Logical reasoning

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.]

We present results on the proof validation behaviors of sixteen U.S. undergraduates after taking an inquiry-based transition-to-proof course. Participants were interviewed individually towards the end of the course using the same protocol used by Selden and Selden (2003). We describe participants’ observed validation behaviors and provide descriptions of their evaluative comments and their sense-making attempts. We make the, perhaps counterintuitive, suggestion that taking an inquiry-based transition-to-proof course emphasizing proof construction, in which validation was modelled, may not enhance students’ abilities to judge the correctness of other students’ proof attempts.

Reasoning abilities of the learners and its development was well-discussed in the world of education. The higher the ability of the person to reason abstractly, the higher the probability that a person will effectively function in the society. Thus, it is the main goal of the K-12 Curriculum of the Department of Education to improve the reasoning abilities and formal reasoning among students in the country. The higher the reasoning ability of a person, the more productive he is. The ability of logical reasoning has an essential function in the academic performance of students and their construction of the concepts. This study aimed to determine the logical reasoning abilities of 150 randomly selected junior high school students. Specifically, this study aimed to determine the logical reasoning abilities namely combinatorial reasoning, controlling variables, correlation reasoning, probabilistic reasoning and proportional reasoning among the grade 10 junior high school students and de...

Educators have long sought to develop critical thinking ability and open-mindedness in their students, as well as to measure this development. Physiological and psychological research studies on practitioners of the Transcendental Meditation (TM) program have showed EEG brain wave coherence to be a measure of the development of orderliness in thinking and expanded awareness. It was hypothesized that EEG phase coherence would provide a neurophysiological correlate of critical thinking ability and open-mindedness. It was also hypothesized that the educational curriculum at Maharishi International University, which as the TM program as its basis, would show seniors to be more developed in critical thinking and open-mindedness than seniors at other universities. Other hypotheses were that critical thinking (as measured by the Watson-Glaser Critical Thinking Appraisal) would correlate positively with open-mindedness (measured using the Rokeach Adult Dogmatism Scale), class, and show no d...

Öz: Doğru düşünme kural ve formlarının bilgisi ya da düşünme yasalarının bilimi olarak tanımlanan mantık, öğrencilere doğru düşünmenin öğretilmesi konusunda başvurulması gereken alanlardan birisi olarak karşımıza çıkmaktadır. Mantıksal düşünme becerisinin gelişimi ve desteklenmesi için mantık biliminin uygulamalarını içeren akıl ilkelerinin ve akıl yürütme yöntemlerinin bir araç olarak kullanılması oldukça önemlidir. Bu araştırmada, ilkokul ve ortaokul öğrencilerinin mantıksal düşünme becerilerinin yaş ve sınıf düzeyi değişkenlerine göre nasıl değiştiğinin incelenmesi amaçlanmıştır. Bu çerçevede ilgili alan yazın taraması sonrası öğrencilerin mantıksal düşünme becerilerini ölçmeye yönelik bir test geliştirilmiş ve pilot uygulama sonrasında geçerlik ve güvenirlik çalışmaları tamamlanmıştır. Analizler sonucunda 18 madde ve 4 boyuttan oluşan bir yapı elde edilmiş ve ortaya çıkan yapının geçerli bir uyum iyiliğine sahip olduğu görülmüştür. Araştırma sonucunda, öğrencilerin mantıksal düşünme becerilerinin mantık testinde yer alan alt boyutlara göre farklılaştığı tespit edilmiştir. Ayrıca öğrencilerin yaşları ve sınıf düzeyleri arttıkça mantıksal düşünme becerilerini yansıtan performansları da artmıştır. Öğrencilerin erken yaşlardan itibaren sistematik düşünme, akıl yürütme, çıkarımda bulunma gibi becerilerini desteklemek amacıyla mantık eğitiminin belli bir ders kapsamında ve formel mantık bilgisini içerecek şekilde verilmesi önerilmektedir.

