Using Syllogistics to Teach Metalogic

Abstract

This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the distinction between contradictories and contraries, and the square of opposition. This strategy helps to improve students understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics.

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

“…Logically speaking, one can argue that contradiction is the 'strongest' Aristotelian relation, since it is the only relation that involves not one, but two identity conditions (cf. the discussion in Section 2) [73,90,91]. Consequently, every bitstring b ∈ B 4 has exactly one contradictory, ¬b, whereas it has multiple contraries and/or subcontraries [50,71].…”

“…In the definition of CD, the condition that b 1 ∧ b 2 = 0000 captures the informal idea that contradictories 'cannot be true together', while the condition that b 1 ∨ b 2 = 1111 captures the informal idea that they 'cannot be false together' [72,73]. In the definitions of C and SC, one of the two identity conditions of CD is negated: contraries can be false together, while subcontraries can be true together.…”

“…In the bitstring representation of B 4 , the level of a bitstring simply corresponds to its number of 1-bits; for example 0100 ∈ L 1 and 1011 ∈ L 3 . The Aristotelian relations are usually defined for formulas of some given logical system [50], but this can easily be generalized to arbitrary Boolean algebras [72,73]. In the specific case of the bitstring representation of B 4 , we say that two bitstrings b 1 , b 2 ∈ {0, 1} 4 are:…”

Metalogical Decorations of Logical Diagrams

“…We thus see that the 16 − 2 = 14 relations of the Aristotelian RDH for OG pairwise collapse into the relations of Béziau's hexagon. 24 In other words, Béziau's hexagon is correct after all (and plays a role analogous to our RDH), if only we are prepared to make the assumption that NCD = ∅. This leads to a more charitable interpretation of Béziau's hexagon, since the only substantial criticism that can now be made is that he should have stated more explicitly that he was working under the assumption that NCD = ∅.…”

“…It should also be emphasized that these metalogical diagrams are not only important from a strictly theoretical perspective, but are also relevant in more practically oriented contexts. For example, it has been argued that the metalogical square can be fruitfully used in teaching metalogic to certain groups of students [24].…”