiments, but rather something that must be conceptually characterized and discussed in terms of its foundational virtues and vices. To assess it, we cannot rely on experimentation but rather on philosophical discussion: realism is not a phlogiston-like notion, despite the efforts of the contemporary quantum orthodoxy to conceive it in Russellian terms as the relics of a bygone age. (shrink)

According to Relational Quantum Mechanics the wave function \ is considered neither a concrete physical item evolving in spacetime, nor an object representing the absolute state of a certain quantum system. In this interpretative framework, \ is defined as a computational device encoding observers’ information; hence, RQM offers a somewhat epistemic view of the wave function. This perspective seems to be at odds with the PBR theorem, a formal result excluding that wave functions represent knowledge of an underlying reality (...) described by some ontic state. In this paper we argue that RQM is not affected by the conclusions of PBR’s argument; consequently, the alleged inconsistency can be dissolved. To do that, we will thoroughly discuss the very foundations of the PBR theorem, i.e. Harrigan and Spekkens’ categorization of ontological models, showing that their implicit assumptions about the nature of the ontic state are incompatible with the main tenets of RQM. Then, we will ask whether it is possible to derive a relational PBR-type result, answering in the negative. This conclusion shows some limitations of this theorem not yet discussed in the literature. (shrink)

I compare deterministic and stochastic hidden variable models of the Bell experiment, exphasising philosophical distinctions between the various ways of combining conditionals and probabilities. I make four main claims. (1) Under natural assumptions, locality as it occurs in these models is equivalent to causal independence, as analysed (in the spirit of Lewis) in terms of probabilities and conditionals. (2) Stochastic models are indeed more general than deterministic ones. (3) For factorizable stochastic models, relativity's lack of superluminal causation does not favour (...) locality over completeness. (4) If we prohibit all superluminal causation, then the violation of the Bell inequality teaches us a lesson, besides quantum mechanics' familiar ones that quantities can lack precise values and that pairs of quantities can lack joint probabilities: namely, some pairs of events are not screened off by their common past. (shrink)

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory, and of related issues such as the Kochen-Specker theorem. This extension has two main parts: the use of von Neumann algebras as a base category (Section 2); and the relation of our generalized valuations to (i) the assignment to quantities of intervals of real numbers, and (ii) the idea of a subobject of the coarse-graining presheaf (Section 3).

Bell’s theorem has fascinated physicists and philosophers since his 1964 paper, which was written in response to the 1935 paper of Einstein, Podolsky, and Rosen. Bell’s theorem and its many extensions have led to the claim that quantum mechanics and by inference nature herself are nonlocal in the sense that a measurement on a system by an observer at one location has an immediate effect on a distant entangled system. Einstein was repulsed by such “spooky action at a (...) distance” and was led to question whether quantum mechanics could provide a complete description of physical reality. In this paper I argue that quantum mechanics does not require spooky action at a distance of any kind and yet it is entirely reasonable to question the assumption that quantum mechanics can provide a complete description of physical reality. The magic of entangled quantum states has little to do with entanglement and everything to do with superposition, a property of all quantum systems and a foundational tenet of quantum mechanics. (shrink)

I note that the logics of the relevant group most closely tied to the research programme in paraconsistency are those without the contraction postulate(A.AB).AB and its close relatives. As a move towards gaining control of the contraction-free systems I show that they are prime (that wheneverA B is a theorem so is eitherA orB). The proof is an extension of the metavaluational techniques standardly used for analogous results about intuitionist logic or the relevant positive logics.

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms (...) are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma. (shrink)

Any attempt to construct a realist interpretation of quantum theory founders on the Kochen-Specker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each (...) context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalised valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalised valuation. A key ingredient throughout is the idea that, in a situation where no normal truth-value can be given to a proposition asserting that the value of a physical quantity A lies in a set D of real numbers , it is nevertheless possible to ascribe a partial truth-value which is determined by the set of all coarse-grained propositions that assert that some function f(A) lies in f(D), and that are true in a normal sense. The set of all such coarse-grainings forms a sieve on the category of self-adjoint operators, and is hence fundamentally related to the theory of presheave. (shrink)

The PBR theorem, which implies that the Einsteinian realist view on quantum mechanics (QM) is inconsistent with predictions of the standard Copenhagen view on QM, has been hailed as one of the most important theorems in the foundations of QM. Here we show that the special measurement, used by Pusey et al. to derive the theorem, is nonexisting from the Einsteinian view on QM.

In a recent paper (Found Phys 54:14, 2024), Carcassi, Oldofredi and Aidala concluded that the \(\psi\) -ontic models defined by Harrigan and Spekkens cannot be consistent with quantum mechanics, since the information entropy of a mixture of non-orthogonal states are different in these two theories according to their information theoretic analysis. In this paper, I argue that this no-go theorem for \(\psi\) -ontic models is false by explaining the physical origin of the von Neumann entropy in quantum mechanics.

