37,058 research outputs found
Rejection in Łukasiewicz's and Słupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Łukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Łukasiewicz and developed by
his student Słupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
Penanaman Karakter Penalaran Matematis dalam Pembelajaran Matematika melalui 1 Pola Pikir Induktif-Deduktif
One of the mathematics learning purposes is using reasoning in patterns and properties, mathematical manipulation in making generalization, compiling evidence, or explaining mathematical ideas and statements. Mathematical learning based on behaviorism has been seen as less successful in instilling character of mathematics reasoning. Therefore, it is necessary to find alternative learning not only teaching but instilling character of mathematical reasoning. This paper offers constructivist in mathematics learning, which is one of the ways to involve the use of inductive-deductive thinking. Activities that involve students learning to use the inductive-deductive mindset needs to be designed and implemented by teachers. Using inductive thinking can be conditioned, especially in the process of understanding a concept or generalization. Deductive thought patterns can be conditioned to improve mathematical reasoning, for example, in the proofing. The mindset of inductive and deductive mathematical reasoning is difficult to separate in the mathematical reasoning therefore it is regarded involving the use of inductive-deductive thinking
An Axiomatic Approach to Liveness for Differential Equations
This paper presents an approach for deductive liveness verification for
ordinary differential equations (ODEs) with differential dynamic logic.
Numerous subtleties complicate the generalization of well-known discrete
liveness verification techniques, such as loop variants, to the continuous
setting. For example, ODE solutions may blow up in finite time or their
progress towards the goal may converge to zero. Our approach handles these
subtleties by successively refining ODE liveness properties using ODE
invariance properties which have a well-understood deductive proof theory. This
approach is widely applicable: we survey several liveness arguments in the
literature and derive them all as special instances of our axiomatic refinement
approach. We also correct several soundness errors in the surveyed arguments,
which further highlights the subtlety of ODE liveness reasoning and the utility
of our deductive approach. The library of common refinement steps identified
through our approach enables both the sound development and justification of
new ODE liveness proof rules from our axioms.Comment: FM 2019: 23rd International Symposium on Formal Methods, Porto,
Portugal, October 9-11, 201
Enhancing students’ generalizations: a case of abductive reasoning
The aim of this paper is to understand how a path of teacher’s actions leads to students’ generalization. Generalization, as a main process of mathematical reasoning, may be inductive, abductive, or deductive. In this paper, we focus on an abductive generalization made by a student. The study is carried out in the third cycle of design of a design-based research involving lessons about linear equations in a grade 7 class. Data is gathered by classroom observations, video and audio recorded, and by notes made in a researcher’s logbook. Data analysis focus on students’ generalizations and on teacher’s actions during whole-class mathematical discussions. The results show a path of teacher’s actions, with a central challenging action, that allowed an extending abductive generalization, and also a subsequent deductive generalization.info:eu-repo/semantics/publishedVersio
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A comparative survey of integrated learning systems
This paper presents the duction framework for unifying the three basic forms of inference - deduction, abduction, and induction - by specifying the possible relationships and influences among them in the context of integrated learning. Special assumptive forms of inference are defined that extend the use of these inference methods, and the properties of these forms are explored. A comparison to a related inference-based learning frame work is made. Finally several existing integrated learning programs are examined in the perspective of the duction framework
From SCAN to Real Data: Systematic Generalization via Meaningful Learning
Humans can systematically generalize to novel compositions of existing
concepts. There have been extensive conjectures into the extent to which neural
networks can do the same. Recent arguments supported by evidence on the SCAN
dataset claim that neural networks are inherently ineffective in such cognitive
capacity. In this paper, we revisit systematic generalization from the
perspective of meaningful learning, an exceptional capability of humans to
learn new concepts by connecting them with other previously known knowledge. We
propose to augment a training dataset in either an inductive or deductive
manner to build semantic links between new and old concepts. Our observations
on SCAN suggest that, following the meaningful learning principle, modern
sequence-to-sequence models, including RNNs, CNNs, and Transformers, can
successfully generalize to compositions of new concepts. We further validate
our findings on two real-world datasets on semantic parsing and consistent
compositional generalization is also observed. Moreover, our experiments
demonstrate that both prior knowledge and semantic linking play a key role to
achieve systematic generalization. Meanwhile, inductive learning generally
works better than deductive learning in our experiments. Finally, we provide an
explanation for data augmentation techniques by concluding them into either
inductive-based or deductive-based meaningful learning. We hope our findings
will encourage excavating existing neural networks' potential in systematic
generalization through more advanced learning schemes.Comment: 19 pages, 4 figures, 14 table
Refinement by interpretation in {\pi}-institutions
The paper discusses the role of interpretations, understood as multifunctions
that preserve and reflect logical consequence, as refinement witnesses in the
general setting of pi-institutions. This leads to a smooth generalization of
the refinement-by-interpretation approach, recently introduced by the authors
in more specific contexts. As a second, yet related contribution a basis is
provided to build up a refinement calculus of structured specifications in and
across arbitrary pi-institutions.Comment: In Proceedings Refine 2011, arXiv:1106.348
Sciduction: Combining Induction, Deduction, and Structure for Verification and Synthesis
Even with impressive advances in automated formal methods, certain problems
in system verification and synthesis remain challenging. Examples include the
verification of quantitative properties of software involving constraints on
timing and energy consumption, and the automatic synthesis of systems from
specifications. The major challenges include environment modeling,
incompleteness in specifications, and the complexity of underlying decision
problems.
