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.

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)

In the traditional formalism of quantum mechanics, a simple direct proof of the Spin Geometry Theorem of Penrose is given; and the structure of a model of the ‘space of the quantum directions’, defined in terms of elementary SU-invariant observables of the quantum mechanical systems, is sketched.

Hardy’s $$\psi$$ -ontology theorem proves the reality of the wave function under the assumption of restricted ontic indifference. It has been conjectured that restricted ontic indifference, which is a very strong assumption from the $$\psi$$ -epistemic view, can be derived from two weaker sub-assumptions: an ontic state assumption and a locality assumption. However, Leifer argued that this derivation cannot go through when considering the existence of the vacuum state in the second-quantized description of quantum states. In this paper, I (...) present a new analysis of Hardy’s theorem. First, I argue that the ontic state assumption is valid in the second-quantized description of quantum states. Second, I argue that the locality assumption is a locality assumption for product states and it is weaker than the preparation independence assumption of the PBR theorem. Third, I argue that Leifer’s objection to the derivation of restricted ontic indifference is invalid. Finally, I argue that although the vacuum state is irrelevant, the existence of the tails of the wave function will block the derivation of restricted ontic indifference from the ontic state assumption and the locality assumption. (shrink)

There is a theorem that shows that it is impossible for an algorithm to jointly satisfy the statistical fairness criteria of Calibration and Equalised Odds non-trivially. But what about the recently advocated alternative to Calibration, Base Rate Tracking? Here, we show that Base Rate Tracking is strictly weaker than Calibration, and then take up the question of whether it is possible to jointly satisfy Base Rate Tracking and Equalised Odds in non-trivial scenarios. We show that it is not, thereby (...) establishing an even more general impossibility theorem. (shrink)

In this work we consider the problem of the approximate hedging of a contingent claim in the minimum mean square deviation criterion. A theorem on martingale representation in case of discrete time and an application of the result for semi-continuous market model are also given.

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)

Lewis proved a Dutch book theorem for Conditionalization. The theorem shows that an agent who follows any credal update rule other than Conditionalization is vulnerable to bets that inflict a sure loss. Lewis’s theorem is tailored to factive formulations of Conditionalization, i.e. formulations on which the conditioning proposition is true. Yet many scientific and philosophical applications of Bayesian decision theory require a non-factive formulation, i.e. a formulation on which the conditioning proposition may be false. I prove a (...) Dutch book theorem tailored to non-factive Conditionalization. I also discuss the theorem’s significance. (shrink)

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

The problem of multiple-computations discovered by Hilary Putnam presents a deep difficulty for functionalism (of all sorts, computational and causal). We describe in out- line why Putnam’s result, and likewise the more restricted result we call the Multiple- Computations Theorem, are in fact theorems of statistical mechanics. We show why the mere interaction of a computing system with its environment cannot single out a computation as the preferred one amongst the many computations implemented by the system. We explain why (...) nonreductive approaches to solving the multiple- computations problem, and in particular why computational externalism, are dualistic in the sense that they imply that nonphysical facts in the environment of a computing system single out the computation. We discuss certain attempts to dissolve Putnam’s unrestricted result by appealing to systems with certain kinds of input and output states as a special case of computational externalism, and show why this approach is not workable without collapsing to behaviorism. We conclude with some remarks about the nonphysical nature of mainstream approaches to both statistical mechanics and the quantum theory of measurement with respect to the singling out of partitions and observables. (shrink)

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)

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)

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)

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.

Metaphysics, geometry, and the problems with diagrams -- The Pythagorean theorem: Euclid I.47 and VI.31 -- Thales and geometry: Egypt, Miletus, and beyond -- Pythagoras and the famous theorems -- From the Pythagorean theorem to the construction of the cosmos out of right triangles.

We study the first-order consequences of Ramsey’s Theorem fork-colourings ofn-tuples, for fixed$n, k \ge 2$, over the relatively weak second-order arithmetic theory$\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of$\mathrm {RCA}^*_0$that does not satisfy$\Sigma ^0_1$induction,$\mathrm {RT}^n_k$is equivalent to its relativization to any proper$\Sigma ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe.We give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$for$n (...) \ge 3$. We show that they form a non-finitely axiomatizable subtheory of$\mathrm {PA}$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _{\ell +3}$fragment for$\ell \ge 1$lies between$\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$and$\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$form a subtheory of$\mathrm {I} \Sigma _2$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _4$fragment is strictly weaker than$\mathrm {B} \Sigma _2$but not contained in$\mathrm {I} \Sigma _1$.Additionally, we consider a principle$\Delta ^0_2$-$\mathrm {RT}^2_2$which is defined like$\mathrm {RT}^2_2$but with both the$2$-colourings and the solutions allowed to be$\Delta ^0_2$-sets rather than just sets. We show that the behaviour of$\Delta ^0_2$-$\mathrm {RT}^2_2$over$\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$is in many ways analogous to that of$\mathrm {RT}^2_2$over$\mathrm {RCA}^*_0$, and that$\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$-$\mathrm {RT}^2_2$is$\Pi _4$- but not$\Pi _5$-conservative over$\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of$\Pi _5$-conservativity is not provable in$\mathrm {RCA}_0 +\mathrm {RT}^2_2$. (shrink)

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure, then $$\mathsf S4$$ S4 is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.

A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

This little gem is stated unbilled and proved in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows:Second Recursion Theorem. Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of arguments with values in V so that the (...) standard assumptions and hold with. Every n-ary recursive partial function with values in V is for some e. For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let.We will abuse notation and write ž; rather than ž() when m = 0, so that takes the simpler formin this case ). (shrink)

We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \ provability predicate of T whose provability logic is precisely the modal logic \. For this purpose, we introduce a new bimodal logic \, and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \.

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)

The two theories that revolutionized physics in the twentieth century, relativity and quantum mechanics, are full of predictions that defy common sense. Recently, we used three such paradoxical ideas to prove “The Free Will Theorem” (strengthened here), which is the culmination of a series of theorems about quantum mechanics that began in the 1960s. It asserts, roughly, that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity. More precisely, (...) if the experimenter can freely choose the directions in which to orient his apparatus in a certain measurement, then the particle’s response (to be pedantic—the universe’s response near the particle) is not determined by the entire previous history of the universe. Our argument combines the well-known consequence of relativity theory, that the time order of space-like separated events is not absolute, with the EPR paradox discovered by Einstein, Podolsky, and Rosen in 1935, and the Kochen-Specker Paradox of 1967 (See [2].) We follow Bohm in using a spin version of EPR and Peres in using his set of 33 directions, rather than the original configuration used by Kochen and Specker. More contentiously, the argument also involves the notion of free will, but we postpone further discussion of this to the last section of the article. Note that our proof does not mention “probabilities” or the “states” that determine them, which is.. (shrink)

In 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness showed that not every modal logic (...) can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided. (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.

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we (...) offer a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics. (shrink)

We investigate the extent to which structures consisting of sequences of forests on the same underlying set are well-quasi-ordered under embeddings.

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)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

We say that a linear ordering is extendible if every partial ordering that does not embed can be extended to a linear ordering which does not embed either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddability, contains no infinite descending chain and no infinite antichain. In this paper we study (...) the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of .We also prove that Fraïssé’s conjecture is equivalent, over , to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+,−}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings.While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just . Moreover, for all these linear orderings, , we prove that any partial ordering, , which does not embed has a linearization, hyperarithmetic in , which does not embed. (shrink)

This paper reassesses the question of whether Craig’s theorem poses a challenge to Quine's empirical underdetermination thesis. It will be demonstrated that Quine’s account of this issue in his paper “Empirically Equivalent Systems of the World” (1975) is flawed and that Quine makes too strong a concession to the Craigian challenge. It will further be pointed out that Craig’s theorem would threaten the empirical underdetermination thesis only if the set of all relevant observation conditionals could be shown to (...) be recursively enumerable — a condition which Quine seems to overlook — and it will be argued that, at least within the framework of Quine’s philosophy, it is doubtful whether this condition is satisfiable. (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.

ABSTRACTCitizens in John Rawls's well-ordered society face an assurance dilemma. They wish to act justly only if they are reasonably sure their fellow citizens will also act justly. According to Rawls, this assurance problem is solved through public reasoning. This paper argues that public reason cannot serve this function. It begins by arguing that one kind of incompleteness public reason faces that most Rawlsians grant is ubiquitous but unproblematic from a normative standpoint is problematic from an assurance perspective: it makes (...) it possible for citizens to argue for policy conclusions that are favored by their private interests, rather than justice. In response, perhaps the thing to do is structure deliberative democratic institutions such that citizens will always be incentivized to use public reasons to only argue for conclusions they believe are favored by justice. The paper proves that this is impossible by extending the Gibbard–Satterthwaite theorem. (shrink)

This paper provides a philosophical analysis of the ongoing controversy surrounding R.A. Fisher's famous ‘fundamental theorem’ of natural selection. The difference between the ‘traditional’ and ‘modern’ interpretations of the theorem is explained. I argue that proponents of the modern interpretation have captured Fisher's intended meaning correctly and shown that the theorem is mathematically correct, pace the traditional consensus. However, whether the theorem has any real biological significance remains an unresolved issue. I argue that the answer depends (...) on whether we accept Fisher's non-standard notion of environmental change, on which the theorem rests; arguments for and against this notion are explored. I suggest that there is a close link between Fisher's fundamental theorem and the modern ‘gene's eye’ view of evolution. Introduction What Does the Fundamental Theorem Say? Key Concepts Explained Alleged Significance of the FTNS Traditional versus Modern Interpretations of the FTNS The Modern Interpretation Illustrated Fisher's Concept of ‘Environmental Change’ Causality and the Modern Interpretation The Significance of the FTNS Re-considered Appendix CiteULike Connotea Del.icio.us What's this? (shrink)

This paper explores the role that difference plays in collective decision-making using the Condorcet jury theorem. Agents facing a dichotomous decision might prove biased toward one of the options facing them. That is, they may be more likely to decide correctly when one of the options is correct than when the other option is. A juror might be more likely to convict a guilty defendant than to acquit an innocent one. Agents may display opposing biases. This paper identifies the (...) optimal committee composition in the presence of such bias. This composition depends upon both the prior probability that each option is correct and the relative costs of the two types of possible error. Optimal committee composition will be expressed in terms of the absolute margin that a group with one form of bias should enjoy over the other. (shrink)

According to the Gottesman–Knill theorem, quantum algorithms that utilize only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this article that this conclusion is misleading. First, the statement of the theorem is, on reflection, already evident when we consider Bell’s and related inequalities in the (...) context of a discussion of computational machines. This, in turn, helps us to understand that the appropriate conclusion to draw from the Gottesman–Knill theorem is not that entanglement is insufficient to enable a quantum performance advantage, but rather that if we limit ourselves to the operations referred to in the Gottesman–Knill theorem, we will not have used the resources provided by an entangled quantum system to their full potential. (shrink)

One of the logic defined by Richard Epstein in a context of an analysis of subject matter relationship is Symmetric Relatedness Logic S. In the monograph [2] we can find some open problems concerning relatedness logic, a Post-style completeness theorem for logic S is one of them. Our paper introduces a solution of this metalogical issue.

We pose and resolve a puzzle about spontaneous symmetry breaking in the quantum theory of infinite systems. For a symmetry to be spontaneously broken, it must not be implementable by a unitary operator in a ground state's GNS representation. But Wigner's theorem guarantees that any symmetry's action on states is given by a unitary operator. How can this unitary operator fail to implement the symmetry in the GNS representation? We show how it is possible for a unitary operator of (...) this sort to connect the folia of unitarily inequivalent representations. This result undermines interpretations of quantum theory that hold unitary equivalence to be necessary for physical equivalence. (shrink)

Condorcet’s voting paradox shows that pairwise majority voting may lead to cyclical majority preferences. In a famous paper, Sen identified a general condition on a profile of individual preference orderings, called triplewise value-restriction, which is sufficient for the avoidance of such cycles. This note aims to make Sen’s result easily accessible. We provide an elementary proof of Sen's possibility theorem and a simple reformulation of Sen’s condition. We discuss how Sen’s condition is logically related to a number of precursors. (...) Finally, we state a necessary and sufficient condition for the avoidance of cycles, and suggest that, although there is still some logical space between that condition and Sen’s sufficient condition, Sen’s condition cannot be further generalized in an appealing way. (shrink)