The meaning of the wave function is an important unresolved issue in Bohmian mechanics. On the one hand, according to the nomological view, the wave function of the universe or the universal wave function is nomological, like a law of nature. On the other hand, the PBR theorem proves that the wave function in quantum mechanics or the effective wave function in Bohmian mechanics is ontic, representing the ontic state of a physical system in the universe. It is usually (...) thought that the nomological view of the universal wave function is compatible with the ontic view of the effective wave function, and thus the PBR theorem has no implications for the nomological view. In this paper, I argue that this is not the case, and these two views are in fact incompatible. This means that if the effective wave function is ontic as the PBR theorem proves, then the universal wave function cannot be nomological, and the ontology of Bohmian mechanics cannot consist only in particles. This incompatibility result holds true not only for Humeanism and dispositionalism but also for primitivism about laws of nature, which attributes a fundamental ontic role to the universal wave function. Moreover, I argue that although the nomological view can be held by rejecting one key assumption of the PBR theorem, the rejection will lead to serious problems, such as that the results of measurements and their probabilities cannot be explained in ontology in Bohmian mechanics. Finally, I briefly discuss three $$\psi$$ -ontologies, namely a physical field in a fundamental high-dimensional space, a multi-field in three-dimensional space, and RDMP (Random Discontinuous Motion of Particles) in three-dimensional space, and argue that the RDMP ontology can answer the objections to the $$\psi$$ -ontology raised by the proponents of the nomological view. (shrink)

L. J. Savage and I. J. Good have each demonstrated that the expected utility of free information is never negative for a decision maker who updates her degrees of belief by conditionalization on propositions learned for certain. In this paper Good's argument is generalized to show the same result for a decision maker who updates her degrees of belief on the basis of uncertain information by Richard Jeffrey's probability kinematics. The Savage/Good result is shown to be a special case of (...) the more general result. (shrink)

A few results generalizing well-known classical model theory ones have been obtained in institution theory these last two decades (e.g. Craig interpolation, ultraproduct, elementary diagrams). In this paper, we propose a generalized institution-independent version of the Beth definability theorem.

Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following: Let T be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula in T built up from the propositional variables $q,p_{1},...,p_{n}$ , using logical connectives and the predicate Pr, has the same "fixed-points" relative to q (that is, formulas $\psi (p_{1},...,p_{n})$ for which for all $p_{1},...,p_{n}\vdash _{T}\phi (\psi (p_{1},...,p_{n}),p_{1},...,p_{n})\leftrightarrow \psi (...) (p_{1},...,p_{n})$ ) of a formula $\phi ^{\ast}$ of the same kind, obtained from φ in an effective way. Moreover, such $\phi ^{\ast}$ is provably equivalent to the formula obtained from φ substituting with $\phi ^{\ast}$ itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in φ, $\phi ^{\ast}$ is the unique (up to provable equivalence) "fixed-point" of φ. Since this result is proved only assuming Pr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma. All the results are proved more generally in the intuitionistic case. (shrink)

This paper addresses arguments that “separability” is an assumption of Bell’s theorem, and that abandoning this assumption in our interpretation of quantum mechanics (a position sometimes referred to as “holism”) will allow us to restore a satisfying locality principle. Separability here means that all events associated to the union of some set of disjoint regions are combinations of events associated to each region taken separately.In this article, it is shown that: (a) localised events can be consistently defined without implying (...) separability; (b) the definition of Bell’s locality condition does not rely on separability in any way; (c) the proof of Bell’s theorem does not use separability as an assumption. If, inspired by considerations of non-separability, the assumptions of Bell’s theorem are weakened, what remains no longer embodies the locality principle. Teller’s argument for “relational holism” and Howard’s arguments concerning separability are criticised in the light of these results. Howard’s claim that Einstein grounded his arguments on the incompleteness of QM with a separability assumption is also challenged. Instead, Einstein is better interpreted as referring merely to the existence of localised events. Finally, it is argued that Bell rejected the idea that separability is an assumption of his theorem. (shrink)

f The purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions (...) from it. (shrink)

In this paper we attempt to discuss what has Kochen-Specker theorem to say about physical invariance and quantum individuality. In particular, we will discuss the impossibility of making reference to objective physical properties within the orthodox formalism of quantum mechanics. Through an analysis of the meaning of physical invariance and quantum contextuality we will derive a Corollary to KS theorem that proves that a vector in Hilbert space cannot be interpreted coherently as an object possessing physical properties. As (...) a consequence, the notion of quantum object can be only defined in terms of nomological properties. We conclude the paper by analyzing the consequences of this Corollary to KS theorem for the ongoing debate about quantum individuality. (shrink)

Following a long-standing philosophical tradition, impartiality is a distinctive and determining feature of moral judgments, especially in matters of distributive justice. This broad ethical tradition was revived in welfare economics by Vickrey, and above all, Harsanyi, under the form of the so-called Impartial Observer Theorem. The paper offers an analytical reconstruction of this argument and a step-wise philosophical critique of its premisses. It eventually provides a new formal version of the theorem based on subjective probability.

