This new volume on logic follows a recognizable format that deals in turn with the topics of mathematical logic, moving from concepts, via definitions and inferences, to theories and axioms. However, this fresh work offers a key innovation in its ‘pyramidal’ graph system for the logical formalization of all these items. The author has developed this new methodology on the basis of original research, traditional logical instruments such as Porphyrian trees, and modern concepts of classification, in which pyramids are the (...) central organizing concept. The pyramidal schema enables both the content of concepts and the relations between the concept positions in the pyramid to be read off from the graph. Logical connectors are analyzed in terms of the direction in which they connect within the pyramid. Additionally, the author shows that logical connectors are of fundamentally different types: only one sort generates propositions with truth values, while the other yields conceptual expressions or complex concepts. On this basis, strong arguments are developed against adopting the non-discriminating connector definitions implicit in Wittgensteinian truth-value tables. Special consideration is given to mathematical connectors so as to illuminate the formation of concepts in the natural sciences. To show what the pyramidal method can contribute to science, a pyramid of the number concepts prevalent in mathematics is constructed. The book also counters the logical dogma of ‘false’ contradictory propositions and sheds new light on the logical characteristics of probable propositions, as well as on syllogistic and other inferences. (shrink)

Roger Swyneshed, in his treatise on insolubles (logical paradoxes), dating from the early 1330s, drew three notorious corollaries of his solution. The third states that there is a contradictory pair of propositions both of which are false. This appears to contradict the Rule of Contradictory Pairs, which requires that in every such pair, one must be true and the other false. Looking back at Aristotle's treatise De Interpretatione, we find that Aristotle himself, immediately after defining the notion of a contradictory (...) pair, gave counterexamples to the rule. Thus Swyneshed's solution to the logical paradoxes is not contrary to Aristotle's teaching, as many of Swyneshed's contemporaries claimed. Dialetheism, the contemporary claim that some propositions are both true and false, is wedded to the Rule, and in consequence divorces denial from the assertion of the contradictory negation. (shrink)

According to the traditional view, the following incompatibility holds true: in reasoning, either there is warrant (certainty) or there is novelty. If there is warrant, there is not novelty: that would be the case of deductive reasoning. If there is novelty, there is not warrant: that would be the case of inductive reasoning. Causal reasoning would belong to the second group because there is novelty and, therefore, there is not warrant in it. I argue that this is false: reasoning may (...) have novelty and, nevertheless, be a deductive one. That is precisely what happens in (some) causal reasoning. And I will develop the following line of argumentation: one thing is to warrant that some state of affairs exists and other thing is to warrant that warrant. So we may have correct deductive reasoning without having certainty of that correction, like in some cases of causal reasoning. (shrink)

It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories always consist of (...) an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations. (shrink)