This position paper proposes sciduction, an approach to tackle these
challenges by integrating inductive inference, deductive reasoning, and
structure hypotheses. Deductive reasoning, which leads from general rules or
concepts to conclusions about specific problem instances, includes techniques
such as logical inference and constraint solving. Inductive inference, which
generalizes from specific instances to yield a concept, includes algorithmic
learning from examples. Structure hypotheses are used to define the class of
artifacts, such as invariants or program fragments, generated during
verification or synthesis. Sciduction constrains inductive and deductive
reasoning using structure hypotheses, and actively combines inductive and
deductive reasoning: for instance, deductive techniques generate examples for
learning, and inductive reasoning is used to guide the deductive engines.
We illustrate this approach with three applications: (i) timing analysis of
software; (ii) synthesis of loop-free programs, and (iii) controller synthesis
for hybrid systems. Some future applications are also discussed
Logical and Spiritual Reflections
Logical and Spiritual Reflections is a collection of six shorter philosophical works, including: Hume’s Problems with Induction; A Short Critique of Kant’s Unreason; In Defense of Aristotle’s Laws of Thought; More Meditations; Zen Judaism; No to Sodom.
Of these works, the first set of three constitutes the Logical Reflections, and the second set constitutes the Spiritual Reflections.
Hume’s Problems with Induction, which is intended to describe and refute some of the main doubts and objections David Hume raised with regard to inductive reasoning. It replaces the so-called problem of induction with a principle of induction. David Hume’s notorious skepticism was based on errors of observation and reasoning, with regard to induction, causation, necessity, the self and freewill. These are here pointed out and critically analyzed in detail – and more accurate and logical theories are proposed. The present work also includes refutations of Hempel’s and Goodman’s alleged paradoxes of induction.
A Short Critique of Kant’s Unreason, which is a brief critical analysis of some of the salient epistemological and ontological ideas and theses in Immanuel Kant’s famous Critique of Pure Reason. It shows that Kant was in no position to criticize reason, because he neither sufficiently understood its workings nor had the logical tools needed for the task. Kant’s transcendental reality, his analytic-synthetic dichotomy, his views on experience and concept formation, and on the forms of sensibility (space and time) and understanding (his twelve categories), are here all subjected to rigorous logical evaluation and found deeply flawed – and more coherent theories are proposed in their stead.
In Defense of Aristotle’s Laws of Thought, which addresses, from a phenomenological standpoint, numerous modern and Buddhist objections and misconceptions regarding the basic principles of Aristotelian logic. Many people seem to be attacking Aristotle’s Laws of Thought nowadays, some coming from the West and some from the East. It is important to review and refute such ideas as they arise.
More Meditations, which is a sequel to the author’s earlier work, Meditations. It proposes additional practical methods and theoretical insights relating to meditation and Buddhism. It also discusses certain often glossed over issues relating to Buddhism – notably, historicity, idolatry, messianism, importation to the West.
Zen Judaism, which is a frank reflection on the tensions between reason and faith in today’s context of knowledge, and on the need to inject Zen-like meditation into Judaism. This work also treats some issues in ethics and theodicy.
No to Sodom, which is an essay against homosexuality, using biological, psychological, spiritual, ethical and political arguments
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