A long tradition of research on schizophrenia has considered defects in logical reasoning one of the core features of this mental disorder. According to this view, reasoning in schizophrenia is severely impaired. Nevertheless, an increasing amount of data coming from the cognitive perspective is making this conclusion more controversial, and seems to be heading in the same direction as a very different kind of tradition, that of psychiatric phenomenology. This approach considers the typical features of schizophrenia in terms of an enhancement of logic, rather than a defect of it (cf. minkowski 1927, Binswanger 1956, Blankenburg 1971). In this paper, we will bring together these two different perspectives in order to explain why the "lack of logic" paradigm seems to fail to understand schizophrenics' reasoning abilities.

The well-known necessary-consequence/necessary-consequent ambiguity—a species of segmentation, scope, and structural ambiguities—is found in conditional sentences such as:

(A) If zero is oblong, then necessarily some square is oblong.

A can read as expressing a proposition to the effect that “some square is oblong” is a necessary consequence of “zero is oblong”. This necessary-consequence segmentation parses A with the adverb as part of the discontinuous modal connective ‘if … then necessarily’. The consequent ‘some square is oblong’ is non-modal. Alternatively, the adverb can be taken instead as part of the consequent yielding the necessary-consequent segmentation: A can read as expressing roughly that the modal proposition “necessarily some square is oblong” is a material consequence of “zero is oblong”. Brackets indicate segmentation:

Necessary-consequence segmentation
[If] zero is oblong, [then necessarily] some square is oblong.

Necessary-consequent segmentation
If zero is oblong, then [necessarily some square is oblong].

Similar confusing ambiguities arise in logic texts where ‘valid’, ‘logical’, ‘necessary’, ‘sound’, ‘derivable’, and the like occur before ‘conclusion’, ‘inference’, ‘deduction’ and the like.

This paper, supplementing [1], compares corresponding segmentations of various conditional sentences. E.g., taking both “necessities” as logical, the following have different truth-values—first true, second false. The second implies the first.

[If] Twain is Clemens, [then necessarily] Clemens is Twain.
If Twain is Clemens, then [necessarily Clemens is Twain].

The first says that “Clemens is Twain” follows logically from “Twain is Clemens”; the second that “Clemens is Twain” is logically necessary if Twain is Clemens.

"Lisanyuk Elena. Ad baculums are no arguments.

Ad baculum argument is traditionally considered to be argumentation fallacy, but the reasons of its fallaciousness have not been fully studied yet. In my paper, I am intended to show that strictly speaking ad baculums are no arguments; instead, verbally expressed force and threat can be used as means of pressure used by a dialogue agent in order to incline the other dialogue agent to behave in a certain way. In this case we deal with another speech acts that differ from persuasion. In other words, ad baculum argument is unacceptable as means of persuasion but this does prevent it from being effectively used it in other, different from argumentation, dialogue types.

Довод к палке традиционно считается ошибкой аргументации, однако за привычностью такого подхода, как правило, нет понимания причин его ошибочности, что, в свою очередь означает, что канон, отклонением от которого является такая ошибка, еще до конца не изучен. В этой статье я намерена показать, что такой разновидности доводов как аргумент к силе, строго говоря, не существует, в том смысле, что сила и угроза, выраженные вербально, могут быть использованы как средства давления или подталкивания, применяемые одним агентом к другому с тем, чтобы склонить последнего к определенной программе действий, однако в этом случае речь идет не об убеждении, но об иных речевых действиях. Иначе говоря, довод к палке как средство убеждения недопустим, однако может эффективно применяться в других, помимо аргументативного, вербальных взаимодействиях."

Reasoning abilities of the learners and its development was well-discussed in the world of education. The higher the ability of the person to reason abstractly, the higher the probability that a person will effectively function in the society. Thus, it is the main goal of the K-12 Curriculum of the Department of Education to improve the reasoning abilities and formal reasoning among students in the country. The higher the reasoning ability of a person, the more productive he is. The ability of logical reasoning has an essential function in the academic performance of students and their construction of the concepts. This study aimed to determine the logical reasoning abilities of 150 randomly selected junior high school students. Specifically, this study aimed to determine the logical reasoning abilities namely combinatorial reasoning, controlling variables, correlation reasoning, probabilistic reasoning and proportional reasoning among the grade 10 junior high school students and determine whether there is a significant difference in students' logical reasoning abilities according to their gender. The respondents answered the Test of Logical Thinking (TOLT). Thirty respondents were interviewed to verify their answers. The findings of the study led to the following conclusions: most students correctly answered problems in probabilistic reasoning and least number of students correctly answered problems in proportional reasoning and combinatorial reasoning and, male and female respondents have equal performances in problems pertaining to combinatorial reasoning, controlling variables, correlational reasoning and probabilistic reasoning but female respondents are better in proportional reasoning than the male respondents.