The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal (...) notation is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)

Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $, we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that (...) is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory, that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic. (shrink)

In this paper, we generalize a result of Brown and Simpson [1] to prove that RCA0+Π0∞-BCT is conservative over RCA0 with respect to the set of formulae in the form ∃!Xφ, where φ is arithmetical. We also consider the conservation of Π00∞-BCT over Σb1-NIA+∇b1-CA.

A variety of inaccurate claims about Gold's Theorem have appeared in the cognitive science literature. I begin by characterizing the logic of this theorem and its proof. I then examine several claims about Gold's Theorem, and I show why they are false. Finally, I assess the significance of Gold's Theorem for cognitive science.

In this paper, we consider some contributions to the model theory of the logic of formal inconsistency $\mathsf{QmbC}$ as a reply to Walter Carnielli, Marcelo Coniglio, Rodrigo Podiacki and Tarcísio Rodrigues’ call for a ‘wider model theory.’ This call demands that we align the practices and techniques of model theory for logics of formal inconsistency as closely as possible with those employed in classical model theory. The key result is a proof that the Keisler–Shelah isomorphism theorem holds for $\mathsf{QmbC}$, (...) i.e. that the strong elementary equivalence of two $\mathsf{QmbC}$ models $\mathfrak{A}$ and $\mathfrak{B}$ is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between $\mathsf{QmbC}$ models, explicitly prove Łoś’ theorem and introduce a useful technique of model-theoretic ‘atomization’ in which the satisfaction sets of non-deterministically evaluated formulae are associated with new predicates. Finally, we consider some of the extensions of $\mathsf{QmbC}$, explicitly showing that Keisler–Shelah holds for $\mathsf{QCi}$ and suggesting that it holds of extensions like $\mathsf{QCila}$ and $\mathsf{QCia}$ as well. (shrink)

This paper reconstructs and evaluates the representation theorem presented by Ramsey in his essay 'Truth and Probability', showing how its proof depends on a novel application of Hölder's theory of measurement. I argue that it must be understood as a solution to the problem of measuring partial belief, a solution that in many ways remains unsurpassed. Finally I show that the method it employs may be interpreted in such a way as to avoid a well known objection to it (...) due to Richard Jeffrey. (shrink)

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the (...) proof, in particular I0+EXP. The method is adapted to obtain a similar completeness result for the Rosser logic. (shrink)

This paper concerns voting with logical consequences, which means that anybody voting for an alternative x should vote for the logical consequences of x as well. Similarly, the social choice set is also supposed to be closed under logical consequences. The central result of the paper is that, given a set of fairly natural conditions, the only social choice functions that satisfy social logical closure are oligarchic (where a subset of the voters are decisive for the social choice). The set (...) of conditions needed for the proof include a version of Independence of Irrelevant Alternatives that also plays a central role in Arrow's impossibility theorem. (Published Online July 11 2006) Footnotes1 Much of this article was written while the author was a fellow at the Swedish Collegium for Advanced Study in the Social Sciences (SCASSS) in Uppsala. I want to thank the Collegium for providing me with excellent working conditions. Wlodek Rabinowicz and other fellows gave me valuable comments at a seminar at SCASSS when an early version of the paper was presented. I also want to thank Luc Bovens, Franz Dietrich, Christian List and an anonymous referee for their excellent comments on a later version. The final version was prepared during a stay at Oxford University for which I am grateful to the British Academy. (shrink)

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms (...) and proof processes of that formal system. Gentzen's proof succeeds by utilisingtransfinite inductionto prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to provethe restricted ordinal theorem, that a descending sequence of ordinals, less thanε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction isfinitist. (shrink)

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with (...) a counting argument. We extend his method to arbitrary similarity types. (shrink)

We show that \\) cannot be proved with one typical application of \\) in an intuitionistic extension of \ to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach’s axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility.

We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for ${\Sigma^0_2}$ formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs.

In this paper we attempt to physically interpret the Modal Kochen–Specker theorem. In order to do so, we analyze the features of the possible properties of quantum systems arising from the elements in an orthomodular lattice and distinguish the use of “possibility” in the classical and quantum formalisms. Taking into account the modal and many worlds non-collapse interpretation of the projection postulate, we discuss how the MKS theorem rules the constraints to actualization, and thus, the relation between actual (...) and possible realms. (shrink)

In a recent article I compared the problem of theory choice, in which scientists must choose between competing theories, with the problem of social choice, in which society must choose between competing social alternatives. I argued that the formal machinery of social choice theory can be used to shed light on the problem of theory choice in science, an argument that has been criticized by Michael Morreau and Jacob Stegenga. This article replies to Morreau’s and Stegenga’s criticisms.