Proof and Reasoning

Quale fenomeno trasversale all’intera esperienza umana, la prova può presentarsi nelle forme più disparate. Nel prosieguo mi propongo di fissare gli aspetti di identità e di differenza delle varie figure probatorie, al fine di selezionare le caratteristiche che compongono la morfologia della prova giudiziaria prevista nell’ordinamento italiano (d’ora in poi: PGOI), in particolare penale, e in ordinamenti analoghi.
Più precisamente, mi concentrerò sulla prova del giudice di primo grado, dato che quella del giudice del gravame di merito presenta caratteristiche
che la rendono irriducibile al modello qui delineato, e dato che il giudice di legittimità non ha il compito di provare, bensì di controllare la correttezza
del ragionamento probatorio esposto nel provvedimento impugnato.
Sul piano metodologico, sebbene nell’epoca del “pensiero liquido” l’idea di sistema sia considerata un’anticaglia, e le tassonomie in cui si traduce siano considerate superflue, a mio avviso entrambe conservano una funzione ordinatrice essenziale. Procederò dunque alla stesura di una tassonomia ad albero delle forme di prova. Più precisamente, dopo aver tratteggiato la nozione generale di “prova” (§ 2), fisserò la summa divisio tra prove singolari e prove collettive (§ 3). Nel regno delle prime, distinguerò la classe delle prove ricostruttive da quella delle prove constatative (§ 4). Nella classe delle prove ricostruttive, distinguerò l’ordine delle prove lato sensu storiche da quello delle prove attuali (§ 5). Nell’ordine delle prove lato sensu storiche, distinguerò la famiglia delle prove istituzionali da quella delleprove stricto sensu storiche (§ 6). Nella famiglia delle prove istituzionali, distinguerò il genere delle prove empiriche da quello delle prove mistiche (§ 7). Nel genere delle prove empiriche, distinguerò la specie delle prove autonome da quella delle prove eteronome (§ 8). Nella specie delle prove autonome, distinguerò la sottospecie delle prove pubbliche da quella delle prove intime (§ 9).

The idea of free evaluation of evidence has a long history. In the contemporary constitutional State it constitutes a fundamental principle of the law of evidence. The paper reviews the main stages of the historical evolution of this idea and looks into the restrictions it implies for the legislator and the judge in the field of criminal trial.

The single most important statement that can be made with regard to the logical status of human and physical geographical reasoning is that it belongs to the class of non-monotonic reasoning. In other words,  geographical reasoning is defeasible (i.e. earlier drawn conclusions can be defeated by the addition of new facts, some of which are generated by the very application of our ideas outside academe) and non-demonstrative (i.e. our beliefs about reality cannot be demonstrated once and for ever; we must do with building provisional arguments based on the current weight of the evidence, which itself changes with the times). Indeed, unlike mathematics, which is a closed world of formalisms that relies heavily on deduction, demonstration, and proof, geography studies and attempts to improve the real, messy world out there. Because that wild, unruly world is always in a process of becoming, the conclusions drawn yesterday may have to be abandoned today, and so on, without end in sight. The core virtue of geographical reasoning is not certainty, but adaptability. As the world changes, so do geography’s entertained beliefs about how to improve it. The point we are trying to make is that an alternative window into the nature of geographical reasoning, besides the one provided by the discipline’s key concepts, offers an opening toward the more subtle problematic of the mechanisms by which the updating of geographical beliefs operate.

In 1933 G ¨ odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G ¨ odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G ¨ odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and-calculus.

On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans.
Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique.
Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie).