"ABSTRACT:
    This study assumes the subject's pursuit of meaning is generally incapacitating and should be suspended. It aims to demonstrate how such a suspension is theoretically accomplished by utilizing Lacan's formulae of sexuation integrated with his work in discourse theory and topology.
    Part I places this study into context by examining scholarship from the established fields of hermeneutics, phenomenology, (post)structuralism, aesthetic theory and psychoanalysis in order to extract out their respective theory of meaning. These theories reveal that an historical struggle with meaning has been underway since the Reformation and reaches near crisis proportions in the 20th century. On the one hand this crisis is mollified by the rise of Heideggerian-Gadamerian hermeneutical phenomenology which questions traditional epistemological approaches to the text using a new ontological conceptualization of meaning and a conscious rejection of methodology. On the other hand this crisis is exacerbated when the ubiquitous nature of meaning is itself challenged by (post)structuralism's discovery of the signifier which inscribes a limit to meaning, and by the domains of sense and nonsense newly opened up by aesthetic theory. These historical developments culminate in the field of psychoanalysis which most consequentially delimits a cause of meaning said to be closely linked to the core of subjectivity.
    Part II extends these findings by rigorously constructing out of the Lacanian sexuated formulae a decidedly non-hermeneutical phenomenological approach useful in demonstrating the sexual nature of meaning. Explicated in their static state by way of an account of their original derivation from the Aristotelian logical square, it is argued that these four formulae are relevant to basic concerns of textual theory inclusive of the hermeneutical circle of meaning. These formulae are then set into motion by integrating them with Lacan's four discourses to demonstrate the breakdown of meaning. Finally, the cuts and sutures of two-dimensional space that is topology as set down in L'étourdit are performed to confirm how the very field of meaning is ultimately suspended from a nonsensical singular point known in Lacanian psychoanalysis as objet a. The contention is that by occupying this point the subject frees himself from the debilitating grip of meaning."

This article is focused on answering the question to what extent one can be a sceptic. Sextus Empiricus’s Outlines of Scepticism serves as a guide. In section 1, it is investigated whether three logical laws have a certain foundation or are subject to doubt. In section 2, Sextus’s way to deal with these laws is examined; the question arises how dogmatic his approach is. After that, a possible ‘reply’ by Sextus to the criticism receives attention. Section 3 is concentrated on a possible alternative to Sextus’s approach. Besides logical laws, some important methods are concerned.

