Distribution and proportion

Peterson, Philip L.

doi:10.1007/bf01048531

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…In the project of setting up a systematic typology of Aristotelian diagrams, it has been shown that there are exactly 18 families of octagons of opposition [42]. In this paper, we will not deal with all these families, but rather focus on some of the most well-known ones: the Moretti-Pellissier octagons [48,49], the Lenzen octagons [50][51][52], the Buridan octagons [53][54][55], the Beziau octagons [56,57], and the Keynes-Johnson octagons [58,59] (see Figures 7 and 8). Once again, these families of octagons (and also the other ones, which we do not deal with in this paper) can straightforwardly be characterized in AD-logic.…”

Section: Characterizing Some Aristotelian Families Of Octagons Of Opp...mentioning

confidence: 99%

The Modal Logic of Aristotelian Diagrams

Frijters

Demey

2023

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In this paper, we introduce and study AD-logic, i.e., a system of (hybrid) modal logic that can be used to reason about Aristotelian diagrams. The language of AD-logic, LAD, is interpreted on a kind of birelational Kripke frames, which we call “AD-frames”. We establish a sound and strongly complete axiomatization for AD-logic, and prove that there exists a bijection between finite Aristotelian diagrams (up to Aristotelian isomorphism) and finite AD-frames (up to modal isomorphism). We then show how AD-logic can express several major insights about Aristotelian diagrams; for example, for every well-known Aristotelian family A, we exhibit a formula χA∈LAD and show that an Aristotelian diagram D belongs to the family A iff χA is validated by D (when the latter is viewed as an AD-frame). Finally, we show that AD-logic itself gives rise to new and interesting Aristotelian diagrams, and we reflect on their profoundly peculiar status.

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Section: Characterizing Some Aristotelian Families Of Octagons Of Opp...mentioning

confidence: 99%

The Modal Logic of Aristotelian Diagrams

Frijters

Demey

2023

Axioms

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“…Syllogism, a type of deductive inference, is based on the relationship among sets and their cardinalities. Because of this, the reasoning process is directly linked with the classical distribution problem [25], i.e., how the elements of the referential fulfill the properties or terms in the statements. Thus, from this point of view, each one of the premises of the syllogism is a restriction that delimits the distribution and the conclusion is another constraint that must be compatible with the distribution described in the premises.…”

Section: General Inference Schema For Syllogistic Reasoningmentioning

confidence: 99%

A fuzzy syllogistic reasoning schema for generalized quantifiers

Pereira-Fariña

Vidal

Daz-Hermida

et al. 2014

Fuzzy Sets and Systems

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In this paper, a new approximate syllogistic reasoning schema is described that expands some of the approaches expounded in the literature into two ways: (i) a number of different types of quantifiers (logical, absolute, proportional, comparative and exception) taken from Theory of Generalized Quantifiers and similarity quantifiers, taken from statistics, are considered and (ii) any number of premises can be taken into account within the reasoning process. Furthermore, a systematic reasoning procedure to solve the syllogism is also proposed, interpreting it as an equivalent mathematical optimization problem, where the premises constitute the constraints of the searching space for the quantifier in the conclusion.1 In [2], an extension of Aristotelian syllogistics considering the problem of the distribution is proposed.

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Alpha-Structures and Ladders in Logical Geometry

De Klerck,

Demey

2024

Stud Logica

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Aristotelian diagrams, such as the square of opposition and other, more complex diagrams, have a long history in philosophical logic. Alpha-structures and ladders are two specific kinds of Aristotelian diagrams, which are often studied together because of their close interactions. The present paper builds upon this research line, by reformulating and investigating alpha-structures and ladders in the contemporary setting of logical geometry, a mathematically sophisticated framework for studying Aristotelian diagrams. In particular, this framework allows us to formulate well-defined functions that construct alpha-structures and ladders out of each other. In order to achieve this, we point out the crucial importance of imposing an ordering on the elements in the diagrams involved, and thus formulate all our results in terms of ordered versions of alpha-structures and ladders. These results shed interesting new light on the prospects of developing a systematic classification of Aristotelian diagrams, which is one of the main ongoing research efforts within logical geometry today.

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Distribution and proportion

Cited by 3 publications

References 11 publications

The Modal Logic of Aristotelian Diagrams

The Modal Logic of Aristotelian Diagrams

A fuzzy syllogistic reasoning schema for generalized quantifiers

Alpha-Structures and Ladders in Logical Geometry

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