By explaining the argument from ignorance in terms of the presumption of innocence, many textbooks in argumentation theory suggest that some arguments from ignorance might share essential features with some types of presumptive reasoning. The stronger version of this view, suggesting that arguments from ignorance and presumptive reasoning are almost indistinguishable, is occasionally proposed by Douglas Walton. This paper explores the nature and limits of the stronger proposal and argues that initial presumptions and arguments from ignorance are not closely connected. There are three main reasons. First, the argument from ignorance, unlike typical presumptive reasoning, is a negative kind of inference. Second, the typical initial presumption is sensitive to a broader set of defeaters and thus assumes a higher (negative) standard of acceptability. Third, in dialectical terms, initial presumption and argument from ignorance bring different attacking rights and obligations. I conclude that Waltonian intuition is unsupported or, at best, is limited only to practical presumptions and practical arguments from ignorance.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

In an empirical study, the impact of exposing Linear Algebra students to a ‘Transparent Pseudo-Proof’ (TPP) of theorems was compared to that of exposing a comparable group of students to the formal proof of the same theorems. Following an analysis of the quantitative data collected in 36 sessions, this paper suggests a distinction between the benefit from employing TPPs to the teaching and learning of algorithmic proofs and that of employing TPPs to non-algorithmic proofs. Based upon analyses of the qualitative data gathered employing several tools, this paper discusses the particular advantages of employing TPPs to non-algorithmic proofs.

El libro Verdad. Controversias abiertas analiza la discusión habida desde un punto de vista histórico y sistemático del concepto “verdad” desde sus origines hasta la actualidad. En nuestro uso cotidiano, el término “verdad” hace referencia a que lo que expresamos está de acuerdo con los hechos o la realidad. También nos referimos a que algo es auténtico o fiel. Muchas actividades humanas dependen de dicho concepto por lo que se asume su naturaleza como concepto. Por ello, “verdad” es abordado en la amplia mayoría de las ciencias, el Derecho o nuestra la vida cotidiana. Para Parménides, la esencia de la verdad se enlaza al desenmascaramiento o la revelación. Así pues, el término está estrechamente emparentado al hallazgo de lo que antes estaba oculto como revela el uso del término griego de aletheia. Consecuentemente, el concepto de verdad se considera esencial por lo que no puede ser descripto en términos que sean más sencillos de entender que el concepto de verdad mismo. En dicho contexto, la verdad se enmarca en una correspondencia entre el lenguaje del pensamiento con una realidad independiente.
Requerimos más de dos mil años para transformar este enfoque analizando el enigma en un marco más adecuado, dilucidando su estructura lógica y semántica con el fin de descubrir sus límites y posibilidades. Verdad se opone a falsedad que asume asimismo un significado formal. Con ello, el lenguaje y las palabras dejan de ser un medio utilizado para determinar lo que es una “verdad” y se transforma en el centro de examen. Se estudia la estructura del lenguaje en el que aparece el predicado “verdad” y el modo como viene adscrito así como las funciones que asume. Esta investigación desplaza las viejas cuestiones acerca de las indagaciones sobre qué cosas son verdaderas o falsas; o, cómo definir, identificar y distinguir la verdad; o, si la verdad es subjetiva u objetiva, relativa o absoluta.
Gracias a la indagación formal se han conseguido seleccionar ciertos patrones racionales que pueden ayudarnos a decidir si una proposición es verdadera o falsa. En dicho sentido “verdad” se estudia en el marco de un lenguaje específico mediante una interpretación específica, considerando un sistema lógico afín y los presupuestos empíricos en los que se desenvuelve. Por esta razón, cualquier enunciado no es meramente considerado como una afirmación contingente sobre algún hecho aislado sino que ha de ser detallado en todas las interpretaciones posibles. Hemos puesto especial atención en desarrollar la teoría semántica de la verdad en la que se analizan los lenguajes abiertos. Según este punto de vista, ningún lenguaje puede contener su propio predicado de verdad por lo que la expresión “es verdadero” sólo se aplica a los enunciados del lenguaje objeto desde el metalenguaje. La razón de esta restricción se debe a que los lenguajes que contienen su propio predicado de verdad son propensos a generar enunciados paradójicos. Una consecuencia inmediata de esta condición es que la propuesta semántica no se puede aplicar a los lenguajes naturales ya que admiten sus propios predicados de verdad.
Por el contrario, se ha propuesto como modelo alternativo escrutar el predicado “verdad” introduciendo una restricción de los lenguajes naturales por lo que se puede probar que pueden contener su propio predicado de verdad sin dar lugar a la contradicción. El libro muestra el procedimiento seguido en la interpretación en los que intervienen lenguajes cerrados. Desde un punto de vista formal dicha propuesta implica que algunas oraciones no son ni verdaderas ni falsas. Esto contradice el principio de la bivalencia según el cual todo enunciado es, o bien, verdadero, o bien, falso. Este principio es una premisa clave para derivar la paradoja del mentiroso que también es abordado en estas páginas. En el capítulo en el que se estudia la distinción entre verdad y demostración se señala que la autorreferencia no puede evitarse ingenuamente, ya que las proposiciones sobre objetos aparentemente no relacionados pueden tener un significado autorreferencial informal. Gödel y Tarski descubrieron que el predicado “verdad” no puede ser definido de manera consistente. Por todo ello, el sistema cerrado acarrea una contradicción ya que el predicado de verdad que se construye es meramente parcial. Considera que ciertas proposiciones como las construidas en la prueba de Tarski con verdaderas pero a su vez inconsistentes. Se sigue pues discutiendo si la prueba de Tarski puede aplicarse a cualquier sistema de verdad parcial ya que hasta la fecha ninguno ha sido consistente con métodos aceptables usados en la lógica matemática.
El libro concluye con el estudio sistemático de algunas consideraciones acerca de los presupuestos empíricos que se presuponen en el uso del predicado verdadero en los modelos estudiados.

https://www.routledge.com/Practices-of-Reason-Fusing-the-Inferentialist-and-Scientific-Image/Koren/p/book/9780367702212

This book offers new insights into the nature of human rational capacities by engaging inferentialism with empirical research in the cognitive sciences.

Inferentialism advocates that humans’ unique kind of intelligence is discursive and rooted in competencies to make, assess and justify claims. This approach provides a rich source of valuable insights into the nature of our rational capacities, but it is underdeveloped in important respects. For example, little attempt has been made to assess inferentialism considering relevant scientific research on human communication, cognition or reasoning. By engaging philosophical and scientific approaches in a productive dialogue, this book shows how we can better understand human rational capacities by comparing their respective strengths and weaknesses. In this vein, the author critically revisits and constructively develops central themes from the work of Robert Brandom and other "language rationalists": the nature of the assertoric practice and its connection to reasoned discourse, the linguistic constitution of the shared space of reasons, the social nature and function of reasoning, the intersubjective roots of social-normative practices and the nature of objective thought.

Practices of Reason will be of interest to scholars and advanced students working in philosophy of mind, philosophy of language and philosophy of logic.

We combined the phases of the van Hiele theory with the methods of Cognitive Apprenticeship and enriched these combinations adding the three following ideas: (a) a special worksheet, named Structured Form Worksheet, which we used when teaching geometry (SFW), (b) a matrix named Reasoning Control Matrix for the Proving Process (RECOMPP), which
helped students with reasoning production and (c) the concepts of simple and partial proof to write the formal proof. We called the above mentioned combination Model of p-m Combinations.
Then, we used this model to teach geometry courses to 15-year old senior high
school (Lyceum) students. In this article we claim that students should be able to write simple and partial proofs before they are taught how to write formal proofs.

Haack opens her paper on “future directions” in scientific testimony with a quotation from Aristotle and an observation from a hundred-year old U.S. Supreme Court ruling! Her reason? Her argument will be that the hankering for exactness where no exactness is possible that Holmes identified as “a source of fallacy throughout the law” still impedes us; and, specifically, that it gets in the way of a clear understanding of the proper role of probabilistic and statistical evidence—such as testimony of random-match DNA probabilities in criminal cases, or epidemiological testimony of increased risk in toxic-tort cases—in legal proof of questions of fact. As a result, the law sometimes asks more of statistics than statistics can give, and sometimes gets less from statistics than statistics could give.