The purpose of the present research was to develop a game, and then to identify and organize the actions of the players while playing this game and solving puzzles. More specifically,
we intended to investigate which mathematical experiences would emerge from the action of playing this game, by means of the immersion and exploration within the realm of this game. We developed and implemented the game in its Alpha version – Wind Phoenix: Tales of Prometheus –, which offered, from the perspective of our hypothesis, to its protagonist the
opportunity to develop and to produce mathematical nowledge through logical and mathematical reasoning, by solving the puzzles and exploring the paces of the game. We have articulated these aspects with our survey issues: [1] What mathematical objects emerge from the actions of the protagonists during their experience in the game? [2] What aspects influence the elaboration and modification of the game and the puzzles? [3] What didactic and methodological applications arise from this kind of game?
The methodology adopted was the Design Based Research (DBR) model, since it presents an interactive and interventionist approach, which allowed us the consideration of the theories we presented through the production and gameplay of this game.
The Embodied Cognition – presented as our theoretical principle – deals with concepts of the Merleau-Ponty’s Phenomenology and reaches the work of Barsalou (2003; 2008), Fauconnier (2000), among others, from which we understand that every action is contextualized. Therefore, in our case, this experience within a game contributes for the production of knowledge upon the logical and mathematical reasoning.
Indeed, the challenges the players accepted and faced – in a contextualized way – in an adventure in this digital game provided engagements with the Mathematics connected with the real life. The overcoming of the process of trial and error and its substitution for processes of problem solving, through the creation of conjectures, data collection and tests and analysis of the effects showed an elaborate capacity of strategic thinking. We also observed a meaningful increase in the capacity of concentration in the actions of the participants, given their attention and observation of the details of the game, such as sceneries, map of the game and its operation or effects/feedbacks. For the next versions, Wind Phoenix will
be implemented using the game engine Unreal 4, adapting its level design, in order to expand the mechanisms proposed in the parameterizations of the puzzles.
The premise of freedom or invitation to play has always been present in our proposal. We have observed that the apparently uncommitted playing presented itself in a pleasant and motivating way. Thus, the use of digital games in learning environments is related to a pedagogical proposal, which must be implied in the choice or production of a game.
We concluded that the players assumed a role in the game and constituted a process of enunciation of logical problems – puzzles. Hence, the development of epistemological games in
3D with commercial quality showed a possibility for us to unveil new paradigms in the production of knowledge and in the construction of mathematical knowledge upon the logical and mathematical reasoning that surpasses our personal experiences.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ must 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. MSC: 11-XX (Number Theory)

Основная цель настоящей работы — показать, что логика является эффективным средством анализа аргументации в нормативных контекстах, поскольку, во-первых, обладает арсеналом инструментов для изучения формальных аспектов нормативных рассуждений. Во-вторых, несмотря на то, что способность рассуждать рациональным образом присуща человечеству в целом, и это в некоторой степени объясняет возможность сочетания в повседневной жизни юридической практики и неосведомленности в вопросах формальной логики, информационные системы и педагогические задачи в области юриспруденции нуждаются в методиках моделирования аргументативных правовых дискурсов. Нужды исследований в области правового дискурса и информационных систем (искусственного интеллекта) определяются не столько задачами создания и поддержания юридических баз данных и экспертных систем в этой области, сколько необходимостью изучения рассуждений «здравого смысла», или практических рассуждений. Педагогическая практика в этой области, как и во всякой иной, направленная на обучение основным принципам и приемам обоснования и аргументации, преследует не только энциклопедические и общеобразовательные цели, но и в значительной мере теоретические. Логика представляется здесь наиболее приемлемым кандидатом, потому что одной из главных ее задач является изучение моделей правильных рассуждений, для чего в рамках логики был выработан специальный аппарат выявления структуры рассуждений (формализация) и сопоставления их с идеальными моделями (доказательство, разрешимость). Другая важная задача логики — распознание и анализ смысловых значений рассуждений и выявление взаимосвязи формальных и содержательных аспектов рассуждений (логическая семантика). Таким образом, выявление формальных структур и содержательных аспектов рассуждений, а также особенностей использования полученных моделей в аргументативных дискуссиях делает логику одним из эффективных средств анализа таких дискурсов.
Для достижения поставленной цели в работе представлен краткий обзор основных подходов к изучению аргументации в нормативных контекстах: логического, риторического, диалектического. Поскольку в фокусе исследования каждого из этих подходов оказывается преимущественно один из критериев приемлемости аргументации, соответственно, формальный, материальный или процедурный, в работе обсуждается специфика каждого из этих аспектов обоснования. На основании этого делается вывод, что потребности универсализации процедуры обоснования, сведения ее к рациональным принципам, формам и аспектам приемлемости как отдельных аргументов, так и аргументации в целом, апеллируют к формальным критериям, характерным для логической обоснованности: непротиворечивости, семантической определенности, полноты (релевантности). Правда, в отличие от строго формальных логических критериев выводимости, непосредственно предъявляемых к формальным системам, данные критерии приемлемости аргументации представляют собой результат инструментального применения логики для реконструкция аргументативного дискурса. Так, требования полноты в формальной системе означает, что всякая общезначимая формула в ней доказуема. Критерий «полноты» в правовой аргументации означает, что всякий приемлемый аргумент является обоснованным.
Важное место в работе отводится рассмотрению таких особенностей правовой аргументации как наличие нестандартных ситуаций в юридической практике, и как следствие, обоснование исключений из установленных норм, а также уточнение норм в процессе законотворчества. Введение уточняющих положений в норму закона приводит к тому, что на действие первоначальной нормы накладываются ограничения (вводятся расширения, отмены). В работе показано, что наиболее адекватным инструментом оказывается немонотонная логика, или логика умолчаний.
Другим испытанием для стандартных логических систем стала задача моделирования рассуждений в условиях противоречивой информации, или в условиях дискуссии, когда аргументы сторон внутренне непротиворечивы, но вместе составляют противоречие. Основной принцип построения модели разрешения таких ситуаций состоит в ранжировании информации по релевантности и создании иерархии посылок.

