Free Rides in Logical Space Diagrams Versus Aristotelian Diagrams

Abstract

In this paper we compare two types of diagrams for the representation of logical relations such as contradiction and contrariety, namely Logical Space diagrams (LSD) and Aristotelian diagrams (AD). The cognitive potential of Free Ride -defined in terms of tracking by consequence (Shimojima 2015) -is shown to hold for LSDs but not for ADs. The latter, however, do exhibit a greater inspection potentialdefined in terms of tracking by correlation. The translational or informational equivalence between LSDs and ADs is contrasted to their lack of computational equivalence and their different degrees of iconicity.

On the Cognitive Potential of Derivative Meaning in Aristotelian Diagrams

“…The overall aim of this paper is to apply the general semantic and cognitive framework for the analysis of diagrams proposed by Shimojima [13] to the analysis of Aristotelian diagrams developed by Demey and Smessaert in the framework of Logical Geometry. In our joint Diagrams 2020 paper [16] we demonstrated the relevance of Shimojima's first cognitive potential -namely that for Free Ride in Inference -in drawing the distinction between consequential constraint tracking by consequence with Logical Space Diagrams and consequential constraint tracking by correlation with Aristotelian diagrams. In the present paper we focus on Shimojima's fourth cognitive potential -namely that for Derivative Meaning -in order to characterise various families of Aristotelian diagrams.…”

“…3(b)which does require length 4 -and the octagons later on, we stick to length 4. 8 The contrariety triangle in the JSB hexagon thus closely resembles the four Aristotelian subdiagrams -left/right triangle, hour glass and bow tie -in [16]. 9 First of all, there are six so-called strong JSBs that form their contrariety triangle like Γ1, i.e.…”

“…With bitstrings of length 4, the source type collection G contains 18 of these Γ n sets. 16 The source type σ = ↓ ↓ counts as an abstraction over G -i.e. ↓ ↓ G. Γ 1 collectively indicates target type set ∆ 1 = { CR(1000,0001), SCR(1110,0111), SA(1000,1110), 15 As to the Apprehension Principle [17] -the structure/content of the visualisation should be readily/accurately perceived/comprehended -the three SC squares in Fig.…”

“…5(b-c) the two JSB squares are almost upside down, and thus less easily perceivable. 16 In the last step in this paper we will consider a very special type of Derivative Meaning. In the examples studied so far -the contrariety and subalternation triangles and the embedded classical squares -Derivative Meaning arose when we shifted the focus from individual Aristotelian relations between pairs of formulas to the observation of shapes or constellations of interconnected relations between three or four formulas inside an Aristotelian diagram.…”

Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

“…In the research program of logical geometry, we study Aristotelian diagrams as objects of independent mathematical interest, regardless of the specific details of their concrete applications in philosophy, artificial intelligence and elsewhere [51][52][53][54][55][56][57][58][59][60]. One of the main insights from logical geometry is that Aristotelian diagrams are highly sensitive to the details of the logical system with respect to which they are defined [52].…”