Ethics is of vital importance to the Swedish educational system, as in many other educational systems around the world.Yet, it is unclear how ethics should be dealt with in school, and prior research and evaluations have found serious problems regarding ethics in education.The field of moral education lacks clear and widely accepted definitions of key concepts, and these ambiguities negatively impact both research and educational practice. This thesis draws a distinction between three approaches to ethics in school – the descriptive (...) ethics approach, the value transmission approach, and the inquiry ethics approach – and studies in what way (if at all) they are prescribed by the national curriculum for the Swedish compulsory school, how they relate to students’ moral reasoning about technology choices and online behaviour, and what pedagogical merits and disadvantages they have. Hopefully, this both contributes to reducing the ambiguities of the field, and to answering the question of how ethics should be dealt with in education. The descriptive ethics approach asserts that school should teach students empirical facts about ethics, such as what views and opinions people have. The value transmission approach holds that school should mediate some set of predefined values to the students and make sure the students come to accept these values.The inquiry ethics approach is the view that school should teach students to reason and think critically about ethics and to engage in ethical inquiry. The role of ethics in the curriculum has not been studied in light of the above distinction, in prior research,and such an investigation is undertaken here.The results suggest that ethics has a prominent, but complicated, role in the Swedish national curriculum. Although no explicit distinction is drawn or acknowledged in the curriculum, all three approaches are prescribed throughout the curriculum, albeit to different degrees. In the general section of the curriculum, the value transmission and inquiry ethics approaches are more extensively prescribed than the descriptive ethics approach. It was found that most of the syllabi contained explicit references to ethics, while some only contained implicit references to ethics, and two syllabi lacked references to ethics altogether. In the syllabi, the inquiry ethics approach is the most dominant, both in the sense of being present in the most syllabi, and in the sense of being more strongly prescribed in many of the syllabi where several approaches occur.The value transmission approach has the weakest role in the syllabi. In total, the inquiry ethics approach is the approach most strongly prescribed by the curriculum. But prior research has shown that inquiry ethics is very rarely implemented in the classroom. In this thesis, it is found that the inquiry ethics and the value transmission approaches are incompatible, given certain reasonable interpretations, which makes the finding that inquiry ethics is rarely implemented less surprising, since value transmission is practiced in schools. Some possible causes, and some consequences, of this is discussed. The students, in their moral reasoning about technology choices, reasoned in accordance with several classical normative theories – including consequentialism, deontological ethics and virtue ethics – and in doing so, they expressed reasoning that in the discussion is found to be in conflict with the values of the value foundation in the curriculum. These findings complement earlier findings, for example that students in their actions contradict the value foundation, by adding that such conflicts also exist in their reasoning. The existence of these conflicts is found to be problematic for a value transmission approach. Many of the students defended very restrictive views on disclosing personal information online, and prior research as well as the present data has shown that adults typically hold views that are very similar to these, concerning how they think that young people ought to act online. On the other hand, youths’ actual online behaviour, as reported in earlier studies, differs considerably from this. In line with this, the students also seemed to endorse a form of private morals view, according to which moral choices are simply up to one’s own taste, which would yield an escape exit from the restrictive views mentioned above, and permit any behaviour. In the discussion, it is argued that this is the result of an attempt at value transmission from the grown-up community, probably including teachers, which might seem to work, since the students claim to hold certain views, but which likely instead constitutes a false security, since these values are not actually accepted, but only paid lip service to, and the adults are therefore wrong in their belief that the students are protected by a certain set of values (that they think the students are upholding), since the students in fact do not uphold, and therefore do not act based upon, these values. This situation risks making the students more vulnerable than had no value transmission attempt been taken in the first place. Hence, the attempted value transmission runs the risk of counteracting its purpose of helping the students acquire a safe online behaviour. Throughout the moral reasoning mentioned above, extensive variations in the students’ reasoning were found, both interpersonally and intrapersonally, both in the decision method and in the rightness criterion dimensions, as well as in between the dimensions.The existence of such variations is a novel finding, and while possible applications in future research are discussed, it is also noted that this existence constitutes a reason to question the successfulness of both the value transmission and the inquiry ethics endeavours of the educational system. The results and discussions described above highlight the importance of investigating the merits of the different approaches. Several arguments that arise from the material of this thesis are presented, evaluated and discussed. The ability of each approach to fulfil some alleged key aims of ethics education is scrutinised; their abilities to educate for good citizenship, to educate for quality of life of the individual, and to facilitate better educational results in other subjects are all investigated, as well as the ability of each approach to help counteract the influence from online extremist propaganda aimed at young people and to promote safe online behaviour in general. It is concluded that the inquiry ethics approach has the strongest support from the material of this thesis. Some consequences for school practice are discussed, and it is concluded that changing the role of ethics in the curriculum would be beneficial, downplaying the role of value transmission and further increasing, and making more explicit and clear, the role of inquiry ethics. It is also shown that there are strong reasons for the inclusion of a new subject in the Swedish compulsory education with special focus on ethics. 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The question of whether the Pyrrhonist adheres to certain logical principles, criteria of justification, and inference rules is of central importance for the study of Pyrrhonism. Its significance lies in that, whereas the Pyrrhonist describes his philosophical stance and argues against the Dogmatists by means of what may be considered a rational discourse, adherence to any such principles, criteria, and rules does not seem compatible with the radical character of his skepticism. Hence, if the Pyrrhonist does endorse them, one must (...) conclude that he is inconsistent in his outlook. Despite its import, the question under consideration has not received, in the vast literature on Pyrrhonism of the past three decades, all the attention it deserves. In the present paper, I do not propose to provide a full examination of the Pyrrhonist’s attitude towards rationality, but to focus on the question of whether he endorses the law of non-contradiction (LNC). However, I will also briefly tackle the question of the Pyrrhonist’s outlook on both the canons of rational justification at work in the so-called Five Modes of Agrippa and the logical rules of inference. In addition, given that the LNC is deemed a fundamental principle of rationality, determining the Pyrrhonist’s attitude towards it will allow us to understand his general attitude towards rationality. (shrink)

With material on his early philosophical views, his contributions to set theory and his work on nominalism and higher-order quantification, this book offers a uniquely expansive critical commentary on one of analytical philosophy’s great ...

I argue there are some objects which do not respect the Law of the Excluded Middle, i.e., which are such that, for some property F, the disjunction Fo v ~Fo fails to be true. I call such objects “odd objects” and present three examples—fictional objects, nonsort objects, and quantum objects. I argue that each of these objects is best understood as violating LEM. I, then, discuss Jessica Wilson’s account of metaphysical indeterminacy. I show how the indeterminacy which arises with odd (...) objects can be accounted for on Wilson’s account. I, then, argue that my Wilson-inspired, but non-LEM-respecting, account of metaphysical indeterminacy is superior to Wilson’s in terms of costs and benefits. (shrink)

La interpretación modal-hamiltoniana fue introducida para resolver ciertos problemas de interpretación vinculados a su ontología y a la medición cuántica. Su regla de actualización establece que todo sistema cerrado tiene su energía bien definida, y debido a la indeterminación entre la energía y el tiempo esto plantea un interrogante respecto a la situación temporal de estos sistemas. En este trabajo se analiza el problema del tiempo en sistemas cerrados y se propone la reconstrucción de un tiempo relacional compatible con la (...) perspectiva holista. (shrink)