I argue against inferentialism about logic. First, I argue against an analogy between logic and chess, before considering a more basic objection to stipulating inference rules as a way of establishing the meaning of logical constants. The objection-the Mushroom Omelette Objection-is that stipulative acts are partly constituted by logical notions, and therefore cannot be used to explain logical thought. I then argue that the same problem also attaches to following existing conventional rules, since either those rules have logical contents, or following those conventional rules is done for logical reasons. Lastly, I compare this argument with other arguments found in Quine's early work, and consider two attempts to reply to Quine.

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.

Contemporary accounts of the possession conditions for basic logical concepts rely on either the subject being primitively compelled by transitions that are isomorphic to the introduction and elimination rules for the logical constant or the rules are an innate property of the subject’s language of thought. Drawing from Heidegger’s work in the philosophy of logic and Boghossian’s work on inference, I argue that any attempt to explain logical inference as a form of rule following will lead to arguments for the impossibility of logic or an infinite regress. An under-explored alternative to defining all inferences as instances of following a rule is to take some inferences as basic – inferences that do not follow rules and provide a starting point for our inferential practices. I propose that neural networks can provide a useful model for understanding basic inferences as a primitive ability since they show how one can infer in accordance with a logical principle without necessarily following a rule. Determining whether a subject possesses a basic logical concept is arrived at then by interpreting their inferential behaviour.

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.]

Edilaine Lins GOUVEIA, Antonio ROAZZI, David P. O’BRIEN, Karina MOUTINHO & Maria da Graça B. B. DIAS
RESUMO; Nos últimos anos, tem havido muito debate acerca da existência ou não de uma lógica mental. Essa idéia tem sofrido inúmeros ataques, tanto por estudiosos que acreditam que todo raciocínio decorre de modelos mentais
(e.g., Johnson-Laird & Byrne, 1993), como por aqueles que defendem que o raciocínio humano é dependente do conteúdo (Holyoak & Cheng, 1995).
Essa controvérsia invadiu revistas internacionais como Psychological Review, Behavioral and Brain Sciences. No entanto, os proponentes da Teoria da Lógica Mental – TLM – crêem que poucos cientistas cognitivos realmente compreendem esta teoria (O’Brien, 1998a). Diante desse quadro, o presente artigo se propõe a trazer essa discussão para o cenário nacional. Serão apresentadas sumariamente algumas teorias sobre o raciocínio dedutivo. A seguir, as principais críticas à existência de uma lógica mental; e a “defesa” dos que proclamam a existência desse tipo de lógica. Por fim, a TLM será discutida mais detalhadamente.
Palavras-chave: Raciocínio lógico, Teoria da Lógica Mental, Modelos Mentais, Teoria dos Esquemas Pragmáticos

ABSTRACT: Nowadays, it has been having a lot of debate concerning the existence of a mental logic. This idea has been suffering countless attacks, as much for specialists that believe that whole reasoning elapses of mental models (e.g.,
Johnson-Laird & Byrne, 1993), as for those who defends that the human reasoning is content dependent (Holyoak & Cheng, 1995). That controversy invaded international magazines like Psychological Review. However, the authors of the Theory of the Mental Logic - TLM - believe that few cognitive scientists really understand this theory (O’Brien, 1998a). In this context, the present article intends to bring this discussion for the national scenery. Some theories about the deductive reasoning will be presented, as well as the main critics to the existence of a mental logic and the “ defense “ of those who advocates in favor of the existence of this logic. Finally, TLM will be discussed.
Key words: Logical reasoning, Theory of the Mental Logic, Mental Models, Theory of Pragmatic Schemas.

