Classical Proof Research Articles

Language is a “quite useless” tool: A rejoinder to Fedorenko, Piantadosi, and Gibson’s “Language is primarily a tool for communication rather than thought”

Contrary to the prevailing assumption that language is “primarily a tool for communication rather than thought”, I argue that language is, to invoke Oscar Wilde, “quite useless”. Arguing from aesthetic philosophy and the minimalist program for linguistic theory, I conject that language, like art, is not “for” anything—it simply is, conforming to aesthetic rather than utilitarian principles. Of course, like art, language can be a powerful instrument of communication, but its function is not that of expressing thought; it creates thoughts, “primarily” for communicating with oneself, engaging in Popperian critical rationalism, making thoughts (e.g., sentences, constructive proofs) to match Platonic objects (e.g., propositions, classical proofs).

(Conceptual) explanations in logic

Abstract To explain phenomena in the world is a central human activity and one of the main goals of rational inquiry. There are several types of explanation: one can explain by drawing an analogy, as one can explain by dwelling on the causes (see e.g. see [Woodward (2004, Making Things Happen: A Theory of Causal Explanation. Oxford University Press, Oxford)]. Amongst these different kinds of explanation, in the last decade, philosophers have become receptive to those explanations that explain by providing the reasons (or the grounds) why a statement is true; these explanations are often called conceptual explanations (e.g. see [Betti (2010, Explanation in metaphysics and Bolzano's theory of ground and consequence. Logique et analyse, 211:281316)]). The main aim of the paper is to propose a logical account of conceptual explanations. We will do so by using the resources of proof theory, in particular sequent rules analogous to deep inferences ([e.g. see Brunnler (2004, Deep Inference and Symmetry in Classical Proofs. Logoc Verlag)]). The results we provide not only shed light on conceptual explanations themselves, but also on the role that logic and logical tools might play in the burgeoning field of inquiry concerning explanations. Indeed, we conclude the paper by underlining interesting links between the present research and some other existing works on explanations and logic that have arise in recent years, e.g. see [Arieli et al. (2022, Explainable logic-based argumentation. Computational Models of Argument, 353:3243); Darwiche and Hirth (2023, On the (complete) reasons behind decisions. Journal of Logic Language and Information, 32:6388); Piazza, Pulcini, and Sabatini (2023, Abduction as deductive saturation: a proof-theoretic inquiry. Journal of Philosophical Logic, 52:15751602)]. For here it is for the empirical scientist to know the fact and for the mathematical to know the reason why (our emphasis) [Aristotle (1993, Posterior Analytics. Oxford University Press, Oxford)].

Imaging endolymphatic space of the inner ear in vestibular migraine

BackgroundVestibular migraine (VM), the most frequent episodic vertigo, is difficult to distinguish from Ménière’s disease (MD) because reliable biomarkers are missing. The classical proof of MD was an endolymphatic hydrops...

The higher dimensional propositional calculus

Abstract In recent research, some of the present authors introduced the concept of an $n$-dimensional Boolean algebra and its corresponding propositional logic $n\textrm{CL}$, generalizing the Boolean propositional calculus to $n\geq 2$ perfectly symmetric truth values. This paper presents a sound and complete sequent calculus for $n\textrm{CL}$, named $n\textrm{LK}$. We provide two proofs of completeness: one syntactic and one semantic. The former implies as a corollary that $n\textrm{LK}$ enjoys the cut admissibility property. The latter relies on the generalization to the $n$-ary case of the classical proof based on the Lindenbaum algebra of formulas and Boolean ultrafilters.

Coarsening Natural Deduction Proofs I: Finding Perfect Proofs

Abstract This paper explores how, given a proof, we can systematically transform it into a proof that contains no irrelevancies and which is as strong as possible. I define a weaker and stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. Using classical core logic to study classical validities and core logic to study intuitionistic validities, I show that every core proof or classical core proof can be transformed into a perfect proof. In a sequel paper, I show how proofs in core logic can also be transformed into gaunt proofs and I observe that this property fails for classical core logic.

PLANE METRICS AND THE UNIFORMIZATION THEOREM ON MINIMAL THE THREE DIMENSIONAL SPHERE

Let (M, g) be a torus with a riemannian metric g. The Uniformization theorem implies the existence of a covering conforma! map <:p : R2 ----> /1.1 such that the group H,t, of diffeo­morphisms p: R2 ---+ R2 such that <Pop= <j,, is a subgroup of the group of translations in R2 and is isomorph to Z2• The classical proof of his theorem uses tools from algebraic topology and PDE's in Sobolev spaces. In this paper we will give an alternative and easier proof of this theorem in the case that M is a minimal immersed torus in the unit three dimensional sphere, S3. We will achieve this theorem by studying the shape operator of minimal torus on the sphere. One of the main ideas of the this paper is to induce the reader to the study of minimal torus in S3• Recall that one important conjecture in differential geometry is Law­son's conjecture. This conjecture states that the only minimal embedded tori in S3 are the Clifford tori.

Abdüllatif Harpûtî’nin İsbât-ı Vâcib Delilleri

Abdullatif Harpūtī (d.1916) was a late Ottoman scholar who worked as a professor of "Kalām" at the Istanbul Darulfunun. Recognizing the deficiencies in the science of kalām and arguing that some changes should be made in terms of method and content, Harpūtī states that the time has come for the third compilation in the science of kalām and that this compilation is obligatory upon the scholars of the century. He himself tried to serve as a model for this period with his book Tanqīḥ al-kalām. His study, which consists of muqaddimah (preface), divinity, prophecy, sam'iyyāt, and imāmate chapters, deals with the currents of thought that emerged in the Western world by bringing evidence against them. Harpūtī states that the scientific data on matters of faith, which are put forward by rational methods, change over time. For this reason, while writing chapters such as divinity and prophecy, he used different information in light of the scientific data of his time. He also included and refuted the views of movements such as materialism, positivism, and atheism, which were newly widespread in his time. In this paper, we have analyzed the proofs used by al Harpūtī on the subject of ithbāt al-wājib. Harpūtī analyzed the proofs of ithbāt al-wājib under the subheadings of "the proofs of philosophers and muhaqqiq theologians", "the path followed by most theologians", "the path followed by the prophets and asfiyān", and "the recent theologian philosophers and ithbāt al-wājib”. These subheadings do not clearly indicate which of the classical proofs he means. By "the proof of the philosophers and the muhaqqiq theologians" Harpūtī meant the evidence of possibility (imkān); by "the path followed by most theologians" he meant the proof of existence (huduth); and by “the path followed by the Prophets and asfiyān" he meant the evidence of purpose and order. Under the title “The recent theologian philosophers and ithbāt al-wājib", he collected the proof of possibility, the proof of motion, and the proofs of purpose and order. The proof of the disposition (fıtrah) that is included in our study is not to be found in the titles. However, from Harpūtī 's explanations, we have determined that he also used this argument. This study aims to reveal the proofs of ithbāt al-wājib used by al Harpūtī . This research is significant for us to comprehend what evidence Harpūtī , who opened the door to the new science of Kalām and is considered one of the important figures of the new science of Kalām, used for proof. The documentation method was used in this research, and the data obtained were analyzed and reflected in the text. As a result of the research, it was determined that al Harpūtī used the proofs of possibility, existence, purpose and order, motion, and disposition.

Doubly adaptive zero-knowledge proofs

In [TCC 2009 and JoC 2011] Lindell and Zarosim defined adaptive-corruption zero knowledge giving to the environment the power to perform post-execution corruption only. Moreover, their zero-knowledge proof system does not enjoy adaptive-input selection since it crucially relies on fixing the inputs already at the onset of the protocol.In this paper, we propose a new definition of adaptive-corruption zero-knowledge proofs that more naturally fits the setting that benefits from adaptive-input selection. We allow the non-rewindable environment to corrupt the prover adaptively based on its visibility of the messages exchanged so far. Then we show a zero-knowledge proof system for NP that is doubly adaptive (i.e., it enjoys adaptive-corruption under our stronger notion and adaptive-input selection) matching the same assumptions and round complexity of classical zero-knowledge proofs.

Dialectica principles via Gödel doctrines

Gödel's Dialectica interpretation was conceived as a tool to obtain the consistency of Heyting arithmetic in the 40s. In recent years, several proof theoretic transformations, based on Gödel's Dialectica interpretation, have been used systematically to extract new content from classical proofs, following a suggestion of Kreisel. Thus, the interpretation has found new relevant applications in several areas of mathematics and computer science. Several authors have explained the Dialectica interpretation in categorical terms. In particular, de Paiva introduced the notion of a Dialectica category as an internal version of Gödel's Dialectica Interpretation in her doctoral work. This was generalised by Hyland and Hofstra, who considered the interpretation in terms of fibrations. In our previous work, we introduced an intrinsic presentation of the Dialectica construction via a generalisation of Hofstra's work, using the notion of Gödel fibration and its proof-irrelevant version, a Gödel doctrine. The key idea is that Gödel fibrations (and doctrines) can be thought of as fibrations generated by some basic predicates playing the role of quantifier-free predicates. This categorification of quantifier-free predicates is crucial not only to show that our notion of Gödel fibration is equivalent to Hofstra's Dialectica fibration in the appropriate way, but also to show how Gödel doctrines embody the main logical features of the Dialectica Interpretation. To show that, we derive the soundness of the interpretation of the implication connective, as expounded by Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely (suitable versions of) the Markov Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight correspondence between the logical system and the categorical framework. This tight correspondence should come in handy not only when discussing the traditional applications of Dialectica, but also when dealing with some newer uses of the interpretation, as in modelling games or concurrency theory. Finally, to complete our analysis, we characterise categories obtained as a result of the Hyland, Johnstone and Pitts tripos-to-topos construction when applied to Gödel doctrines.

Numerical Stabilities of Vasicek and Geometric Brownian Motion Models

Stochastic differential equations (SDEs) are very often used as models for a large number of phenomena in the physical, economic and management sciences. They generalize the notion of ordinary differential equations, taking into account a white additive and multiplicative noise term, to model random trajectories such as stock market prices or particles movements, on the quantum scale, subject to diffusion phenomena. In rare cases, it is generally impossible to have explicit solution to these equations. In this case, the numerical approach, presenting itself under various aspects, is the only favorable outcome. However, the stability of numerical schemes for stochastic differential equations solution is much more significant. In this paper, we establish and make a classical proof of the mean and mean-square stabilities of the numerical SDEs schemes for Vasicek and Geometric Brownian motion models.

Classical reasoning in the justification paradigm

Abstract Artemov, building upon a tradition beginning with Kolmogorov and Gödel, developed a paradigm for understanding Constructive Reasoning in terms of classical proofs. Kolmogorov–Gödel–Artemov constructivism flies in the face of the usual understanding of Constructive Reasoning as being distinguished from Classical Reasoning in terms of its theory of truth. Is there something that stands to traditional Classical Reasoning as Kolmogorov–Gödel–Artemov constructivism stands to Constructive Reasoning? In this paper, we develop an affirmative answer to this question by presenting a justification account of Classical Reasoning in terms of explicit justification. The traditional truth paradigm account of Classical Reasoning leads to the well-known paradoxes of material implication. We show that the justification account of Classical Reasoning avoids this problem.1

Quantum Proofs of Proximity

We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property Π and reject inputs that are ε-far from Π, while only probing a minuscule portion of their input.We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016).Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable. A consequence of this framework is a QMA algorithm to verify the Parity of an n-bit string with O(n2/3) queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.

Доказательства бытия Бога – случай Достоевского

This theme was one of the main ones for Dostoevsky, he turned to it throughout his life and work. As everyone knows, its essence has taken the form of catchphrases like „If there is no God, then we can do everything.” The theme has given rise to a huge number of interpretations. Some of them go back to the discourse of proofs of the existence of God as it has developed in the history of philosophy. In Christian theology, the argument relating to the question of the existence of God has received the general definition of “ontological proof of the existence of God.” In general, the scheme of reasoning in the ontological proof can be represented as follows. An absolutely perfect being, such as God appears, must have the totality of perfections; existence is one of these perfections; God, as a perfect being, must have existence not only in our imagination, but also in reality; therefore, God exists. In the Western European philosophical tradition, Descartes, Leibniz, Kant, Hegel, and others turned to ontological proof. Kant, examining the classical proofs of the existence of God, showed that the existence of a thing cannot be seen in the content of its concept. Thus, it is impossible neither to prove nor disprove the existence of God. The appeal to this topic in the general cultural Russian discourse, and in the emerging Russian philosophical discourse, was due to two circumstances. First, the influence of European classical philosophy. One can recall Pushkin’s lines: “By the name of Vladimir Lensky, / With a soul directly Goettingent, / A handsome man in full bloom of years, / An admirer of Kant and a poet,” or Belinsky’s mental anguish about the Hegelian formula “everything real is reasonable, everything reasonable is real”. Secondly, the impact of Western technogenic civilization on Russian society, traditional in its type. One of the manifestations of this influence was the increased tension along the line “faith – knowledge” (“religion – science”). The article attempts to translate Dostoevsky’s plots into axiology and the theory of values. The general conclusion that follows from the analysis of Dostoevsky’s work in the context of the article can be formulated as follows: the denial of God leads to the legalization of his surrogate likenesses.

Quantum generalizations of the polynomial hierarchy with applications to QMA(2)

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, \(\rm{QCPH}\), uses classical proofs, and the second, \(\rm{QPH}\), uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, \(\rm{Q\Sigma_3}\), into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for \(\rm{QMA(2)}\), the variant of Quantum Merlin-Arthur (\(\rm{QMA}\)) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if \(\rm{QCPH = QPH}\) (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in \({\rm P}^{{\rm pp}^{{\rm pp}}}\)). Second, because \(\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}\), \(\rm{QMA(2)}\) is strictly contained in NEXP unless \(\rm{QMA(2)}=\rm{Q\Sigma_3}\) (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').

Convergent Nested Alternating Minimization Algorithms for Nonconvex Optimization Problems

We introduce a new algorithmic framework for solving nonconvex optimization problems, that is called nested alternating minimization, which aims at combining the classical alternating minimization technique with inner iterations of any optimization method. We provide a global convergence analysis of the new algorithmic framework to critical points of the problem at hand, which to the best of our knowledge, is the first of this kind for nested methods in the nonconvex setting. Central to our global convergence analysis is a new extension of classical proof techniques in the nonconvex setting that allows for errors in the conditions. The power of our framework is illustrated with some numerical experiments that show the superiority of this algorithmic framework over existing methods. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant 800240] and the Israel Science Foundation [Grant 2480/21].

From Magnitudes to Geometry and Back: De Zolt's Postulate.

A crucial trend of nineteenth‐century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon.” This proposition, known as De Zolt's postulate, was conceived as a strictly geometrical expression of the general principle of magnitudes “the whole is greater than the part.” On the one hand, we illustrate this striving for purity in the foundations of geometry by analysing David Hilbert's classical proof of De Zolt's postulate. On the other hand, we connect this geometrical problem with the first axiomatizations of the concept of magnitude by the end of the nineteenth century. In particular, we argue that a recent result in the logical analysis of the concept of magnitude casts new light on Hilbert's proof. We also outline an alternative development of a theory of magnitude that includes a proof of De Zolt's postulate in an abstract setting.

Carleman estimates and the contraction principle for an inverse source problem for nonlinear hyperbolic equations

The main aim of this paper is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the inverse problem. To find this fixed point, we define a recursive sequence with an arbitrary initial term by the same manner as in the classical proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution with the exponential rate. Therefore, our new method can be considered as an analog of the contraction principle. We rigorously study the stability of our method with respect to noise. Numerical examples are presented.

Optimal transport pseudometrics for quantum and classical densities

This paper proves variants of the triangle inequality for the quantum analogues of the Wasserstein metric of exponent 2 introduced in Golse et al. (2016) [13] to compare two density operators, and in Golse and Paul (2017) [14] to compare a phase space probability measure and a density operator. The argument differs noticeably from the classical proof of the triangle inequality for Wasserstein metrics, which is based on a disintegration theorem for probability measures, and uses in particular an analogue of the Kantorovich duality for the functional defined in Golse and Paul (2017) [14]. Finally, this duality theorem is used to define an analogue of the Brenier transport map for the functional defined in Golse and Paul (2017) [14] to compare a phase space probability measure and a density operator.

Effective resistance is more than distance: Laplacians, Simplices and the Schur complement

This article reviews and discusses a geometric perspective on the well-known fact in graph theory that the effective resistance is a metric on the nodes of a graph. The classical proofs of this fact make use of ideas from electrical circuits or random walks; here we describe an alternative approach which combines geometric (using simplices) and algebraic (using the Schur complement) ideas. These perspectives are unified in a matrix identity of Miroslav Fiedler, which beautifully summarizes a number of related ideas at the intersection of graphs, Laplacian matrices and simplices, with the metric property of the effective resistance as a prominent consequence.

A simple and constructive proof to a generalization of Lüroth's theorem

A generalization of L{\"u}roth's theorem expresses that every transcendence degree 1 subfield of the rational function field is a simple extension. In this note we show that a classical proof of this theorem also holds to prove this generalization.