Saving the square of opposition

Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cognitive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leaking O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute the effects of U-restriction

Questions and Answers about Oppositions

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth

739 observed NEAs and new 2-4m survey statistics within the EURONEAR network

We report follow-up observations of 477 program Near-Earth Asteroids (NEAs) using nine telescopes of the EURONEAR network having apertures between 0.3 and 4.2 m. Adding these NEAs to our previous results we now count 739 program NEAs followed-up by the EURONEAR network since 2006. The targets were selected using EURONEAR planning tools focusing on high priority objects. Analyzing the resulting orbital improvements suggests astrometric follow-up is most important days to weeks after discovery, with recovery at a new opposition also valuable. Additionally we observed 40 survey fields spanning three nights covering 11 sq. degrees near opposition, using the Wide Field Camera on the 2.5m Isaac Newton Telescope (INT), resulting in 104 discovered main belt asteroids (MBAs) and another 626 unknown one-night objects. These fields, plus program NEA fields from the INT and from the wide field MOSAIC II camera on the Blanco 4m telescope, generated around 12,000 observations of 2,000 minor planets (mostly MBAs) observed in 34 square degrees. We identify Near Earth Object (NEO) candidates among the unknown (single night) objects using three selection criteria. Testing these criteria on the (known) program NEAs shows the best selection methods are our epsilon-miu model which checks solar elongation and sky motion and the MPC's NEO rating tool. Our new data show that on average 0.5 NEO candidates per square degree should be observable in a 2m-class survey (in agreement with past results), while an average of 2.7 NEO candidates per square degree should be observable in a 4m-class survey (although our Blanco statistics were affected by clouds). At opposition just over 100 MBAs (1.6 unknown to every 1 known) per square degree are detectable to R=22 in a 2m survey based on the INT data, while our two best ecliptic Blanco fields away from opposition lead to 135 MBAs (2 unknown to every 1 known) to R=23.Comment: Published in Planetary and Space Sciences (Sep 2013

The Square of Opposition: Innovations in Teaching Logic

Teaching classical logic can often be challenging, especially when working with students who lack any prior experience with the more technical aspects of critical thinking. The abstraction of statements into logical symbols and the implementation of various diagramming methods can be enough to frustrate novice logicians, leading to a lack of hope and sometimes failure of mastery. The unique difficulties in teaching classical logic can, in addition, exacerbate tricky pedagogical issues that arise on a day to day basis in the critical thinking classroom. For example, it can be challenging to convey complex information in a meaningful way when dealing with classes over fifty students. Oftentimes this leads to presenting lectures utilizing a very general approach to content delivery. Yet, not all students will respond to a generalized lecture approach to logic and critical thinking; in all likelihood, a majority of students do not learn well in such an environment. This project seeks to explore alternative methods for teaching logic and critical reasoning through the development of one innovative technique aimed at mastery of a specific, classical topic. The focus of the project is the “square of opposition,” a model of reasoning as old as Aristotle’s philosophy that nevertheless forms a foundational part of beginning logic pedagogy even today. Despite the square’s longevity as a method of foundational reasoning, its intricacies often elude novice logicians, who struggle when attempting to make immediate inferences using the traditional representation of the square. As a result, a new visual model for both representing the reasoning of the square of opposition and using it to make inferences has been developed. “Dimo’s Square” evolved in response to students who did not demonstrate mastery of the traditional model of teaching the square of opposition and has many benefits that the classical square does not. The trajectory of Dimo’s Sqaure, an original alternative to traditional teaching of the square of opposition, is outlined in the project paper. Firstly, the traditional model for presenting and teaching the square of opposition is presented. Second, “Dimo’s Square,” is introduced and explained. Third, a comparative analysis of Dimo’s Square and the traditional square of opposition is undertaken, highlighting significant differences between them (such as Dimo’s Sqaure requiring fewer rules to be memorized and more intuitive operational patterns). Finally, formal proofs showing the logical equivalence of Dimo’s Square and the traditional square of opposition are provided, demonstrating the logical equivalence of the models, which augers well for the pedagogical superiority of Dimo’s Square, in which nothing is lost conceptually despite enhanced technical mastery

The Square of Opposition with “most” and “many”

Our concern is with constructing a traditional square of opposition into which “most” and “many” are integrated. Our basic position is in favor of traditional formal logic that originated with Aristotle, but the present discussion have taken liberties with recent developments of formal semantics to such an extent that they make contribution to more understanding of what the square of opposition looks like. The concept of directionality of monotonicity and especially our tests contribute to our conclusion. Still our discussion finds crucial insight in Keynes, one of formal logicians of a century ago. We propose as a conclusion a traditional square of opposition in which a near universal and near particular are neatly incorporated into a traditional proto-type square in such a way that they are entirely wrapped up.departmental bulletin pape

Potentiality and Contradiction in Quantum Mechanics

Following J.-Y.B\'eziau in his pioneer work on non-standard interpretations of the traditional square of opposition, we have applied the abstract structure of the square to study the relation of opposition between states in superposition in orthodox quantum mechanics in \cite{are14}. Our conclusion was that such states are \ita{contraries} (\ita{i.e.} both can be false, but both cannot be true), contradicting previous analyzes that have led to different results, such as those claiming that those states represent \ita{contradictory} properties (\ita{i. e.} they must have opposite truth values). In this chapter we bring the issue once again into the center of the stage, but now discussing the metaphysical presuppositions which underlie each kind of analysis and which lead to each kind of result, discussing in particular the idea that superpositions represent potential contradictions. We shall argue that the analysis according to which states in superposition are contrary rather than contradictory is still more plausible