Reasoning abilities of the learners and its development was well-discussed in the world of education. The higher the ability of the person to reason abstractly, the higher the probability that a person will effectively function in the society. Thus, it is the main goal of the K-12 Curriculum of the Department of Education to improve the reasoning abilities and formal reasoning among students in the country. The higher the reasoning ability of a person, the more productive he is. The ability of logical reasoning has an essential function in the academic performance of students and their construction of the concepts. This study aimed to determine the logical reasoning abilities of 150 randomly selected junior high school students. Specifically, this study aimed to determine the logical reasoning abilities namely combinatorial reasoning, controlling variables, correlation reasoning, probabilistic reasoning and proportional reasoning among the grade 10 junior high school students and de...

The questions in this section are based on the reasoning contained in brief statements or passages. For some questions, more than one of the choices could conceivably answer the question. However, you are to choose the best answer; that is, the response that most accurately and completely answers the question. You should not make assumptions that are by commonsense standards implausible, superfluous, or incompatible with the passage.

This paper aims the aim of the present study is to ascertain the extend of logical reasoning of standard IX students as predictor of their academic performance in mathematics and to find out the significant relationship between Logical reasoning and academic performance in Mathematics. The descriptive method with a survey design was adopted. A sample of 540 standard ninth students in Kanchipuram region from 13 high/higher secondary schools was selected through stratified random sampling technique. The tools developed by the investigators was Logical Reasoning questionnaire for High School Students (LRQHSS) consisting of 25 items despite of being considered the half yearly examination math mark of students as academic performance. Findings revealed that as many as 67.8% of variances could be predicated from students’ Logical reasoning on academic performance of students in Mathematics and significant correlation between logical reasoning and academic performance in Mathematics.

Abstract
At a time in the history of mankind when man became aware of his environment the changes he observed in the phenomena prompted him to wonder. Then he tried to speculate on what could be at the base of reality. This quest for knowledge came in two distinct forms: First some consider it only in the material aspect of being where as others based it on both material and immaterial aspects. Though it is relatively old for people of the developed nations, the discipline is also relatively new in the sense that the study comes with its discoveries in consonance with the age that studies it. The task of this paper is to examine the philosophy of mathematics and its relevance to national development.

ABSTRACT

It is truism in human life that man lives in an odd world in which he strives to live a religious life, but human value play a very contrary role to the quest for re religiosity. Consequently it becomes almost impossible to attain perfection in religion vis-a-vis man diverse interest in human value. This paper is focused at the modern era of humanism or philosophical anthropocentrism whereby man gives primacy to human value denying God his existence. This is exemplified in man's quest for materialistic and political pursuit in our contemporary Nigerian Society.

To learn to write well-reasoned persuasive arguments, students need in situ help thinking through the complexity and complications of an issue, making inferences based on evidence, and hierarchically grouping and logically sequencing ideas. They rely on teachers to make this happen. In this article, the authors explain the framework they used and describe how they taught reasoning to students at an alternative high school, where they recorded what happened.

Existence and uniqueness are standard questions in cases where definite descriptions are used. In his Proslogion Anselm of Canterbury uses definite and non-definite descriptions of God: He is " id/aliquid quo maius cogitari non potest " (and similar). While Anselm's proof for the existence of God is widely discussed, including its relations to those famous descriptions, this is not the case for the question of uniqueness. Is there at most one perfect being or might there be more than one? 'Of course there can at most be one', one might want to answer – but what reasons for this claim can be found in the Proslogion? In this celebrated work of philosophical theology, Anselm does not prove uniqueness explicitly. But one may try a proof " in the spirit of the Proslogion " .

Inspired by Bermúdez's notion of proto-logic, I would like to fathom what the true proto-logic could be like. But this will be approached only in a negative way of figuring out what it could not be. I shall argue that it could not be purely deductive by exploiting the recent researches in logic of maps. This will allow us to reorient the search for proto-logic, starting with animal abduction. I will also suggest that protologic won't get off the ground without proto-geometry. These negative results will shed some lights on some further conceptual and historical issues around the language of thought hypothesis to arrive at the true proto-logic.