The current volume explores a variety of topical themes within the general field of religion -- defined as the search for the truth concerning the nature of one's relationship with Being/Reality. Among the topics explored are: Evolution, the origins of faith, conceptual viruses, suffering, irreligion, the new atheism, nihilism, sacredness, the nature of the self, mythology, Carl Jung, Joseph Campbell, free will, epistemology, and spiritual abuse. All of the foregoing topics are critically examined against a backdrop that helps orient the discussion. More specifically, most human beings wonder, in one way or another, about the nature of reality ... that is, they seek to deal, as best they can, with the reality problem. The Final Jeopardy challenge refers to the task of trying to work toward providing a best-effort final response to the reality problem that resonates with, and reflects to varying degrees, the character of reality, before the sands in the hourglass of time run out in a person's life.

Los problemas de demostración, vistos como una instancia de la resolución de problemas, demandan en el individuo que los afronta que ponga en juego distintos conocimientos y habilidades instrumentales cuando cuenta con el apoyo de la geometría dinámica. Sin embargo, como se muestra en esta investigación, el conjunto de conocimientos con los que un individuo cuenta y su grado de instrumentalización del software no se convierten en los únicos aspectos relevantes en el marco del proceso de resolución. Aspectos metacognitivos como el control, la regulación y la evaluación de las acciones ejecutadas pueden llevar a que un problema de demostración sea resuelto pertinentemente, aun cuando no se cuente con un gran repertorio teórico o que, inclusive, aun cuando se cuente con los conocimientos pertinentes y necesarios, el resultado obtenido o proceso de resolución no sea afortunado. Apoyados en dos grupos con un nivel de formación matemática distinta, mostramos que la metacognición se convierte en un aspecto que puede llevar a un grupo, con un conocimiento matemático reducido, a obtener mejores resultados que un grupo que cuenta con un conocimiento profundo de la disciplina.

This work presents the rhetorical tools needed to meet three of the challenges surrounding the writing of history : how to produce an acceptable discourse on a polemical subject ; how to put the intuitions of the phase of discovery into words ; how to learn lessons from history and formulate them.


L’ouvrage offre des outils rhétoriques pour la maîtrise de trois enjeux de l’écriture de l’histoire: comment produire un discours acceptable sur un sujet polémique ? Comment mettre en mots les intuitions de la phase de découverte ? Peut-on tirer des leçons de l’histoire et comment les formuler ?

Accertare BARD il dolo vuol dire che il giudice deve “captare” oltre ogni ragionevole dubbio la «sostanza psicologica» della previsione e della volontà senza – per quanto è possibile – ricorrere a presunzioni anche nei contesti precettivi intrisi di elementi valutativi. Un compito, un obiettivo difficile da realizzare, ancora più difficile nel caso del dolo eventuale; maggiori difficoltà in parte irrisolte, forse irrisolvibili. Nonostante i progressi compiuti, i “modelli” di accertamento elaborati, sinora, dalla letteratura penalistica e dalla giurisprudenza non offrono “prestazioni” del tutto appaganti. Non sfugge a questi limiti nemmeno la sentenza delle Sezioni Unite sul c.d. caso Thyssen che, nonostante il poderoso apparato teorico-motivazionale, sembra persino echeggiare formule del diritto intermedio proponendo (o ri-proponendo) ”meta-massime” di esperienza secondo le quali in base a ciò che di regola accade non si risale direttamente alle due articolazioni della sostanza psicologica del dolo (la previsione e la volontà) ma vengono individuati i contrassegni che di solito consentono di dirimere l’alternativa dolo eventuale/colpa con previsione. In ogni caso, fluidità, intercambiabilità, genericità di questi criteri vincolano poco, consegnando la formazione del libero convincimento sul dolo eventuale al convincimento libero del giudice. Il dolo eventuale sembra quindi davvero una tra le più acuminate spine nel fianco della formula BARD, cui, senza indulgere in atteggiamenti di «rassegnato distacco» o di «paralizzante pessimismo», con un approccio di “umano realismo” in questo ambito non sembra di poter chiedere «più di quanto essa possa offrire».

CFP with instruction for authors and complete stylesheet (including how to cite Kant). Additional information:

Invited contributors: Martin Bunte - Mario Caimi - Gabriele Gava - Michelle Grier - Jindrich Karásek - James Kreines - Jacinto Rivera de Rosales Chacón - Udo Thiel - Marcus Willaschek.

There are no topical restrictions. All topics related to Kant’s Transcendental Dialectic are welcome. We especially look forward to papers that emphasize the positive, non-destructive side of the TD, and re-evaluate its importance. We also especially look forward to contributions that in any aspect and in any part of the paper deal with Marcus Willaschek’s book Kant on the Sources of Metaphysics: The Dialectic of Pure Reason, Cambridge: Cambridge University Press 2018. This is optional, but Marcus Willaschek will answer objections and comment further suggestions regarding his book in his paper.

Investors largely invest their funds after considering a lot of factors on the investment outlets. There are large number of investment outlets are available for the investing community. Investors are sometimes assumes risk and sometimes avoids risk. Return maximization is the big objective all class of investors. With a view to study the investor’s attitude towards selecting investment outlets in Tamilnadu, this study has been initiated. This study collects a sample of 100 investors in Tamilnadu. A well designed and structured questionnaire is administered to collect data from the respondents. The questionnaire has been divided into four parts, first part seeks personal profile, second part consists of preference towards investment outlets, third part encompasses with investment attributes and fourth part enlists purpose and motive of investment. The collected data has been analyzed with simple percentage analysis and chi-square test. In order to test the worthiness of data, chi-square test has been employed to test the relationship between personal profile of the investors and factors associated with the selection of investment outlets. The collected data from the investors are properly analyzed and interpretations made accordingly.

"Hogyan tudja elkerülni a szkepticizmus az öncáfolat csapdáját?" - teszi fel a kérdést Pöntör Jenő Tőzsér János szkeptikus könyve kapcsán. Meglátása szerint bár Tőzsér beleesett ebbe a csapdába, mégis van rá mód, hogy a szkepticizmus ne legyen öncáfoló. Az alábbiakban először is azt mutatom meg, hogy bár Tőzsér filozófiával szembeni szkepticizmusa valóban öncáfoló, ennek okai mélyebbek, mint Pöntör feltételezi. Majd pedig amellett érvelek, hogy a náluk megjelenő szkepticizmus nem kerülheti el az öncáfolatot a Pöntör vázolta módon, és bár lehet úgy szkeptikus valaki hozzájuk hasonló módon, hogy nem öncáfoló, ezt csak olyan veszteségek árán teheti, amelyek révén szkepticizmusából csupán egy nagyon is törékeny és voltaképp önkényes pozíció marad.

Newton's claim to provide experimental proofs is often criticized. It is argued that his proofs are based on hypotheses and not inferred from the experiment alone. This criticism, however, applies a hypothetico-deductive analysis to Newton's experimental reasoning. Such an analysis is not consistent with Newton's own understanding of his proof method. The following reconstruction of Newton's proof method is intended to do justice to his understanding by applying the conception of iconic proofs to Newton's proof by experiments. The main purpose of this analysis is to explain Newton's dictum that the experiment alone serves as the source of evidence from which his theorems are derived. After a general distinction between symbolic and iconic proofs and its illustration by Euclidian proofs and Aristotelian syllogisms, I will apply this distinction to Newton's experimental proofs and analyze Newton's proof of the heterogeneity of sunlight by his experimentum crucis as an iconic proof. Finally, I show that this experiment and its underlying method is still prominent in Newton's Opticks.

This paper considers how proof comprehension, proof construction, proof validation, and proof evaluation have been described in the literature. It goes on to discuss relations between and amongst these four concepts—some from the literature, some conjectural. Lastly, it raises some teaching implication questions and suggests a few possible answers.

Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as a way to explore the similarities and differences between diagrams and text (visual thinking versus language thinking). Traditional text-based proofs are considered (by many to be) more rigorous than diagrams alone. In this paper we focus on human perceptual-cognitive characteristics that may encourage textual modes for proofs because of the ergonomic affordances of text relative to diagrams. We sug- gest that visual-spatial perception of physical objects, where an object is perceived with greater acuity through foveal vision rather than peripheral vision, is similar to attention navigating a conceptual visual-spatial structure. We suggest that at- tention has foveal-like and peripheral-like characteristics and that textual modes appeal to what we refer to here as foveal-focal attention, an extension of prior work in focused attention.

The use of formal languages in the specification of upper ontologies helps establishing precise and unambiguous definitions for its concepts and relations. In this context, the use of formal languages with support of proof-assistant software systems has potential of aiding the specification author in various aspects. We describe a study case in which the Isabelle/HOL language and the Isabelle proof-assistant environment are used to formalize a simplified version of the UFO-A ontology of endurants, to verify and correct the original axiomatization, and to optimize the specification theory's axioms and signature.

We outline here a certain history of ideas concerning the relation between intuitions and their external verification and consider its potential for detrivializing the concept of virtuality. From Descartes and Leibniz onward to 19th-century geometry and the concept of "invariant" that it shares with 19th-century psychology, we follow the thread of what might be informally called an "opera-tional" conception of the virtual, an intuition progressively developed in the 20th century from of group theoretical thinking into "functorial" thinking (in the context of category theory), and eventually intuitions for the concept of "univalence" (homotopy type theory) and its implications for the meaning of equality and identity. At each turn, skeptical arguments haunt this history's modes of exteriorization, proof, and verification; we consider the later Wittgenstein's worries concerning rule following and the apparent unbridgeable gap between formal theory and informal practice. We show how the development of mathematical intuitions and formalisms in the last century and the discovery of deep connection between intuitionistic logic and computation have begun to respond to some of these concerns and favour a conception of virtuality that is operational, constructive, pragmatic, and hospitible to scientific detrivialization.

Los estándares de prueba se insertan en un proceso de valoración racional y son los criterios que indican cuándo se ha conseguido la prueba de un hecho, o sea cuándo está justificado aceptar como verdadera la hipótesis que lo describe. Puesto que esto ocurrirá cuando el grado probabilidad de esa hipótesis sea suficiente, la construcción de un estándar de prueba implica hacer dos cosas: primero hay que decidir qué grado de probabilidad se requiere, y después hay que formular objetivamente el estándar, o sea hay que establecer los criterios objetivos que indican que ese grado de probabilidad se ha alcanzado. La primera es una decisión política o valorativa consistente en determinar cuál de los dos errores que pueden cometerse en la fijación de los hechos se considera más asumible [el de aceptar como verdadero (o dar por probado) lo que es falso, o el de no aceptar como verdadero (o dar por no probado) lo que es verdadero] y en qué grado estamos dispuestos a asumirlo. La segunda cuestión (la formulación objetiva del estándar) está muy vinculada al modelo de valoración (y por tanto del concepto de probabilidad) que se maneje. Así, mientras en los esquemas matemáticos los estándares se formularán simplemente <<cuantificando>> el grado de probabilidad exigible para probar un hecho, en los esquemas basados en el grado de confirmación se tendrán que formular estableciendo <<exigencias de confirmación>> más o menos severas.
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The standards of proof fall within a rational evaluation of evidence and are the criteria for determining where or under which circumstances a hypothesis on facts can be considered as proven. The construction of a standard of proof entails two tasks: first, it is necessary to decide which degree of probability will be required for considering a hypothesis on facts as being proven; secondly, the standard must be objectively formulated. The first decision is a matter of policy, as it consists of determining which of the two possible errors, when trying the facts, is more acceptable: a) that of considering what is false as being true (or proven), or b) that of not considering what is true as being true (or considering it not proven).  The second question (that of objectively formulating the standard) will necessarily depend on the model for assessing evidence (and therefore on the concept of probability) that we consider. In mathematical models the standards shall be formulated by merely <<quantifying>> the degree of probability required, whereas in the confirmation models they must be formulated by establishing <<specific requirements of confirmation>> more or less severe.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal.  Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]