Nel contesto Mi sembra che stia accadendo questo, ditemi se siete d'accordo. In giro si sente dire: "Ci sta". È un nuovo modo per dire "è normale", "lo trovo giusto", "naturale", "sono d'accordo". Tutti noi troviamo naturale inserire un oggetto, un'azione, una frase in un certo contesto, nell'ambito del quale quella tal cosa s'inquadra bene, mentre fuori dal contesto proprio no: non c'entra, è come "i cavoli a merenda". Siamo umani e limitati (se qualcuno storce il naso di fronte a questa affermazione, magari dicendo "parla per te" è già un brutto segno) e abbiamo bisogno di un contesto nel quale inquadrare l'oggetto del nostro giudizio. Eppure, c'è chi vuol comunque vuole prescindere da questa esigenza, dichiarandosi superiore a queste necessarie limitazioni. Chi? Molto semplice: ad esempio, chi tradisce la moglie o il marito, il compagno o la compagna, chi ritiene superfluo sposarsi, chi pensa di poter uccidere il prossimo impunemente, chi evade le tasse. Insomma, chi pecca, direbbe la Chiesa, chi "cerca la cosa giusta nel posto sbagliato", direbbe sant'Agostino. Cosa penseresti se vedessi un signore seduto al bar che di fronte alla sua colazione intingesse nel cappuccino al posto della brioche una... pistola? O se la sera lo vedessi lavarsi i denti col rasoio da barba? O un giovanotto di buona presenza non laureato e senza aver mai ricoperto ruoli di grande responsabilità nella vita civile occupare un posto modesto, insignificante, come quello di Ministro degli Esteri della Repubblica Italiana? O una coppia composta da due uomini che chiedono a una donna di concepire un figlio con il seme di uno dei due, per poi portare avanti la gestazione e alla nascita restituire al padre il figlio che non vedrà mai più? O una nave porta-container sollevarsi dal mare e prendere il volo fino a sparire dall'orizzonte? Assurdità? Cose dell'altro mondo? Talvolta la realtà supera la fantasia e queste cose potrebbero accadere davvero (alcune sappiamo che accadono), per quanto sembrino incredibili. Possiamo anche non dire nulla se al ristorante vediamo qualcuno mangiare l'aragosta bevendo la Coca Cola, ma di certo ci ribelleremmo se un quindicenne passeggero di un volo aereo pretendesse di prendere i comandi dell'aereo che ci porta in America, giusto per provare l'ebbrezza di farlo, solo perché l'ha già fatto con la PlayStation. Noi umani abbiamo bisogno del contesto. Non è una cosa di poco conto: è una questione di maturità, oltre che di umanità. Per pilotare un aereo ci vogliono determinate caratteristiche e molte ore di volo. Per fare il Ministro degli Esteri, lo stesso. Per formare una famiglia, pure. Mi dispiace dover dare a qualcuno questa cattiva notizia: nessuno di noi è Dio. Bella forza, direte, lo sappiamo. Sicuro? Lo sappiamo tutti o c'è invece qualcuno-o moltiche pensano di poter prescindere dal contesto? C'è qualcuno che si sente Dio? Oppure, magari non si sente Dio, ma non ammette niente al di sopra di se stesso, né la propria famiglia, né la comunità civile, né la propria città o il proprio paese: Niente! Non è un problema: un buon psichiatra si trova. Nessuno di noi può fare tutto quello che gli pare, anche se c'è gente intelligentissima, come ad esempio certi primari medici, che non se ne rende conto. Ne conoscete qualcuno? Mi direte: "Certo, lo sappiamo, non siamo mica bambini!"

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.]

In memory of Alfred Tarski 1901-1983 and Alonzo Church 1903-1995 on the 40th anniversary of their classic works:
Logic, Semantics, Metamathematics and Introduction to Mathematical Logic.


Information-theoretic approaches [114] to formal logic analyse the "common intuitive" concept of propositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity.